WIRELESS NETWORKING
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WIRELESS NETWORKING

The Morgan Kaufmann Series in Networking Series Editor, David Clark, M.I.T.

Wireless Networking Anurag Kumar, D. Manjunath, and Joy Kuri

GMPLS: Architecture and Applications Adrian Farrel and Igor Bryskin

Bluetooth Application Programming with the Java APIs, Essentials Edition Timothy J. Thompson, Paul J. Kline, and C. Bala Kumar

Network Security: A Practical Approach Jan L. Harrington

Internet Multimedia Communications Using SIP Rogelio Martinez Perea Information Assurance: Dependability and Security in Networked Systems Yi Qian, James Joshi, David Tipper, and Prashant Krishnamurthy Network Simulation Experiments Manual, 2e Emad Aboelela Network Analysis, Architecture, and Design, 3e James D. McCabe Wireless Communications & Networking: An Introduction Vijay K. Garg Ethernet Networking for the Small Ofﬁce and Professional Home Ofﬁce Jan L. Harrington IPv6 Advanced Protocols Implementation Qing Li, Tatuya Jinmei, and Keiichi Shima Computer Networks: A Systems Approach, 4e Larry L. Peterson and Bruce S. Davie Network Routing: Algorithms, Protocols, and Architectures Deepankar Medhi and Karthikeyan Ramaswami Deploying IP and MPLS QoS for Multiservice Networks: Theory and Practice John Evans and Clarence Filsﬁls Trafﬁc Engineering and QoS Optimization of Integrated Voice & Data Networks Gerald R. Ash

Content Networking: Architecture, Protocols, and Practice Markus Hofmann and Leland R. Beaumont Network Algorithmics: An Interdisciplinary Approach to Designing Fast Networked Devices George Varghese Network Recovery: Protection and Restoration of Optical, SONET-SDH, IP, and MPLS Jean Philippe Vasseur, Mario Pickavet, and Piet Demeester Routing, Flow, and Capacity Design in Communication and Computer Networks Michal Pióro and Deepankar Medhi Wireless Sensor Networks: An Information Processing Approach Feng Zhao and Leonidas Guibas Communication Networking: An Analytical Approach Anurag Kumar, D. Manjunath, and Joy Kuri The Internet and Its Protocols: A Comparative Approach Adrian Farrel Modern Cable Television Technology: Video, Voice, and Data Communications, 2e Walter Ciciora, James Farmer, David Large, and Michael Adams Bluetooth Application Programming with the Java APIs C. Bala Kumar, Paul J. Kline, and Timothy J. Thompson

IPv6 Core Protocols Implementation Qing Li, Tatuya Jinmei, and Keiichi Shima

Policy-Based Network Management: Solutions for the Next Generation John Strassner

Smart Phone and Next-Generation Mobile Computing Pei Zheng and Lionel Ni

MPLS Network Management: MIBs, Tools, and Techniques Thomas D. Nadeau

Developing IP-Based Services: Solutions for Service Providers and Vendors Monique Morrow and Kateel Vijayananda Telecommunications Law in the Internet Age Sharon K. Black

Internetworking Multimedia Jon Crowcroft, Mark Handley, and Ian Wakeman Understanding Networked Applications: A First Course David G. Messerschmitt

Internet QoS: Architectures and Mechanisms Zheng Wang

Integrated Management of Networked Systems: Concepts, Architectures, and Their Operational Application Heinz-Gerd Hegering, Sebastian Abeck, and Bernhard Neumair

TCP/IP Sockets in Java: Practical Guide for Programmers Michael J. Donahoo and Kenneth L. Calvert

Virtual Private Networks: Making the Right Connection Dennis Fowler

TCP/IP Sockets in C: Practical Guide for Programmers Kenneth L. Calvert and Michael J. Donahoo

Networked Applications: A Guide to the New Computing Infrastructure David G. Messerschmitt

Multicast Communication: Protocols, Programming, and Applications Ralph Wittmann and Martina Zitterbart

Wide Area Network Design: Concepts and Tools for Optimization Robert S. Cahn

Optical Networks: A Practical Perspective, 2e Rajiv Ramaswami and Kumar N. Sivarajan

MPLS: Technology and Applications Bruce Davie and Yakov Rekhter High-Performance Communication Networks, 2e Jean Walrand and Pravin Varaiya

For further information on these books and for a list of forthcoming titles, please visit our website at http://www.mkp.com

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WIRELESS NETWORKING Anurag Kumar D. Manjunath Joy Kuri

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Morgan Kaufmann Publishers is an imprint of Elsevier

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Morgan Kaufmann Publishers is an imprint of Elsevier. 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA ∞ This book is printed on acid-free paper.

Copyright © 2008, Anurag Kumar, D. Manjunath and Joy Kuri. Published by Elsevier Inc. All rights reserved. The right of author names to be identiﬁed as the authors of this work have been asserted in accordance with the copyright, Designs and Patents Act 1988. Designations used by companies to distinguish their products are often claimed as trademarks or registered trademarks. In all instances in which Morgan Kaufmann Publishers is aware of a claim, the product names appear in initial capital or all capital letters. Readers, however, should contact the appropriate companies for more complete information regarding trademarks and registration. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means—electronic, mechanical, photocopying, scanning, or otherwise—without prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, E-mail: [email protected] You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data Kumar, Anurag. Wireless networking / Anurag Kumar, D. Manjunath, Joy Kuri. p. cm. – (The Morgan Kaufmann series in networking) Includes bibliographical references. ISBN 0-12-374254-4 1. Wireless LANs. 2. Wireless communication systems. 3. Sensor networks. I. Manjunath, D. II. Kuri, Joy. III. Title. TK5105.78.K86 2008 621.384–dc22 2007053011 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-374254-4 For information on all Morgan Kaufmann publications, visit our website at www.mkp.com or www.books.elsevier.com Printed and bound in the United States of America 08 09 10 11 12

5 4 3 2 1

Contents Preface

xiii

1 Introduction

1

1.1

Networking as Resource Allocation

1

1.2

A Taxonomy of Current Practice

3

1.3

Technical Elements

9

1.4

Summary and Our Way Forward

2 Wireless Communication: Concepts, Techniques, Models 2.1

2.2

12 15

Digital Communication over Radio Channels

16

2.1.1

Simple Binary Modulation and Detection

17

2.1.2

Getting Higher Bit Rates

20

2.1.3

Channel Coding

23

2.1.4

Delay, Path Loss, Shadowing, and Fading

25

Channel Capacity

32

2.2.1

Channel Capacity without Fading

32

2.2.2

Channel Capacity with Fading

35

2.3

Diversity and Parallel Channels: MIMO

36

2.4

Wideband Systems

42

2.4.1

CDMA

42

2.4.2

OFDMA

45

2.5 Additional Reading 3 Application Models and Performance Issues

48 53

3.1

Network Architectures and Application Scenarios

54

3.2

Types of Trafﬁc and QoS Requirements

56

3.3

Real-Time Stream Sessions: Delay Guarantees

60

3.3.1

CBR Speech

60

3.3.2

VBR Speech

61

3.3.3

Speech Playout

63

viii

Contents

3.4

3.5

3.3.4

QoS Objectives

65

3.3.5

Network Service Models

67

Elastic Transfers: Feedback Control

67

3.4.1

Dynamic Control of Bandwidth Sharing

69

3.4.2

Control Mechanisms: MAC and TCP

70

3.4.3

TCP Performance over Wireless Links

72

Notes on the Literature

4 Cellular FDM-TDMA

78 81

4.1

Principles of FDM-TDMA Cellular Systems

81

4.2

SIR Analysis: Keeping Cochannel Cells Apart

86

4.3

Channel Reuse Analysis: Hexagonal Cell Layout

92

4.3.1

Cochannel Cell Groups

93

4.3.2

Calculating Nreuse

94

4.3.3

D Ratio: Simple Analysis, Cell Sectorization R

96

4.4

Spectrum Efﬁciency

4.5

Channel Allocation and Multicell Erlang Models

4.6

99 101

4.5.1

Reuse Constraint Graph

101

4.5.2

Feasible Carrier Requirements

103

4.5.3

Carrier Allocation Strategies

103

4.5.4

Call Blocking Analysis

104

4.5.5

Comparison of FCA and MPA

106

Handovers: Techniques, Models, Analysis

112

4.6.1

Analysis of Signal Strength Based Handovers

112

4.6.2

Handover Blocking, Call Dropping: Channel Reservation

115

4.7

The GSM System for Mobile Telephony

117

4.8

Notes on the Literature

119

5 Cellular CDMA

125

5.1 The Uplink SINR Inequalities

126

5.2 A Simple Case: One Call Class

130

5.2.1

Example: Two BSs and Collocated MSs

130

5.2.2

Multiple BSs and Uniformly Distributed MSs

131

5.2.3

Other Cell Interference: Hard and Soft Handover

134

5.2.4

System Capacity for Voice Calls

139

5.3 Admission Control of Multiclass Calls

140

5.3.1

Hard and Soft Admission Control

141

5.3.2

Soft Admission Control Using Chernoff’s Bound

141

5.4 Association and Power Control for Guaranteed QoS Calls

145

Contents

ix

5.5

Scheduling Elastic Transfers

149

5.6

CDMA-Based 2G and 3G Cellular Systems

154

5.7

Notes on the Literature

155

5.8

Appendix: Perron-Frobenius Theory

156

6 Cellular OFDMA-TDMA

161

6.1

The General Model

162

6.2

Resource Allocation over a Single Carrier

163

6.2.1

Power Control for Optimal Service Rate

165

6.2.2

Power Control for Optimal Power Constrained Delay

171

6.3

Multicarrier Resource Allocation: Downlink

178

6.3.1

Single MS Case

178

6.3.2

Multiple MSs

181

6.4

WiMAX: The IEEE 802.16 Broadband Wireless Access Standard

183

6.5

Notes on the Literature

183

7 Random Access and Wireless LANs 7.1 7.2

7.3

187

Preliminaries

188

Random Access: From Aloha to CSMA

189

7.2.1

Protocols without Carrier Sensing: Aloha and Slotted Aloha

190

7.2.2

Carrier Sensing Protocols

199

CSMA/CA and WLAN Protocols

201

7.3.1

Principles of Collision Avoidance

201

7.3.2

The IEEE 802.11 WLAN Standards

204

7.3.3

HIPERLAN

211

7.4

Saturation Throughput of a Colocated IEEE 802.11-DCF Network

213

7.5

Service Differentiation and IEEE 802.11e WLANs

222

7.6

Data and Voice Sessions over 802.11

225

7.6.1

Data over WLAN

226

7.6.2

Voice over WLAN

230

7.7

Association in IEEE 802.11 WLANs

234

7.8

Notes on the Literature

235

8 Mesh Networks: Optimal Routing and Scheduling 8.1 8.2

243

Network Topology and Link Activation Constraints

244

8.1.1

Link Activation Constraints

244

Link Scheduling and Schedulable Region

247

8.2.1

Stability of Queues

250

8.2.2

Link Flows and Link Stability Region

254

x

Contents 8.3

Routing and Scheduling a Given Flow Vector

257

8.4

Maximum Weight Scheduling

264

8.5

Routing and Scheduling for Elastic Trafﬁc

273

8.5.1

Fair Allocation for Single Hop Flows

277

8.5.2

Fair Allocation for Multihop Flows

280

8.6

Notes on the Literature

9 Mesh Networks: Fundamental Limits 9.1

9.2

9.3

287 291

Preliminaries

292

9.1.1

Random Graph Models for Wireless Networks

293

9.1.2

Spatial Reuse, Network Capacity, and Connectivity

296

Connectivity in the Random Geometric Graph Model

297

9.2.1

Finite Networks in One Dimension

298

9.2.2

Networks in Two Dimensions: Asymptotic Results

302

Connectivity in the Interference Model

309

9.4

Capacity and Spatial Reuse Models

315

9.5

Transport Capacity of Arbitrary Networks

318

9.6

9.7

Transport Capacity of Randomly Deployed Networks

322

9.6.1

Protocol Model

322

9.6.2

Discussion

331

Notes on the Literature

10 Ad Hoc Wireless Sensor Networks (WSNs)

333 337

10.1 Communication Coverage

339

10.2 Sensing Coverage

341

10.3 Localization

348

10.4 Routing

353

10.5 Function Computation

359

10.6 Scheduling

368

10.6.1 S-MAC

369

10.6.2 IEEE 802.15.4 (Zigbee)

370

10.7 Notes on the Literature

Appendices A Notation and Terminology A.1 Miscellaneous Operators and Mathematical Notation A.2 Vectors and Matrices A.3

Asymptotics: The O, o, and ∼ Notation

A.4 Probability

372

375 377 377 377 377 379

Contents

xi

B A Review of Some Mathematical Concepts

381

B.1 Limits of Real Number Sequences

381

B.2 A Fixed Point Theorem

382

B.3 Probability and Random Processes

382

B.3.1

Useful Inequalities and Bounds

382

B.3.2

Convergence Concepts

384

B.3.3

The Borel-Cantelli Lemma

385

B.3.4

Laws of Large Numbers and Central Limit Theorem

385

B.3.5

Stationarity and Ergodicity

386

B.4 Notes on the Literature C Convex Optimization

387 389

C.1 Convexity

389

C.2 Local and Global Optima

389

C.3 The Karush-Kuhn-Tucker Conditions

390

C.4 Duality

391

D Discrete Event Random Processes

393

D.1 Stability Analysis of Discrete Time Markov Chains (DTMCs)

393

D.2 Continuous Time Markov Chains

394

D.3 Renewal Processes

398

D.3.1 Renewal Reward Processes

398

D.3.2 The Excess Distribution

399

D.3.3 Markov Renewal Processes

399

D.4 Some Topics in Queuing Theory

401

D.4.1 Little’s Theorem

401

D.4.2 Poisson Arrivals See Time Averages (PASTA)

402

D.5 Some Important Queuing Models

403

D.5.1 The M/G/c/c Queue

403

D.5.2 The Processor Sharing Queue

404

D.6 Notes on the Literature

405

Bibliography

407

Index

417

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Preface Another Book on Wireless Networking? The availability of high performance, low power, and low cost digital signal processors, and advances in digital communication techniques over the radio frequency spectrum have resulted in the widespread availability of wireless network technology for mass consumption. Several excellent books are now available that deal with the area of wireless communications, where topics of recent interest include multiple-input-multiple-output (MIMO) systems, space time coding, orthogonal frequency division multiplexing (OFDM), and multiuser detection. Wireless networks are best known in the context of ﬁrst- and secondgeneration mobile telephony (AT&T’s analog AMPS system in the ﬁrst generation, and the GSM and CDMA digital systems in the second generation). There are books that provide coverage of such wireless networks, and also those that combine a comprehensive treatment of physical layer wireless communication with that of cellular networks. In the last decade, however, there has been an explosion in the development and deployment of new wireless network technologies, and in the conceptualization of, and research in, a variety of newer ones. From the ubiquitous WiFi coffee shop and airport networks to the emerging WiMAX systems, which promise broadband wireless access to mobile users, the menu of wireless access networks promises to become so comprehensive that wired access from user devices may soon become a relic of the past. Research on wireless mesh networks (so-called ad hoc wireless networks), which started in the 1970s, is being pursued with renewed vigor due to the availability of inexpensive and interoperable mobile wireless devices. In addition, the widespread use of wireless sensor networks (in conjunction with emerging standards such as Zigbee and IEEE 802.15.4) is a clear and present possibility. Thus the variety and scale of wireless networks is unprecedented, and, in teaching courses in our institutions, we have felt the need for a comprehensive analytical treatment of wireless networking, keeping in mind the technical developments in the past, the present, and the future. This book is the outcome of our efforts to address this need. The foremost aspect of networking, wireline or wireless, is the design of efﬁcient protocols that work. Taking the view that the devil is in the detail, protocols with “working code” often gain widespread acceptance. With the increasing variety in networks and applications, and also in their scale, complex interactions (e.g., between devices using a particular protocol, or between protocols at the various layers) need to be understood. Although computer

xiv

Preface

simulation is a useful vehicle for understanding the performance of protocols, it is not always sufﬁcient, because, once again, the devil is in the detail. The assumptions made in deriving simulation models play an important role in the results that are obtained. If a simulation program simply encodes the standard, then running the simulation only provides a plethora of numbers, with no new insights being gained. Further, large simulation models, although possibly closer to reality, take a lot of effort to develop and debug, and are slow to execute, thus rendering them not very useful in the early stages of experimentation with algorithms. This is where analytical models become very important. First, the process of deriving such models from the standards, or from system descriptions, provides very useful insights. Second, the analytical models can be used to help verify large simulation programs, by providing exact results for subcases of the model being simulated. Third, research in analytical modeling is necessary to develop models that can be programmed into simulators, so as to increase simulation speed. Finally, the analytical approach is very important for the development of new and efﬁcient protocols, and there is a trend toward optimization via reverse engineering of well-accepted protocols. In addition to the variety of networks and protocols that need to be understood, there is a large body of fundamental results on wireless networks that have been developed over the last ﬁfteen years that give important insights into optimal design and the limits of performance. Examples of such results include distributed power control in CDMA networks, optimal scheduling in wireless networks (with a variety of optimization objectives involving issues such as network stability, performance, revenue, and fairness), transmission range thresholds for connectivity in a wireless mesh network, and the transport capacity of these networks. Further, the imminence of sensor networks has generated a large class of fundamental problems in the areas of stochastic networks and distributed algorithms that are intrinsically important and interesting. This book aims (1) to provide an analytical perspective on the design and analysis of the traditional and emerging wireless networks, and (2) to discuss the nature of, and solution methods to, the fundamental problems in wireless networking. For the sake of completeness, traditional voice telephony over GSM and CDMA wireless access networks also is covered. The approach is via various resource allocation models that are based on simple models of the underlying physical wireless communication.

About the Book and the Viewpoint After the speciﬁcation of the protocols and the veriﬁcation of their correctness, we believe that networking is about resource allocation. In wireless networks, the resources are typically spectrum, time, and power. That theme pervades much of this book in our quest for models for performance analysis, for developing design insights, and also for exploring the fundamental limits. Once a problem has been analytically formulated, we draw upon a wide variety of techniques

Preface

xv

to analyze it. In this process we will use techniques drawn from, among others, probability theory, stochastic processes, constrained optimization and duality, and graph theory. We believe it is necessary to make forays into these areas in order to bring their power to bear on the problem at hand. However, we have attempted to make the book as self-contained as possible. Wherever possible, we have used only elementary concepts taught in basic courses in engineering mathematics. A brief overview of most of the advanced mathematical material that we use is provided in the appendix. Also, wherever possible we have avoided the theorem–proof approach. Instead, we have developed the theorems or results and then formally stated them. After the introductory chapter, we begin the presentation of the main material of the book in Chapter 2 by giving an overview of the physical layer issues that are so much more important to understand wireless networks than they are for wireline networks. Wireless networks are viewed as being either access networks or mesh networks. In access networks mobile wireless nodes connect to an infrastructure node, and in mesh networks they form an independent internet and may or may not connect to an infrastructure network. Access networks are covered in Chapters 4 through 7 and mesh networks are covered in Chapters 8 through 10. The wireless networking aspect of the book begins in Chapter 3. Like in our earlier book, Communication Networking: An Analytical Approach, we precede the discussion on access networks by listing the issues and setting the performance objectives of a wireless network in Chapter 3. FDM-TDMA cellular networks (of which GSM networks are a major example) are discussed in Chapter 4, with the focus on signal-to-interference ratio analysis, on channel allocation, and on the call blocking and call dropping performance. Chapter 5 is on CDMA networks where the main emphasis is on interference management via power allocation. Whereas the trafﬁc model in Chapter 4 and in much of Chapter 5 is an arrival process of calls, each with a rate requirement, in Chapter 6, on OFDMA access networks, we consider buffered models, and discuss power allocation over time and over carriers with the objectives of stability and mean delay. In Chapter 7, we discuss the performance of distributed allocation of channel time in wireless LANs. We begin our discussion of mesh networks in Chapter 8 by considering optimal routing and scheduling in a given mesh network. One can view this class of problems as the optimal allocation of time and space in a network. In Chapter 9 we explore fundamental limits of this time and space allocation to the ﬂows. Chapter 10 is on the emerging area of sensor networks, a rich ﬁeld of research issues including connectivity and coverage properties of stochastic networks, and distributed computation. Some of the material in Chapter 5 and most of the material in Chapters 6 through 10 are being covered in a wireless networking textbook for the ﬁrst time. We have not obtained new results for the book but we have trawled the literature to pick out the fundamental results and those that are illustrative of the issues

xvi

Preface

and complexities. Wherever possible, we have simpliﬁed the models for pedagogic convenience.

Using the Book This is a graduate text, though a ﬁnal year undergraduate course could be supplemented with material from this text. Some understanding of networking concepts is assumed. A quick introduction may also be obtained from Chapter 2 of our earlier book, Communication Networking: An Analytical Approach. Most of the chapters are self-contained and we believe that an instructor can pick and choose the chapters. A course that needs to cover voice and data access networks (including cellular networks and wireless LANs) could be based on Chapters 4 through 7. One can say that these chapters are tied closely to real networks. Chapters 8 through 10 are of a more fundamental and abstract nature. A course with a more current research emphasis could be built around Chapters 6 through 10. The publisher maintains a website for this book at www.mkp.com. We maintain a website for the book at ece.iisc.ernet.in/∼anurag/books. These websites contain errata, additional problems, PostScript ﬁles of the ﬁgures used in the book, and other instructional material. An instructor’s manual containing solutions to all the exercises and problems and some supplementary problems is also available from the authors. Arthur Clarke had said that the communications satellite will make inevitable the United Nations of the Earth. Wireless communication and networking are making these United Nations ﬂatter, and possibly more democratic with unbridled opportunities for all. So let’s “unwire, cut the cord, and go wireless.” And, while we do it, let us step back a bit and understand them from the ground up!

Acknowledgments We are grateful to Onkar Dabeer and P.R. Kumar who reviewed the complete manuscript and provided us invaluable comments and criticisms. Saswati Sarkar was visiting us on her sabbatical when we were writing this book; her insightful comments on several chapters of the book have helped immensely in improving the accuracy of the material. Prasanna Chaporkar, N. Hemachandra, U. Jayakrishnan, Koushik Kar, Biplab Sikdar, Chandramani Kishore Singh, and Rajesh Sundaresan read various chapters of the book at our request. They found many errors and rough edges, and we remain grateful for their time and efforts. Many students helped with reviewing the problems and verifying their solutions; these include Onkar Bharadwaj, Avhishek Chatterjee, Sudeep Kamath, Pallavi Manohar, and K. Premkumar. We would like to thank Chandrika Sridhar, our ever-helpful lab secretary in [email protected], who typed parts of an initial draft from the lecture notes of the ﬁrst author, and also prepared all the problems and solutions, both from handwritten manuscripts.

Preface

xvii

This book has been developed out of the two-part survey article “A Tutorial Survey of Topics in Wireless Networking,” by Anurag Kumar and D. Manjunath, published in Sa¯ dhana¯ , Indian Academy of Sciences Proceedings in Engineering Sciences, Vol. 32, No. 6, December 2007. We are grateful to the publishers of Sa¯ dhana¯ for permitting us to use several extracts and ﬁgures from our survey article. Parts of an early draft of this book have been used by Ed Knightly (Rice University), and Utpal Mukherji (Indian Institute of Science). We hope that the ﬁnished version will meet their expectations for the courses they teach. Book writing grants have been provided by the Centre for Continuing Education of the Indian Institute of Science to the ﬁrst author and by the Curriculum Development Program of the Indian Institute of Technology, Bombay, to the second author. Finally, we are grateful to our families for bearing patiently our absence from the regular call of duty at home, in the evenings, holidays, and weekends during the several months over which this book was developed. Anurag Kumar I.I.Sc., Bangalore

D. Manjunath I.I.T., Bombay

Joy Kuri I.I.Sc., Bangalore

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CHAPTER 1 Introduction

T

he idea of sending information over radio waves (i.e., wireless communication) is over a hundred years old. When several devices with radio transceivers share a portion of the radio spectrum to send information to each other, we say that we have a wireless communication network, or simply a wireless network. In this chapter we begin by developing a three-layered view of wireless networks. We delineate the subject matter of this book—that is, wireless networking—as dealing with the problem of resource allocation when several devices share a portion of the RF spectrum allocated to them. Next, we provide a taxonomy of current wireless networks. The material in the book is organized along this taxonomy. Then, in this chapter, we identify the common basic technical elements that underlie any wireless network as being (1) physical wireless communication; (2) neighbor discovery, association, and topology formation; and (3) transmission scheduling. Finally, we provide an overview of the contents of the remaining nine chapters of the book.

1.1

Networking as Resource Allocation

Following our viewpoint in [89] we view wireline and wireless communication networks in terms of the three-layered model shown in Figure 1.1. Networks carry the ﬂows of information between distributed applications such as telephony, teleconferencing, media-sharing, World Wide Web access, e-commerce, and so on. The points at which distributed information applications connect to the generators and absorbers of information ﬂows can be viewed as sources and sinks of trafﬁc (see Figure 1.1). Examples of trafﬁc sources are microphones in telephony devices, video cameras, and data, voice, or video ﬁles (stored on a computer disk) that are being transmitted to another location. Examples of trafﬁc sinks are telephony loudspeakers, television monitors, or computer storage devices. As shown in Figure 1.1, the sources and sinks of information and the distributed applications connect to the communication network via common information services. The information services layer comprises all the hardware and software required to facilitate the necessary transport services, and to attach the sources or sinks to the wireless network; for example, voice coding, packet buffering and playout, and voice decoding, for packet telephony; or similar

2

1

Introduction

sources and/or sinks distributed applications

Information Services

Resource Allocation

Resource Allocation

Algorithms

Algorithms

Shared Radio Spectrum

Wireline Bit Carrier Infrastructure

(a portion of the RF spectrum)

Figure 1.1 A conceptual view of distributed applications utilizing wireline and wireless networks. Wireless networking is concerned with algorithms for resource allocation between devices sharing a portion of the radio spectrum. On the other hand, in wireline networks the resource allocation algorithms are concerned with sharing the ﬁxed resources of a bit transport infrastructure.

facilities for video telephony or for streaming video playout; or mail preparation and forwarding software for electronic mail; or a browser for the World Wide Web. We turn now to the bottom layer in Figure 1.1. In wireline networks the information to be transported between the endpoints of applications is carried over a static bit-carrier infrastructure. These networks typically comprise highquality digital transmission systems over copper or optical media. Once such links are properly designed and conﬁgured, they can be viewed as “bit pipes,” each with a certain bit rate, and usually a very small bit error rate. The bit carrier infrastructure can be dynamically reconﬁgured on the basis of trafﬁc demands, and such actions are a part of the cloud labeled “resource allocation algorithms” in the ﬁgure. The left side of the bottom layer in Figure 1.1 corresponds to wireless networks. Typically, each wireless network system is constrained to operate in some portion of the RF spectrum. For example, a cellular telephony system may be assigned 5 MHz of spectrum in the 900 MHz band. Information bits are transported between devices in the wireless network by means of some physical wireless communication technique (i.e., a PHY layer technique, in terms of the ISO-OSI model) operating in the portion of the RF spectrum that is assigned to the network. It is well known, however, that unguided RF communication between mobile wireless devices poses challenging problems. Unlike wireline communication, or even point-to-point, high-power microwave links between dish antennas mounted on tall towers, digital wireless communication between mobile devices has to deal with a variety of time-varying channel impairments

1.2 A Taxonomy of Current Practice

3

such as obstructions by steel and concrete buildings, absorption in partition walls or in foliage, and interference between copies of a signal that traverse multiple paths. In order to combat these problems, it is imperative that in a mobile or ad hoc wireless network the PHY layer should be adaptable. In fact, in some systems multiple modulation schemes are available, and each of these may have variable parameters such as the error control codes, and the transmitter powers. Hence, unlike a wired communication network, where we can view networking as being concerned with the problems of resource sharing over a static bit carrier infrastructure, in wireless networking, the resource allocation mechanisms would include these adaptations of the PHY layer. Thus, in Figure 1.1 we have actually “absorbed” the physical wireless communication mechanisms into the resource allocation layer. Hence, we can deﬁne our view of wireless networking as being concerned with all the mechanisms, procedures, or algorithms for efﬁcient sharing of a portion of the radio spectrum so that all the instances of communication between the various devices obtain their desired quality of service (QoS).

1.2 A Taxonomy of Current Practice In this book, instead of pursuing an abstract, technology agnostic approach, we will develop an understanding of the various wireless networking techniques in the context of certain classes of wireless networks as they exist today. Thus we begin our treatment by taking a look at a taxonomy of the current practice of wireless networks. Figure 1.2 provides such a taxonomy. Several commonly used terms of the technology will arise as we discuss this taxonomy. These will be highlighted by the italic font, and their meanings will be clear from the context. Of course, the attendant engineering issues will be dealt with at length in the remainder of the book. Fixed wireless networks include line-of-sight microwave links, which until recently were very popular for long distance transmission. Such networks basically comprise point-to-point line-of-sight digital radio links. When such links are set up, with properly aligned high gain antennas on tall masts, the links can be viewed as point-to-point bit pipes, albeit with a higher bit error rate than wired links. Thus in such ﬁxed wireless networks no essentially new issues arise than in a network of wired links. On the other hand the second and third categories shown in the ﬁrst level of the taxonomy (i.e., access networks and ad hoc networks) involve multiple access where, in the same geographical region, several devices share a radio spectrum to communicate among themselves (see Figure 1.3). Currently, the most important role of wireless communications technology is in mobile access to wired networks. We can further classify such access networks into two categories: one in which resource allocation is more or less static (akin to circuit multiplexing), and the other in which the trafﬁc is statistically multiplexed, either in a centralized manner or by distributed mechanisms.

4

1

Introduction

wireless networks

fixed networks

ad hoc networks (wireless mesh networks)

mobile access networks

circuit multiplexing (e.g., GSM cellular networks)

centralized statistical multiplexing (e.g., CDMA cellular networks: IS 95, cdma2000, WCDMA; and IEEE 802.16 “WIMAX” networks)

distributed statistical multiplexing (e.g., IEEE 802.11 WLANs)

wireless internets

sensor networks

Figure 1.2 A taxonomy of wireless networks.

WLAN AP

Wireline Network

base stations Cellular Network

Figure 1.3 The left panel shows some access networks (a cellular telephony network, and a wireless local area network (WLAN), where the access is via an AP (access point)), and the right panel shows a mesh wireless network of portable computers.

1.2 A Taxonomy of Current Practice

5

Cellular wireless networks were introduced in the early 1980s as a technology for providing access to the wired phone network to mobile users. The network coverage area is partitioned into regions (with diameters ranging from hundreds of meters to a few kilometers) called cells, hence the term “cellular.” In each cell there is a base station (BS), which is connected to the wired network, and through which the mobile devices in the cell communicate over a one hop wireless link. The cellular systems that have the most widespread deployment are the ones that share the available spectrum using frequency division multiplexed time division multiple access (FDM-TDMA) technology. Among such systems by far the most commercially successful has been the GSM system, developed by a European consortium. The available spectrum is ﬁrst partitioned into a contiguous up-link band and another contiguous down-link band. Each of these bands is statically or dynamically partitioned into reuse subbands, with each cell being allocated such a subband (this is the FDM aspect). The partitioning of the up-link and down-link bands is done in a paired manner so that each cell is actually assigned a pair of subbands. Each subband is further partitioned into channels or carriers (also an FDM aspect), each of which is digitally modulated and then slotted in such a way that a channel can carry up to a ﬁxed number of calls (e.g., 8 calls) in a TDM fashion. Each arriving call request in a cell is then assigned a slot in one of the carriers in that cell; of course, a pair of slots is assigned in paired up-link and down-link channels in that cell. Thus, since each call is assigned dedicated resources, the system is said to be circuit multiplexed, just like the wireline phone network. These are narrowband systems (i.e., users’ bit streams occupy frequency bands just sufﬁcient to carry them), and the radio links operate at a high signal-to-interference-plus-noise-ratio (SINR), and hence careful frequency planning (i.e., partitioning of the spectrum into reuse subbands, and allocation of the subbands to the cells) is needed to avoid cochannel interference. The need for allocation of frequency bands over the network coverage area (perhaps even dynamic allocation over a slow timescale), and the grant and release of individual channels as individual calls arrive and complete, requires the control of such systems to be highly centralized. Note that call admission control, that is, call blocking, is a natural requirement in an FDM-TDMA system, since the resources are partitioned and each connection is assigned one resource unit. Another cellular technology that has developed over the past 10 to 15 years is the one based on code division multiple access (CDMA). In these networks, the entire available spectrum is reused in every cell. These are broadband systems, which means that each user’s bit stream (a few kilobits per second) occupies the entire available radio spectrum (a few megahertz). This is done by spreading each user’s signal over the entire spectrum by multiplying it by a pseudorandom sequence, which is allocated to the user. This makes each user’s signal appear like noise to other users. The knowledge of the spreading sequences permits the receivers to separate the users’ signals, by means of correlation receivers. Although no frequency planning is required for CDMA systems, the performance is interference limited as every transmitted signal is potentially an interferer for

6

1

Introduction

every other signal. Thus at any point of time there is an allocation of powers to all the transmitters sharing the spectrum, such that their desired receivers can decode their transmissions, in the presence of all the cross interferences. These desired power levels need to be set depending on the locations of the users, and the consequent channel conditions between the users and the base stations, and need to be dynamically controlled as users move about and channel conditions change. Hence tight control of transmitter power levels is necessary. Further, of course, the allocation of spreading codes, and management of movement between cells needs to done. We note that, unlike the FDM-TDMA system described earlier, there is no dedicated allocation of resources (frequency and time-slot) to each call. Indeed, during periods when a call is inactive no radio resources are utilized, and the interference to other calls is reduced. Thus, we can say that the trafﬁc is statistically multiplexed. If there are several calls in the system, each needing certain quality of service (QoS) (bit rate, maximum bit error rate), then the number of calls in the system needs to be controlled so that the probability of QoS violation of the calls is kept small. This requires call admission control, which is an essential mechanism in CDMA systems, in order that QoS objectives can be achieved. Evidently, these are all centrally coordinated activities, and hence even CDMA cellular systems depend on central intelligence that resides in the base station controllers (BSCs). Until recently, cellular networks were driven primarily by the needs of circuit multiplexed voice telephony; on demand, a mobile phone user is provided a wireless digital communication channel on which is carried compressed telephone quality (though not “toll” quality) speech. Earlier, we have described two technologies for second generation (2G) cellular wireless telephony. Recently, with the growing need for mobile Internet access, there have been efforts to provide packetized data access on these networks as well. In the FDM-TDMA systems, low bit rate data can be carried on the digital channel assigned to a user. As is always the case in circuit multiplexed networks, ﬂexibility in the allocation of bandwidth is limited to assigning multiple channels to each user. Such an approach is followed in the GSM-GPRS (General Packet Radio Service) system, where, by combining multiple TDM slots on an FDM carrier, shared packet switched access is provided to mobile users. A further evolution is the EDGE (Enhanced Data rates for GSM Evolution) system, where, in addition to combining TDM slots, higher order modulation schemes, with adaptive modulation, are utilized to obtain shared packet switched links with speeds up to 474 Kbps. These two systems often are viewed, respectively, as 2.5G and 2.75G evolutions of the GSM system. These are data evolutions of an intrinsically circuit switched system that was developed for mobile telephony. On the other hand there is considerable ﬂexibility in CDMA systems where there is no dedicated allocation of resources (spectrum or power). In fact, both voice and data can be carried in the packet mode, with the user bit rate, the amount of spreading, and the allocated power changing on a packetby-packet basis. This is the approach taken for the third generation (3G) cellular systems, which are based entirely on CDMA technology, and are meant to carry multimedia trafﬁc (i.e., store and forward data, packetized telephony, interactive

1.2 A Taxonomy of Current Practice

7

video, and streaming video). The most widely adopted standard for 3G systems is WCDMA (wideband CDMA), which was created by the 3G Partnership Project (3GPP), a consortium of standardization organizations from the United States, Europe, China, Japan, and Korea. Cellular networks were developed with the primary objective of providing wireless access for mobile users. With the growth of the Internet as the de facto network for information dissemination, access to the Internet has become an increasingly important requirement in most countries. In large congested cities, and in developing countries without a good wireline infrastructure, ﬁxed wireless access to the Internet is seen as a signiﬁcant market. It is with such an application in mind that the IEEE 802.16 standards were developed, and are known in the industry as WiMAX. The major technical advance in WiMAX is in the adoption of several high performance physical layer (PHY) technologies to provide several tens of Mbps between a base station (BS) and ﬁxed subscriber stations (SS) over distances of several kilometers. The PHY technologies that have been utilized are orthogonal frequency division multiple access (OFDMA) and multiple antennas at the transmitters and the receivers. The latter are commonly referred to as MIMO (multiple-input-multiple-output) systems. In an OFDMA system, several subchannels are statically deﬁned in the system bandwidth, and these subchannels are digitally modulated. In order to permit up-link and down-link transmissions, time is divided into frames and each frame is further partitioned into an up-link and a down-link part (this is called time division duplexing (TDD)). The BS allocates time on the various subchannels to various down-link ﬂows in the down-link part of the frame and, based on SS requests, in the up-link part of the frame. This kind of TDD MAC structure has been used in several earlier systems; for example, satellite networks involving very small aperture satellite terminals (VSATs), and even in wireline systems such as those used for the transmission of digital data over cable television networks. WiMAX speciﬁcations now have been extended to include broadband access to mobile users. We now discuss the third class of networks in the mobile access category in the ﬁrst level of the taxonomy shown in Figure 1.2—distributed packet scheduling. Cellular networks have emerged from centrally managed point-to-point radio links, but another class of wireless networks has emerged from the idea of random access, whose prototypical example is the Aloha network. Spurred by advances in digital communication over radio channels, random access networks can now support bit rates close to desktop wired Ethernet access. Hence random access wireless networks are now rapidly proliferating as the technology of choice for wireless Internet access with limited mobility. The most important standards for such applications are the ones in the IEEE 802.11 series. Networks based on this standard now support physical transmission speeds from a few Mbps (over 100s of meters) up to 100 Mbps (over a few meters). The spectrum is shared in a statistical TDMA fashion (as opposed to slotted TDMA, as discussed, earlier, in the context of ﬁrst generation FDM-TDMA systems). Nodes contend for the channel, and possibly collide. In the event of a collision, the colliding nodes back

8

1

Introduction

off for independently sampled random time durations, and then reattempt. When a node is able to acquire the channel, it can send at the highest of the standard bit rates that can be decoded, given the channel condition between it and its receiver. This technology is predominantly deployed for creating wireless local area networks (WLANs) in campuses and enterprise buildings, thus basically providing a one hop untethered access to a building’s Ethernet network. In the latest enhancements to the IEEE 802.11 standards, MIMO-OFDM physical layer technologies are being employed in order to obtain up to 100 Mbps transmission speeds in indoor environments. With the widespread deployment of IEEE 802.11 WLANs in buildings, and even public spaces (such as shopping malls and airports), an emerging possibility is that of carrying interactive voice and streaming video trafﬁc over these networks. The emerging concept of fourth-generation wireless access networks envisions mobile devices that can support multiple technologies for physical digital radio communication, along with the resource management algorithms that would permit a device to seamlessly move between 3G cellular networks, IEEE 802.16 access networks and IEEE 802.11 WLANs, while supporting a variety of packet mode services, each with its own QoS requirements. With reference to the taxonomy in Figure 1.2, we now turn to the category labeled “ad hoc networks” or “wireless mesh networks.” Wireless access networks provide mobile devices with one-hop wireless access to a wired network. Thus, in such networks, in the path between two user devices there is only one or at most two wireless links. On the other hand a wireless ad hoc network comprises several devices arbitrarily located in a space (e.g., a line segment, or a two-dimensional ﬁeld). Each device is equipped with a radio transceiver, all of which typically share the same radio frequency band. In this situation, the problem is to communicate between the various devices. Nodes need to discover neighbors in order to form a topology, good paths need to be found, and then some form of time scheduling of transmissions needs to be employed in order to send packets between the devices. Packets going from one node to another may need to be forwarded by other nodes. Thus, these are multihop wireless packet radio networks, and they have been studied as such over several years. Interest in such networks has again been revived in the context of multihop wireless internets and wireless sensor networks. We discuss these brieﬂy in the following two paragraphs. In some situations it becomes necessary for several mobile devices (such as portable computers) to organize themselves into a multihop wireless packet network. Such a situation could arise in the aftermath of a major natural disaster such as an earthquake, when emergency management teams need to coordinate their activities and all the wired infrastructure has been damaged. Notice that the kind of communication that such a network would be required to support would be similar to what is carried by regular public networks; that is, pointto-point store and forward trafﬁc such as electronic mails and ﬁle transfers, and low bit rate voice and video communication. Thus, we can call such a network a

1.3 Technical Elements

9

multihop wireless internet. In general, such a network could attach at some point to the wired Internet. Whereas multihop wireless internets have the service objective of supporting instances of point-to-point communication, an ad hoc wireless sensor network has a global objective. The nodes in such a network are miniature devices, each of which carries a microprocessor (with an energy efﬁcient operating system); one or more sensors (e.g., light, acoustic, or chemical sensors); a low power, low bit rate digital radio transceiver; and a small battery. Each sensor monitors its environment and the objective of the network is to deliver some global information or an inference about the environment to an operator who could be located at the periphery of the network, or could be remotely connected to the sensor network. An example is the deployment of such a network in the border areas of a country to monitor intrusions. Another example is to equip a large building with a sensor network comprising devices with strain sensors in order to monitor the building’s structural integrity after an earthquake. Yet another example is the use of such sensor networks in monitoring and control systems such as those for the environment of an ofﬁce building or hotel, or a large chemical factory.

1.3 Technical Elements In the previous section we provided an overview of the current practice of wireless networks. We organized our presentation around a taxonomy of wireless networks shown in Figure 1.2. Although the technologies that we discussed may appear to be disparate, there are certain common technical elements that constitute these wireless networks. The efﬁcient realization of these elements constitutes the area of wireless networking. The following is an enumeration and preliminary discussion of the technical elements. 1. Transport of the users’ bits over the shared radio spectrum. There is, of course, no communication network unless bits can be transported between users. Digital communication over mobile wireless links has evolved rapidly over the past two decades. Several approaches are now available, with various tradeoffs and areas of applicability. Even in a given system, the digital communication mechanisms can be adaptive. First, for a given digital modulation scheme the parameters can be adapted (e.g., the transmit power, or the amount of error protection), and, second, sophisticated physical layers actually permit the modulation itself to be changed even at the packet or burst timescale (e.g., if the channel quality improves during a call then a higher order modulation can be used, thus helping in store and forward applications that can utilize such time varying capacity). This adaptivity is very useful in the mobile access situation where the channels and interference levels are rapidly changing.

10

1

Introduction

2. Neighbor discovery, association and topology formation, routing. Except in the case of ﬁxed wireless networks, we typically do not “force” the formation of speciﬁc links in a wireless network. For example, in an access network each mobile device could be in the vicinity of more than one BS or access point (AP). To simplify our writing, we will refer to a BS or an AP as an access device. It is a nontrivial issue as to which access device a mobile device connects through. First, each mobile needs to determine which access devices are in its vicinity, and through which it can potentially communicate. Then each mobile should associate with an access device such that certain overall communication objectives are satisﬁed. For example, if a mobile is in the vicinity of two BSs and needs certain quality of service, then its assignment to only a particular one of the two BSs may result in satisfaction of the new requirement, and all the existing ones. In the case of an access network the problem of routing is trivial; a mobile associates with a BS and all its packets need to be routed through that BS. On the other hand, in an ad hoc network, after the associations are made and a topology is determined, good routes need to be determined. A mobile would have several neighbors in the discovered topology. In order to send a packet to a destination, an appropriate neighbor would need to be chosen, and this neighbor would further need to forward the packet toward the destination. The choice of the route would depend on factors such as the bit rate achievable on the hops of the route, the number of hops on the route, the congestion along the route, and the residual battery energies in devices along the route. We note that association and topology formation is a procedure whose timescale will depend on how rapidly the relative locations of the network nodes is changing. However, one would typically not expect to associate and reassociate a mobile device, form a new topology, or recalculate routing at the packet timescale. If mobility is low, for example in wireless LANs and static sensor networks, one could consider each ﬁxed association, topology, and routing, and compute the performance measures at the user level. Note that this step requires a scheduling mechanism, discussed as the next element. Then that association, topology, and routing would be chosen that optimizes, in some sense, the performance measures. In the formulation of such a problem, ﬁrst we need to identify one or more performance objectives (e.g., the sum of the user utilities for the transfer rates they get). Then we need to specify whether we seek a cooperative optimum (e.g., the network operator might seek the global objective of maximizing revenue) or a noncooperative equilibrium. The latter might model the more practical situation, since users would tend to act selﬁshly, attempting to maximize their performance while reducing their costs. Finally, whatever the solution of the problem, we need an algorithm (centralized or distributed) to compute it online.

1.3 Technical Elements

11

If the mobility is high, however, the association problem would need to be dynamically solved as the devices move around. Such a problem may be relatively simple in a wireless access network, and, indeed, necessary since cellular networks are supposed to handle high mobility users. On the other hand such a problem would be hard for a general mesh network; highly mobile wireless mesh networks, however, are not expected to be “high performance” networks. 3. Transmission scheduling. Given an association, a topology, and the routes, and the various possibilities of adaptation at the physical layer, the problem is to schedule transmissions between the various devices so that the users’ QoS objectives are met. In its most general form, the schedule dynamically needs to determine which transceivers should transmit, how much they should transmit, and which physical layer (including its parameters, e.g., transmit power) should be used between each transceiver pair. Such a scheduler would be said to be cross-layer if it took into account state information at multiple layers; for example, channel state information, as well as higher layer state information, such as link buffer queue lengths. Note that a scheduling mechanism will determine the schedulable region for the network; that is, the set of user ﬂow rates of each type that can be carried so that each ﬂow’s QoS is met. In general, these three technical elements are interdependent and the most general approach would be to jointly optimize them. For example, in a mobile Internet access network the mobile devices are associated with base stations. The channel qualities between the base stations and the mobile devices determine the bit rates that can be sustained, the transmission powers required, and transmission schedule required to achieve the desired QoS for the various connections. Thus, the overall problem involves a joint optimization of the association, the physical layer parameters, and the transmission schedule. In addition to the preceding elements that provide the basic communication functionality, some wireless networks require other functional elements that could be key to the networks’ overall utility. The following are two important ones, which are of special relevance to ad hoc wireless sensor networks. • Location determination. In an ad hoc wireless sensor network the nodes

make measurements on their environment, and then these measurements are used to carry out some global computation. Often, in this process it becomes necessary to determine from which location a measurement came. Sensor network nodes may be too small (in terms of size and available energy) to carry a GPS (global positioning system) receiver. Some applications may require the nodes to be placed indoors, where GPS signals may not penetrate. Hence GPS-free techniques for location determination become important. Even in cellular networks, there is a requirement in some countries that, if needed, a mobile device should be

12

1

Introduction

geographically locatable. Such a feature can be used to locate someone who is stranded in an emergency situation and is unaware of the exact location. • Distributed computation. This issue is speciﬁc to wireless sensor net-

works. It may be necessary to compute some function of the values measured by sensors (e.g., the maximum or the average). Such a computation may involve some statistical signal processing functions such as data compression, detection, or estimation. Since these networks operate with very simple digital radios and processors, and have only small amounts of battery energy, the design of efﬁcient self-organizing wireless ad hoc networks and distributed computation schemes on them is an important emerging area. In such networks there is communication delay and also data loss; hence existing algorithms may need to be redesigned to be robust to information delay and loss.

1.4

Summary and Our Way Forward

We began with a discussion of our view of networking as resource allocation. Figure 1.1 summarizes our view. This was followed by a taxonomy of current wireless practice in Section 1.2. Next, the common technical elements that underlie the apparently disparate technologies were abstracted and discussed in Section 1.3. Before we can proceed to the core topic of this book—resource allocation to meet speciﬁed QoS objectives—we will need to understand basic models of, and notions associated with, the wireless channel. Along with this, the important techniques employed in digital communication will be covered in Chapter 2. These concepts will be like the building blocks in terms of which our resource allocation problems will be posed, and answers sought. Essentially, in Chapter 2, our discussion will be conﬁned to the so-called PHY layer. However, before commencing our study of resource allocation problems, we will pause and take a look at the applications that usually are carried on communication networks. Our objectives will be to understand the characteristics of the bit streams or the packet streams generated by various applications (the top layer of Figure 1.1), as well as the performance requirements the streams demand. This will be the topic of Chapter 3. Beginning with Chapter 4, we will consider, one by one, the different wireless networks shown at the second level of our taxonomy (Figure 1.2). In each case, the emphasis will be on posing and solving resource allocation problems speciﬁc to that type of network. In Chapter 4, narrowband cellular systems will be studied. Power, bandwidth, and time are the resources here, and the principal objective is to maintain the signal-to-interference ratio (SIR) at an adequately high level. Our discussion will give rise to several important concepts, including

1.4

Summary and Our Way Forward

13

frequency reuse, sectorization, spectrum efﬁciency, handover blocking, and channel reservation. Continuing with cellular access networks, we will focus on CDMA systems in Chapter 5. The distinguishing feature here is that of universal frequency reuse. As before, the main theme is to assign power so as to ensure that the signalto-interference-plus-noise ratio (SINR) is adequately high. We will see how the notions of other-cell interference, power control, and hard as well as soft handover arise in this context. In Chapter 6, we will turn to OFDMA-TDMA systems, where power, frequency, and time constitute the basic resources to be allocated. Unlike FDMTDMA and CDMA systems, where to each ﬂow a ﬁxed bit rate is assigned, in OFDMA-TDMA systems, the resources are assigned dynamically over time, depending on time varying user requirements and channel conditions. Generally speaking, the objective is to maximize the aggregate bit capacity of a time-varying channel, subject to a constraint on the average power. The important notion of the water-ﬁlling power allocation will emerge from our discussions. In Chapter 7, the focus shifts to random-access systems and, in particular, IEEE 802.11 WLANs. The principal resource here is channel time, and distributed control of access to the channel is of interest. In a system of n colocated WLAN nodes, what is the saturation throughput that each can achieve? We will analyze this important question. Various issues pertaining to the transport of voice and data trafﬁc over WLANs will also be discussed. Continuing with our discussion of the various networks according to our taxonomy, we will study multihop wireless mesh networks in Chapters 8 and 9. In Chapter 8, we assume that a wireless mesh network is given. On this network, we will address the fundamental question of optimal routing and link scheduling of packet ﬂows for a given set of source-destination pairs. Again, the basic resources here are bandwidth, time, and power, and it is of interest to know which nodes should get access to the bandwidth at what times so as to achieve the objective of maximizing throughput. Our analysis will lead to the notions of optimal scheduling and routing. We ﬁrst consider open loop ﬂows. The ﬂow rates may be given or they may be unknown. For the latter case, the important maximum weight scheduling is described in detail. We also consider routing and scheduling for elastic ﬂows so as to maximize a network utility function. In Chapter 9, we will address some fundamental questions that arise in the context of wireless mesh networks. First, we ask, what is the minimum power level that nodes can use while ensuring that the network of nodes remains connected? After a suitable deﬁnition of the network capacity we also obtain the capacity of arbitrary and random networks. Although asymptotic analyses provide interesting insights, wherever possible, we also consider ﬁnite networks. Finally, in Chapter 10, we will turn to wireless sensor networks. Apart from power and bandwidth, each sensor itself can be considered as a resource now.

14

1

Introduction

A variety of new problems arise; for example, if sensors are deployed in a random manner over a given area, how many of them are required so that every point in the area is sensed by not less than k sensors? As mentioned before in Section 1.3, wireless sensor networks often have special needs; for example, localization and distributed computation. Resource allocation problems for meeting such objectives will also be discussed.

CHAPTER 2 Wireless Communication: Concepts, Techniques, Models

W

e recall from Figure 1.1 in Chapter 1 that, when studying wireless networks, we will not take the links as given bit carriers but will be concerned with the sharing of the wireless spectrum resource as well. The strictly layered approach would view the wireless physical layer as providing a bit carrier service to the link layer. The link layer just offers packets to the physical layer, which does the best it can. If on the other hand, there is interaction between the layers and the link layer can be aware of the time varying quality of the wireless communication, then it could prioritize, schedule, defer, or discard packets in order to attempt to meet the QoS requirements of the various ﬂows. It is therefore important to obtain an understanding of how digital radio communication is performed, and the issues, constraints, and trade-offs that are involved. The material in this chapter is well established and is available in great detail and in much more generality in many books on digital communications. An excellent up-to-date coverage of this topic is provided in [131] and [43]. Readers familiar with digital wireless communication can skip this chapter with no loss of continuity.

Overview Our approach to modeling, analyzing, and designing resource allocation in wireless networks will be based on simple models of the techniques that are used for carrying bit streams over wireless channels. Because of their place in the seven-layer OSI model, these are also called physical layer techniques or, as an abbreviation, PHY techniques. In this chapter we will provide these models, and show how they arise. In Section 2.1 we will study, in some detail, the simplest binary modulation over a very simple radio channel in which the only phenomenon that corrupts the user’s data is additive noise. We will see that the receiver can make errors when attempting to extract the transmitted bits from the noisy received signal, and we will relate the bit error rate (BER) to the received signal-to-noise ratio (SNR). We will see how higher bit rates can be obtained by using higher order constellations into which blocks of user bits can be mapped. We will brieﬂy discuss how adding redundant bits at the transmitter, or channel coding, can be used to reduce the BER

16

2 Wireless Communication: Concepts, Techniques, Models

at the expense of a reduction in the user level bit rate. Then, in Section 2.1.4, we will understand other ways in which propagation over a radio channel can corrupt the user’s data: these are path loss, shadowing, and multipath fading. The latter two are stochastic phenomena, and we will see how they are modeled. Section 2.1 will close with an understanding of how random fading causes a deterioration in the BER achievable for a given SNR. In Section 2.2 we will explain the idea of channel capacity, and we will provide Shannon’s formula for the capacity of an additive white Gaussian channel. The idea of the ergodic capacity of a fading channel will also be introduced. In Section 2.3 we will study how diversity can mitigate the effect of a fading channel. Diversity can be obtained in various ways, one of them being by the use of multiple receive antennas. We will then see that multiple transmit and receive antennas (i.e., MIMO antenna systems) can also provide a capacity gain by making the channel look like several independent parallel channels. Recent mobile wireless access networks have relied heavily on the techniques of code division multiple access (CDMA), and also, more recently, orthogonal frequency division multiple access (OFDMA). In these systems, the resources (e.g., bandwidth and time) are not statically partitioned over the users. Instead, the available spectrum is shared dynamically between the users, with the resource allocation being dynamically adjusted as the user demands and channel conditions vary over time. We study CDMA and OFDMA in Section 2.4.1 and in Section 2.4.2, respectively.

2.1

Digital Communication over Radio Channels

The primary resource that is shared in a wireless network is the radio spectrum. We will limit ourselves to the situation in which the communicating nodes share a radio spectrum of bandwidth1 W , centered at the carrier frequency fc (see Figure 2.1). W 2fc

W

0

fc

f

Figure 2.1 The nodes in a wireless network share a portion of the radio spectrum.

1 The term bandwidth has varied and confusing usage in the wireless networking literature. The RF spectrum

in which a system operates has a bandwidth. When a digital modulation scheme is used over this spectrum then a certain bit rate is provided; often this aggregate bit rate may also be referred to as bandwidth, and we may speak of users sharing the bandwidth. This latter usage is unambiguous in the wire-line context. In multiaccess wireless networks, however, users would be sharing the same RF spectrum bandwidth, but would be using different modulation schemes and thus obtaining different (and time varying) bit rates, rendering the use of a phrase such as “bandwidth assigned to a user” very inappropriate.

2.1

Digital Communication over Radio Channels

17

C1p(t )

C2p(t 2T ) 101101

100101 Modulator

Channel

Demodulator

noise

Figure 2.2 A sequence of pulses is modulated with the bits to be transmitted. The √ basic pulse is p(t). Notice that the bit sequence 101101 is transmitted as + E p(t), s √ √ √ − Es p (t − T ), + Es p (t − 2T ),. . ., + Es p (t − 5T ). There is an error in the third bit, so that, after detection, the received sequence is 100101.

It is assumed that fc >> W ; for example, fc = 2.4 GHz and W = 5 MHz. All communication between any pair of nodes in the network can utilize this entire spectrum.

2.1.1 Simple Binary Modulation and Detection As shown in Figure 2.2, digital communication is achieved over the given radio spectrum by modulating a sequence of pulses by the given bit pattern. The pulse, p(t) (also called the baseband pulse), is chosen so that when translated to the carrier fc its spectrum ﬁts into the given radio spectrum; that is, in this case, the spectrum W W 1 of the baseband pulse will occupy the frequencies − 2 , + 2 . Taking T = W , it , +W , and is such is possible to deﬁne a pulse p(t), that is bandlimited2 to − W 2 2 that p(t − kT), k ∈ {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}, constitute an orthonormal set, +∞ +∞ that is, −∞ p(t)p(t − kT)dt = 0 for k = 0. Further, −∞ p2 (t)dt = 1, that is, the energy of the pulse is 1. The pulses are repeated every T seconds. In the situation depicted in Figure 2.2, √ the modulation is very simple: each pulse√in the pulse train is multiplied by + Es if the bit to be transmitted is 1, and by − Es if the bit to be sent is 0. Notice that the energy of the modulated pulse becomes Es . It is said that the modulator maps √ √ bits into channel symbols. Thus, in this example, the symbol set is − Es , + Es . In general, there could be more than just two possible symbols; for example, four symbols would permit two incoming bits to be mapped into each channel symbol. Continuing our simple 2 Mathematically, a pulse, p(t), that is bandlimited (e.g., to (− W , + W )) occupies inﬁnite time. Practically, 2 2

a pulse that is chosen for a digital modulation scheme has negligible energy beyond a small multiple of T on either side of its main lobe.

18

2 Wireless Communication: Concepts, Techniques, Models

example, let Ck denote the symbol into which the k-th bit is mapped. When the pulses are repeated every T seconds, the modulated pulse stream can be written as X(t) =

∞

Ck p(t − kT)

(2.1)

k=−∞

Given this continuous time signal, and recalling the orthonormality of the various shifts of p(t) by kT , it is easy to see that the following operation recovers the information carrying sequence Ck . +∞ Ck = X(t)p(t − kT)dt −∞

The baseband signal X(t) is then translated to the radio spectrum shown in Figure 2.1 by multiplying it with a sinusoid at the carrier frequency. The resulting signal is S(t) =

∞ √ 2 Ck p(t − kT) cos(2πfc t)

(2.2)

k=−∞

√ The multiplication by 2 is to make the energy in the modulated symbols equal3 to Es . Thus, the symbol energy in the transmitted signal is Es Joules/symbol, and since the symbol rate is T1 symbols/second, the transmitted signal power is therefore Es T Watts. In Figure 2.2 we do not show the translation of the signal by the carrier. It is as if the channel has been shifted to the baseband. As shown in Figure 2.2, as the modulated signal passes through the channel, and is processed in the front-end of the receiver, it is corrupted by noise. This is taken to be zero mean additive white Gaussian noise (AWGN), which means that noise just adds to the signal and is a Gaussian random process with a power spectrum that is constant over the passband of the channel (hence the term “white,” since all frequencies (“colours”) have the same power). The signal occupies a band of W Hz around the carrier frequency fc ( W Hz below and W Hz above ±fc ; 2 2 see Figure 2.1). Hence, we need only be concerned with noise that occupies this band. Such bandpass white Gaussian noise, with a power spectral density of N20 , is mathematically represented as (see [113]) N(t) = U(t) cos(2πfc t)

(2.3)

√

3 To see why we have chosen the symbols C to be ±√E , and the reason for the factor 2, notice s k +∞ (Ck )2 p2 (t) cos2 (2πfc t) dt which can be shown to be that the energy in each transmitted pulse is 2 −∞ equal to Es .

2.1

Digital Communication over Radio Channels

19

where the process U(t) is a zero Gaussian process with power spectral mean white W W density N0 , bandlimited to − 2 , + 2 . We can view the noise process U(t) as a baseband noise process that is translated to the carrier frequency and placed in the passband of the channel. It can now be shown (see this chapter’s Appendix) that the previously described modulation scheme, and the additive white Gaussian noise model, along with receiver processing, results in the following symbol-by-symbol channel model that relates the source symbol sequence Ck and the predetection statistic Yk , from which the source symbol sequence has to be inferred. Yk = Ck + Zk

(2.4)

where Zk is a sequence of i.i.d. zero mean Gaussian random variables with variance N20 . Figure 2.3 depicts the probability density of Yk under the two possible values of Ck . These are both Gaussian densities with variance N20 . The detector concludes that the bit sent was 0 if the value of Yk is smaller than the threshold and 1 if the value of Yk is more than the threshold. An error occurs if 1 is sent and Yk falls below the threshold, and vice versa. When the source produces 0s and 1s with equal probabilities then the threshold is midway between the means of the two densities, that is, the threshold is 0. The probability of error if a 0 was sent is then given by:

Pr(Yk > 0 | 0 was sent) = Q

2Es N0

depends on signal energy

probability density of value at detector if ‘0’ was sent

depends on noise energy

2 Es

threshold

Es

Figure 2.3 The probability densities of the statisticYk under the two possible symbols.

20

2 Wireless Communication: Concepts, Techniques, Models

x2 where Q(τ) : = τ∞ √1 e− 2 dx. This can be seen to be the same as the probability 2π of error if a 1 was sent. Hence the probability of error of the binary modulation scheme that we have described, under AWGN, is given by

2Es Perror − AWGN = Q (2.5) N0

Note that in this example, since each symbol is used to send one bit, the error rate obtained is also the bit error rate (BER). In Problem 2.1 we ﬁnd that Perror − AWGN decreases exponentially with NEs . In particular, for BERs of 10−3 and 10−6 the Es N0

0

values required are approximately 7 dB and 10.5 dB, respectively. We note that if 1500 byte packets have to be transmitted over a wireless link, then in order to obtain a packet error probability of 0.01, we need BER ≤ 10−6 . We see that the probability of correct detection depends on NEs , which is 0 the ratio of the symbol energy to the noise power spectral density. Increasing the symbol energy increases the separation between the two Gaussian probability densities in Figure 2.3, and hence, for given noise variance, reduces the probability of Yk falsely crossing the threshold. Similarly, decreasing the noise reduces the width of the two Gaussian probability densities, thus also reducing the error probability for a given signal energy. 2.1.2 Getting Higher Bit Rates In the simple example in Section 2.1.1, since each pulse is modulated by one of two possible symbols, and the symbol rate is T1 , the bit rate is therefore 1 bps. One of the goals in designing a digital communication system over a radio T spectrum is to use this spectrum to carry as high a bit rate as possible. With the binary modulation example in mind there are two possibilities for increasing the bit rate. 1. Increase the symbol rate; that is, decrease T . 2. Increase the number of possible symbols, from 2 to M > 2. log M

Then, in general, the bit rate will be given by T2 . There are, however, limits on both these possibilities. Note that if the pulse bandwidth is limited to W , the channel bandwidth, 2 then the pulse duration will not be time limited, and in fact the received signal in a symbol interval will be the sum of the pulse in that interval and parts of pulses in neighboring intervals. The pulses therefore have to be appropriately designed to take care of this effect. This leads to the so-called Nyquist criterion, which limits the pulse rate to no more than W (i.e., T1 ≤ W ).

2.1

Digital Communication over Radio Channels

21

Before we proceed, it is useful to make an observation. We saw in Section 2.1.1 that the probability of error for that binary signaling system depended on the ratio NEs . If the signaling rate is T1 , then the average power in the transmitted 0

signal is Es × T1 . The noise power in the channel bandwidth is W N0 . Hence the Es signal power to noise power ratio (SNR) is given by TWN . If, in addition, the 0 Es symbol rate is such that T × W = 1, then the SNR is just N0 . Thus we see that for this example the probability of error depends on the SNR. This is sometimes called the predetection SNR, as it is the SNR before the receiver attempts to decide which symbol was sent. Let us now consider the other alternative for increasing the bit rate; that is, increasing the number of possible symbols that can modulate the pulses. Figure 2.4(a) shows the binary symbol set that we have already discussed. This is called binary pulse amplitude modulation (PAM), or 2-PAM. An example of the simplest possibility is shown in Figure 2.4(b); this is called 4-PAM. Since each of the 2-bit patterns 00, 01, 10, 11 can be mapped to one of the symbols, this scheme can transmit 2 bits per symbol. However, in order to achieve a particular probability of error with a given noise power, the distance between the symbols has to be retained as in the binary case; to see this consider Figure 2.3, add a Gaussian density for each new symbol added, and then consider the probability of error between neighboring symbols. This means that the symbol energy when transmitting the left-most and right-most symbols in Figure 2.4(b) will be 32 times larger than that for the other two symbols. This in turn implies a larger average signal power, and hence a larger SNR (assuming the same noise power) for achieving the same probability of error. Yet another alternative is shown in Figure 2.5(a) where we have two-dimensional symbols. Each symbol can be written in the form ce jθ , with c = 1

π 3π and θ ∈ 0, 2 , π, 2 . This symbol set is called QPSK (quadrature phase shift

⎯ 2√ Es

⎯ √ Es

(a)

Figure 2.4

⎯ 23√Es

⎯ 2√ Es

⎯ √ Es

⎯ 3√Es

(b)

Some symbol sets: (a) binary antipodal, (b) 4-level amplitude modulation.

22

2 Wireless Communication: Concepts, Techniques, Models

(a)

Figure 2.5 added.

(b)

(a) A complex symbol set with 4 symbols; (b) the symbol set with noise

keying) since all the symbols have the same amplitude but they have different phases. Now, instead of the form in (2.2), the transmitted signal takes the general form S(t) =

∞ √ 2 Ck cos(Θk )p(t − kT) cos(2πfc t) k=−∞ ∞ √ Ck sin(Θk )p(t − kT) sin(2πfc t) − 2

(2.6)

k=−∞

Here, the sequence (Ck , Θk ) depends on the modulating bits. Thus, basically, the x-coordinate (i.e., Ck cos(Θk )) of the symbol modulates the carrier cos(2πfc t) and the y-coordinate (i.e., Ck sin(Θk )) of the symbol modulates −sin(2πfc t), which is

also called the quadrature carrier (since it is π2 out of phase with the in-phase carrier). The bandpass additive noise N(t) has the general form N(t) = U(t) cos(2πfc t) − V(t) sin(2πfc t)

where U(t) and V(t) are independent Gaussian processes with power zero mean W , . We can interpret U(t) and V(t) as spectral density N0 , bandlimited to − W 2 2 the in-phase and quadrature noise processes, respectively. In fact, we notice that the QPSK signal shown in (2.6) is the superposition of two orthogonal 2-PAM signals; the in-phase and quadrature signals √ are both 2-PAM signals.√After down conversion (multiplying the signal by 2 cos(2πfc t) and also by − 2 sin(2πfc t) and ﬁltering out the high frequency terms), and multiplication and integration with the pulse p(t), we will obtain the following pair of statistics: (i)

(i)

(q)

(q)

Yk = Ck cos(Θk ) + Zk Yk = Ck sin(Θk ) + Zk

2.1

Digital Communication over Radio Channels

23

where (i) and (q) denote the in-phase and quadrature components. The sequences (q) (i) Zk and Zk are independent, and each is a sequence of i.i.d. zero mean Gaussian random variables with variance N20 . We can write this more compactly by using (q) (q) complex numbers. Deﬁne Yk = Yk(i) + jYk and Zk(i) + jZk . Then, we can write Yk = Xk + Zk

(2.7)

where Xk = Ck ejΘk is the k-th channel symbol. We say that the sequence of complex random variables Zk are circularly symmetric complex Gaussian. In Figure 2.5(b) we show the received symbols after corruption by noise; the noise now has the two-dimensional Gaussian density that is circularly symmetric about each symbol. Notice from the geometry in Figure 2.5(a) that, by utilizing both dimensions, for a given probability of error, a smaller symbol spacing can be used than for the symbol set in Figure 2.4(b), and hence a given BER can be achieved with less average power. Thus, we have noisy observations of the two coordinates of the transmitted complex symbol, from which the transmitted symbol has to be detected. Since in each symbol only one of the phases is used (and the other is 0, owing to the simple QPSK symbol set), the average signal power is that of a 2-PAM signal. As is evident from Figure 2.5 many more symbol sets are possible. If the amplitude as well as phase of the symbols can vary then it is called QAM (quadrature amplitude modulation), whereas if only the phase can vary then it is called a PSK symbol set. Symbol sets are also called constellations. The probability of error of all the digital modulation and demodulation schemes based on the basic ideas discussed earlier can be expressed as a function of the SNR at the receiver. 2.1.3 Channel Coding In a given situation, owing to physical limitations it may not be possible to increase the SNR so as to achieve the desired BER. The application being transported on the wireless link may require a lower BER in order to achieve reasonable performance. For example, if the link is used to transport packets and the packet length is L bits, then a BER of yields a packet error rate of 1 − (1 − )L . We will see in Section 3.4.3 that a high packet error rate can seriously affect the performance of TCP transfers. Hence, this may place a minimum BER requirement on the link. For a given digital modulation scheme, the BER as seen by the data source can be reduced by channel coding. The simplest viewpoint is shown in Figures 2.6 and 2.7. The channel with the given modulation scheme is viewed as an error prone binary channel. Blocks of the incoming bits of length K are coded into codewords of length N(> K), thus introducing redundancy. If the code length and the codes are judiciously chosen, even after the channel introduces errors, an errored codeword can be expected to stay close to the original codeword. In Figure 2.7 we show source bit strings of length K being mapped into code blocks of length N . Since the number of possible code strings (2N ) is larger than

24

2 Wireless Communication: Concepts, Techniques, Models error control coder adds redundant bits

Figure 2.6

binary channel (introduces bit errors)

error control decoder extracts transmitted bits from received code words

Channel coding: adding redundant bits to protect against channel errors.

code words

set of possible blocks of length K (2K blocks)

set of possible blocks “sphere” of highly probable of length N errored code words N (2 blocks)

Figure 2.7 A channel code maps source bit strings into longer code bit strings (or codewords); decoding involves identifying the codeword nearest to the received bit string.

the number of possible source strings (2K ), the code words can be chosen so that there is sufﬁcient spacing between them. Now even if the channel causes errors, the errored codewords will occupy spheres of high probability around the transmitted codewords. Hence, by using nearest codeword decoding, the transmitted codeword, and hence the original source string, can be inferred with a small residual error probability. The trade-off is that the information bit rate of K the communication link becomes N , which is less than 1 information bit per code bit. This is called the rate of the code, denoted by R. One trivial way of improving error performance is to increase N , because this results in the codewords being spaced farther apart; but this reduces the information rate. It is possible, however, to increase K with N , keeping the information rate R constant, while reducing the bit error rate to arbitrarily small values. Shannon’s noisy channel coding theorem states that there is a number C , called the channel capacity, such that if R < C , then, as the block length increases, an arbitrarily small bit error rate can be achieved (of course, at the cost of a large block coding delay). If we attempt to use R > C , then the bit error rate cannot be reduced to 0. Recall our analysis of the two-level modulation carried out earlier in this chapter. We recall that for bit error rates of 10−3 and 10−6 the NEs0 values required

2.1

Digital Communication over Radio Channels

25

were approximately 7 dB and 10.5 dB, respectively. As an example, with a high quality rate 12 code, the required NEs can be reduced by 2 dB for 10−3 and by 5 dB 0

for 10−6 . Of course, the user bit rate drops to 12 bit per symbol. This reduction in Es is called coding gain. N0 A coder is followed by a digital modulation scheme that maps code bits into channel symbols. As discussed in Section 2.1.1, the modulator maps a certain number of code bits (e.g., 2 in 4-QPSK) into each channel symbol. Thus the capacity of the overall system (coder—modulator—channel—demodulator— decoder) can be expressed in terms of bits per symbol. At this point, it is obvious that in order to achieve this capacity the receiver must know the channel coding and modulation scheme that the transmitter is using. Shannon also provided the fundamental relationship between the channel capacity (C ) and the signalto-noise ratio for an additive white Gaussian noise channel. We will introduce this relationship later in this chapter. First we need to study models for signal power attenuation between the transmitter and the receiver. 2.1.4 Delay, Path Loss, Shadowing, and Fading In the previous discussion we assumed that the transmitted signal was contaminated by only additive white Gaussian noise. This yielded the simple model shown in (2.4). However, in practical channels, signals undergo attenuation and delay. In wireless channels, because of propagation over multiple paths, and mobility of the scatters or of the communicating devices, the attenuation can vary with time and the relative location between the transmitter and the receiver. We have seen that the BER performance of a digital communication system depends on the received SNR. Hence, we are interested in the received signal power after the signal has passed through the channel. Radio waves are scattered by the objects on which they impinge. Hence, unless a very narrow antenna beam is used, the receiver’s antenna receives the transmitted signal along several paths. There is often a direct or line-of-sight path, and there are several paths along which the signal reaches the receiver after one or more reﬂections from various objects. Energy is lost in reﬂections, and is absorbed by media through which the signal passes (partitions and walls). Hence the received signal is a sum of attenuated and delayed versions of the original signal.

Delay Spread and Intersymbol Interference Superposition of the delayed signals from the various paths can cause a symbol from one path to overlap with a neighboring symbol from another path. Let us examine this issue ﬁrst. These are electromagnetic signals and hence they travel at the speed of light; let us take the propagation time to be roughly 0.33 μsec per 100 meters. Hence, this is the kind of delay that can be expected if the various path lengths differ by no more than 100 m. If the symbol time is several μseconds (e.g., 100,000 symbols per second) then there will not be signiﬁcant overlap between the neighboring symbols, and

26

2 Wireless Communication: Concepts, Techniques, Models

we can assume that the symbols are still separately discernible, except that each is multiplied by a complex “attenuation.” If this happens then the channel is said to have ﬂat fading. We will understand the term “ﬂat” when we interpret this phenomenon in the frequency domain. Then, motivated by (2.7), we can write the k-th received symbol after down conversion as Yk = Gk Xk + Ik + Zk

(2.8)

where the various new terms are understood as follows. 1. Gk is the random attenuation of the k-th symbol. Gk , k ≥ 1, is a complex valued random process. Thus, a transmitted symbol is not only attenuated, but can also be rotated. Note that the symbol energy is multiplied by |Gk |2 . Let us write Hk = |Gk |2 ; Hk , k ≥ 1, is a random process, and we need to characterize it in order to understand the effect of the channel on the received signal power, and hence the SNR. We note that the Hk are also called channel gains. 2. Ik is a complex random variable that models the interference (from other transmissions in the same or nearby spectrum4 ). We recall that Zk is a sequence of complex random variables that models the additive noise (for example, the thermal noise in the electronic circuitry of the receiver) and is taken to be a white Gaussian random process. A commonly used simpliﬁcation is to use the same model even for the interference process, with the noise and interference processes being modeled as being independent. The BER then becomes a function of the signal to interference plus noise ratio (SINR). For a transmitter receiver pair, the difference between the smallest signal delay and the largest signal delay is called the delay spread, Td . For example, if the path lengths differ by no more than 100s of meters then the delay spread would be in 100s of nanoseconds. When the delay spread is not very small compared to the symbol time then the superposition of the signals received over the variously delayed paths at the receiver results in intersymbol interference (ISI). We then obtain the following linear model: Jd −1

Yk =

Gk ( j)Xk−j + Ik + Zk

(2.9)

j=0

For every k, Gk ( j), 0 ≤ j ≤ Jd − 1, are complex random variables that model the way the channel attenuates and phase shifts the transmitted symbols. Gk ( j) models 4 Note that we are taking the simpliﬁed approach of treating other users’ signals as interference. More generally, it is technically feasible to extract multiple users’ symbols even though they are superimposed. This is called multiuser detection.

2.1

Digital Communication over Radio Channels

27

the inﬂuence that the input j symbols in the past has on the channel output at k. Thus, in general, a channel has memory; in the model, the memory extends over Jd symbols. The memory arises as a consequence of there existing several paths from the transmitter to the receiver, with the different paths having different delays. The notation shows that the channel gain at the k-th symbol could be a function of the symbol index k; this models the fact that fading is a time-varying phenomenon. As the devices involved in the communication move around, the radio channel between them also keeps changing. The delay spread, Td , has been explained previously as a time domain concept. It can also be viewed in the frequency domain as follows. The symbols Xk are carried over the RF spectrum by ﬁrst multiplying them with a (baseband) pulse of bandwidth approximately W (e.g., 200 KHz), and then upconverting the resulting signal to the carrier frequency (e.g., 900 MHz) (recall (2.2)). The delay spread in the channel (i.e., Td ) can be such that superposition of variously delayed versions of some frequency components in the baseband pulse can cancel out. In such a case, some of the frequency components in the pulses can get selectively attenuated, resulting in the corruption of the symbols they carry; this is called 1 frequency selective fading. On the other hand, if Td > W then all the frequencies fade together and we have ﬂat fading. The assumption of ﬂat fading is reasonable for a narrowband system, where the available radio spectrum is channelized and each bit stream occupies one channel. Then the symbol duration becomes larger than the delay spread, and the model of (2.8) is applicable. This will be the channel model that we will use when analyzing FDM-TDMA cellular systems in Chapter 4. On the other hand, consider the situation in which Wc is small compared 1 to the system bandwidth (Td is large compared to W ); that is, the channel is frequency selective. Then, in relation to the model in (2.9), and recalling that 1 the intersymbol interval is W , we observe that frequency selectivity corresponds to the channel memory extending over more than 1 symbol, and hence to the existence of ISI. Thus, when high bit rates are carried over wideband channels (i.e., large W ) then techniques have to be used to combat ISI, or to avoid it altogether. We will encounter CDMA and OFDMA later in this chapter, as two wideband systems that actually exploit delay spread or frequency selectivity to achieve diversity (a concept explained in Section 2.3). In some systems, we can combat ISI by passing the received signal through a channel equalizer, which can compensate for the various channel delays, making the overall system (i.e., the channel followed by the equalizer) appear like a ﬁxed delay channel. In a mobile wireless situation, owing to mobility, the paths that a signal takes between a transmitter and a receiver may keep changing; hence a channel equalizer needs to be adaptive. In some systems the problem of signals arriving over multiple paths is turned into an advantage. If the paths can be

28

2 Wireless Communication: Concepts, Techniques, Models

resolved, and if they fade independently, then their signals can be combined to reduce the probability of error, for a given received signal-to-noise ratio. Such a receiver is said to exploit multipath diversity.

A Characterization of the Power Attenuation Process It follows from the linear model with ﬂat fading, shown in Equation 2.8, that the received sequence, Yk , k ≥ 1, is also a complex valued random process. The problem for the receiver, on receiving the sequence of complex numbers Yk , k ≥ 1, is to carry out a detection of which symbols Xk , k ≥ 1, were sent and hence which user bits were sent. This problem is particularly challenging in mobile wireless systems since the channel is randomly changing with time. The analysis and design of modulation schemes often is based on the analysis of received signal power to noise power ratios. Hence, it is important to have an effective but simple model of the channel power attenuation process, Hk . The process {Hk } is characterized by writing it in terms of three multiplicative components, that is, Hk =

dk d0

−η

· Sk · R2k

(2.10)

Let us write the marginal terms of the stationary random processes in this expression by dropping the symbol index k. We will now discuss each of these terms. −η The term dd is the path loss factor. Here, d is the distance between the 0 transmitter and the receiver when the k-th symbol is being received, d0 is the “far ﬁeld” reference distance beyond which this model is applicable, and η is the path loss exponent, which is typically in the range 2 to 5. The value of d0 relates to the antenna dimensions and the propagation environment. For distances less than d0 , a different path loss exponent may be used, or, when d0 is very small, we may assume no path loss. If the attenuation is measured at various points at a distance d from the transmitter, then the attenuation will be found to be random, owing to variations in the terrain, and in the media through which the signal may have passed. Empirical studies have shown that this randomness is captured well if the second ξ factor S, in (2.10), has the form 10− 10 , with ξ being a Gaussian random variable with mean 0 and variance σ 2 . This is called the shadowing component of the attenuation, and, since log10 of this term has a Gaussian (or normal) distribution, it is called log-normal shadowing. It is often convenient to express values of power and power ratios in the decibel (dB) unit which is obtained by taking 10 log10 of the value. Hence the shadowing attenuation in signal power is 10 log10 S = −ξ dB, which is zero mean Gaussian with variance σ 2 . A typical value of σ is 8 dB. Considering two standard deviations above and below the mean, this value means

2.1

Digital Communication over Radio Channels

29

that, a high with2×8 probability, shadowing can result in a variation of channel gain of −2×8 10 10 times to 0.025 ≈ 10 times the mean path loss. 40 ≈ 10 Shadow fading is spatially varying, and hence if there is relative movement between the transmitter and the receiver then shadow fading will vary. The correlation in the shadow fading in dB between two points separated by a −

D

distance D is given by σ 2 e D0 , where D0 is a parameter that depends on the terrain. Some measurements have given D0 = 500 m for suburban terrains, and D0 = 50 m for urban terrains. Hence if the distance is varying by a few meters per second (note that 36 Kmph = 10 meters/second) then the shadowing will vary over seconds, which means that the variations will occur over hundreds of thousands of symbols. We now turn to the third factor, R2 , in the expression for attenuation in (2.10). Typical carrier frequencies used in mobile wireless networks are 900 MHz, 1.8 GHz (e.g., these two frequency bands are used in cellular wireless telephony systems), or 2.4 GHz (e.g., used in IEEE 802.11 wireless LAN systems). Hence, the carrier wave periods are a few picoseconds. Thus, when the transmitted signal arrives over several paths then very small differences in the path lengths (a few centimeters) can cause large differences in the phases of the carriers that are being superimposed. Thus, although these time delays may not result in ISI, the superposition of the delayed carriers results in constructive and destructive carrier interference, leading to variations in signal strength. This phenomenon is called multipath fading. This is a random attenuation that has strong autocorrelation over a time duration called the coherence time, Tc ; that is, the attenuations at two time instants separated by more than the coherence time are weakly correlated. The coherence time is related to the Doppler frequency, fd , which is related to the carrier frequency, fc , the speed of movement, v, and the speed of light, c, by fd = fc vc . Roughly, the coherence time is the inverse of the Doppler frequency. For example, if the carrier frequency is 900 MHz, and v = 20 meters/sec, then fd = 60 Hz, leading to a coherence time of 10s of milliseconds. In the indoor ofﬁce or home environment, the Doppler frequency could be just a few Hz (e.g., 3 Hz), with coherence times of 100 s of milliseconds. The marginal distribution of R2 depends on whether all the signals arriving at the receiver are scattered signals, or if there is a line-of-sight signal as well. In the former case, assuming uniformly distributed arrival of the signal from all directions, the distribution of R2 is exponential with mean E R2 , that is, fR2 (x) =

2 1 e −x/E R E R2

The distribution of the amplitude attenuation (i.e., R) is Rayleigh; hence this is also called Rayleigh fading. On the other hand if there is a line-of-sight component so

30

2 Wireless Communication: Concepts, Techniques, Models

that a fraction K K+ 1 of the signal arrives directly, and the remaining signal arrives uniformly over all directions, then K+1 fR2 (x) = 2 e E R

−K− (K+1)x 2

( )

E R

where I0 (x) =

1 2π

2π

⎛ ⎞ K(K + 1)x ⎠ I0 ⎝2 E R2

e−x cos(θ) dθ

0

This is called the Ricean distribution. With this characterization of the attenuation in the received signal power we can now write the received SNR (denoted by Ψrcv ) in terms of the ratio of the transmitted signal power to the received noise power (denoted by Ψxmt ). We have Ψrcv = Ψxmt · H = Ψxmt ·

d d0

−η

−ξ

· 10 10 · R2

(2.11)

Then, in dB, we can write the received SNR as (Ψrcv )dB = (Ψxmt )dB + 10 log10 H = (Ψxmt )dB − 10η log10

d d0

− ξ + 10 log10 R2

(2.12)

BER with Fading We now turn to the calculation of the performance of the wireless link in the presence of fading. We have seen that, although the transmitter may send at a ﬁxed power, in the presence of fading, the received power, and hence the received SNR, is time varying. The rate of variation of the SNR depends on the mobility of the receiver. A receiver that moves short distances over the duration of a “conversation” (e.g., a voice call, or a ﬁle transfer) would sample the distribution of the Rayleigh fading but would see roughly constant values of path loss and shadowing. On the other hand a receiver that makes large movements during a call duration would see variations in all the three attenuation factors during the call. Let us consider the former situation. In this case the in-call performance depends on the value of path loss and shadow fading sampled by the call, and on the distribution of Rayleigh fading, but the performance across calls depends on the variation in path loss and shadowing as well. We would like the performance not to fall below some value. For example, there could be a desired upper bound

2.1

Digital Communication over Radio Channels

31

on BER; exceedance of this bound would be termed an outage. Let us examine this point in the context of the binary modulation scheme discussed in Section 2.1.1. The BER for this modulation scheme was given by (2.5): Perror − AWGN (Ψrcv ) = Q

2Ψrcv

If the path loss and shadowing factors during a call are ﬁxed, then we can calculate the in-call, BER averaged over the fading, as follows:

∞

0

Perror − AWGN

d d0

−η

· 10

−ξ 10

· γ · Ψxmt fR2 (γ)dγ

where, as mentioned earlier, 2 for Rayleigh fading, fR2 (·) is the exponential probability density with mean E R . Let us write the SNR during the call, with the fading averaged out, as Ψrcv :=

d d0

−η

−ξ · 10 10 · E R2 Ψxmt

In many cases it can be shown that the preceding integral expression for in-call BER can be simpliﬁed to the form Perror−fading Ψrcv

for some function Perror − fading . For example, for the binary modulation scheme Ψrcv , discussed earlier, it can be shown that Perror − fading Ψrcv = 12 1 − 1+Ψ rcv

which for large Ψrcv can be observed to decrease reciprocally with SNR (i.e., as 1 ), rather than exponentially, as for unfaded AWGN (see Problem 2.1; see also Ψ rcv

Problem 2.4). During a call, we can write the average SNR (with the averaging being over the fading), Ψrcv , in dB as

Ψrcv

dB

= Ψxmt E R2

dB

− 10η log10

d d0

−ξ

The term Ψxmt E R2 dB is the Rayleigh faded SNR “referred to” d0 . We see that the received SNR, in dB, ata distance d from the transmitter is Gaussian with mean 2 d Ψxmt E R dB − 10η log10 d and variance σ 2 . In order to achieve a certain BER, 0 say, , the received SNR will be required to be above a threshold, say, β; that is, Ψrcv > β ⇒ Perror − fading Ψrcv <

32

2 Wireless Communication: Concepts, Techniques, Models

Violation of this requirement would be called an outage, the probability of which we would like to limit to Poutage . We note that, since we assumed that during a call the path loss and shadowing are ﬁxed, Poutage is the outage probability across calls; that is, the fraction of calls that experience a BER larger than . The BER and outage requirement can then be expressed in the following form: Pr Ψrcv dB < (β)dB < Poutage

Equivalently, Pr Ψxmt E R2

dB

− 10η log10

d d0

− ξ < (β)dB

< Poutage

Let us look at an example. Given that dd0 = 10, η = 3, the shadowing standard deviation σ = 8 dB, the received SNR threshold is β = 10 dB, and Poutage = 0.01, the requirement just displayed is satisﬁed if

Ψxmt E R2

dB

− 30 − 2.3 × 8 = 10

where the factor 2.3 is obtained from a table of the Gaussian distribution. This yields Ψxmt E R2 = 58.4 dB dB

2.2

Channel Capacity

2.2.1 Channel Capacity without Fading Consider the following simple version of the general linear model that was shown in (2.9) Yk = Xk + Zk

(2.13)

where we notice that we have removed the model of ISI, the multiplicative fading term, and also the additive interference term, leaving just a model in which the output random variable at symbol k is the input symbol Xk with an additive noise term Zk . Thus, there is no attenuation of the transmitted symbol, but there is perturbation by additive noise. When the Xk are taken from a one-dimensional constellation (as in the beginning of Section 2.1), then the model for the random process Zk , k ≥ 1, is that these are i.i.d. Gaussian random variables with mean 0 and variance σ 2 (see Equation 2.4). This is called an additive white Gaussian noise (AWGN) channel. The information bits are mapped to the channel symbols

2.2

Channel Capacity

33

Xk , which are corrupted by additive noise. The observations Yk have to be used to infer which symbols were transmitted. Suppose that the input symbols have the following power constraint: 1 lim |xk |2 ≤ P n→∞ n n

(2.14)

k=1

that is, the average energy per symbol is bounded by P Joules/symbol. This is a practical constraint as power ampliﬁers operate well only in certain limited power ranges. Also, microwave radiations can be harmful to the body; hence there are safety regulations on how much power can be radiated by radio transmitters. Further, when several systems coexist then intersystem interference needs to be managed. Hence, some form of power constraint usually is required in wireless communication systems. If the input symbols are allowed to be only real numbers, then Shannon’s celebrated Noisy Channel Capacity Theorem states that the maximum rate at which information can be transmitted over this AWGN channel, in bits/symbol, is given by Prcv 1 bits/symbol C = log2 1 + 2 2 σ

(2.15)

where, Prcv is the received signal power per symbol, and Pσrcv is the received 2 signal-to-noise power ratio. Evidently, here, in the no fading case, we have Prcv = P. What this result means is that this rate can be achieved with the bit error rate going to zero as channel coding is done over longer and longer blocks, with the block length going to ∞. In Section 2.1.1, we derived the symbol-by-symbol channel model by starting with a continuous time model for a modulation scheme that used only real valued symbols. Let us now apply this formula to derive the capacity of that system. We saw that the additive noise sequence has variance N20 . If the power constraint on the transmitted signal (i.e., S(t) in (2.2)) is P Watts, then the power constraint per P symbol is P = PT = W Joules/symbol. Since we are assuming no channel loss, using (2.15), we obtain the capacity C=

1 2P log2 1 + bits/symbol 2 N0 W

(2.16)

If, in (2.13), the input symbols are complex numbers, then the additive noise is modeled as a sequence of complex valued random variables, which is taken to be a sequence of i.i.d. zero mean, circularly symmetric Gaussian random variables with variance σ 2 (recall (2.7)). This means that the real and the imaginary parts

34

2 Wireless Communication: Concepts, Techniques, Models

are independent sequences of zero mean i.i.d. Gaussian random variables with the 2 same variance, σ2 . The capacity formula then takes the simple form Prcv bits/symbol C = log2 1 + 2 (2.17) σ where Prcv is the average received power per symbol. Without channel loss Prcv = P. Now let us apply this to the modulation with complex symbols that led to the channel model in (2.7). There Zk are i.i.d. zero mean circularly symmetric Gaussian with variance with the real and imaginary parts have variance N20 . Then, without channel loss, and a power constraint P on the transmitted continuous time P signal, the constraint on the average received energy per symbol is W , yielding the channel capacity P bits/symbol C = log2 1 + (2.18) N0 W It is instructive to compare the expressions (2.16) and (2.18); see Problem 2.5. We note that these capacity expressions gave the answer in bits per symbol. Often, in analysis it is better to work with natural logarithms. With this in mind we can rewrite (2.17) as Prcv C = ln 1 + 2 nats/symbol σ Since ln x = log2 x × ln 2, the capacity in nats per symbol is obtained by multiplying the capacity in bits per symbol by ln 2 ≈ 0.693. If the symbol rate is T1 then, for the AWGN channel with complex symbols, Shannon’s formula yields the bit rate Prcv 1 log2 1 + bits/second T N0 W where Prcv is average power in the received signal. For the system bandwidth W , the bit rate, therefore, is limited to Prcv W log2 1 + (2.19) bits/second N0 W An important measure of performance of a digital modulation scheme is bits/Hz; that is, the number of information bits that can be carried per Hertz of system bandwidth. Let us write Prcv = Eb × C , where we can call Eb the received energy per bit. (2.19) can then be written as C Eb C = log2 1 + W N0 W C W

2.2

Channel Capacity

35

from which we obtain C

Eb 2W − 1 = C N0 W

The quantity on the left is the ratio of the received energy per bit to the power spectral density of the additive noise, and is called the signal-to-noise ratio per bit. We C 2W − 1 Eb C conclude that, in order to achieve W bits/Hz, we require an N of at least . C 0

For example, for

C W

W

= 1 bit/Hz (a typical number for a FDM-TDMA system such

as GSM), the minimum value of schemes need larger values of

Eb N0

Eb , N0

= 1 or 0 dB. Practical modulation and coding

as seen in the examples earlier in this chapter.

2.2.2 Channel Capacity with Fading How does a time varying channel attenuation affect the Shannon capacity formula? If the channel attenuation is h, and the noise is AWGN, then, for transmitted power Pxmt , the channel capacity is given by (2.19): hPxmt W log2 1 + N0 W Suppose that the transmitter is unaware of the extent of the channel fading, and uses a ﬁxed power and a ﬁxed modulation and coding scheme. Suppose also that the fading level varies slowly. Then, for a given level of fading, the receiver must know h in order for the communication to achieve the Shannon capacity. To see this, let us look at Figure 2.4(b). √ If the channel’s power attenuation is h, the received symbols are multiplied by h. This results in the symbols being “squeezed” together or spread apart. Obviously, the detection thresholds will need to depend on the level of fading. Suppose that Hk is a stationary and ergodic process. It can then be shown that if the transmitter cannot adapt its coding and modulation, but the receiver can exactly track the fading, then the channel capacity with fading is given by hPxmt gH (h)dh Cfading − CSIR = W log2 1 + (2.20) WN0 where gH (·) is the marginal density of the channel attenuation process Hk . For example, gH (h) is exponential for Rayleigh fading (see Section 2.1.4). The acronym CSIR stands for channel state (or side) information at the receiver. Thus the transmitter can encode at any ﬁxed rate R < Cfading−CSIR , and for large enough code blocks the error rate can be made arbitrarily small, provided the receiver can track the channel. It is important to bear in mind that this is an ideal result; to

36

2 Wireless Communication: Concepts, Techniques, Models

achieve it, the channel fades will have to be averaged over and this will result in large coding delays. xmt In Problem 2.6 we see that Cfading − CSIR ≤ W log2 1 + E(H)P , that is, the WN0 capacity with fading is less than that with no fading with the same average SNR. With fading, there will be times when the SNR is higher than the average and times when the SNR will be lower than the average. Yet this result shows that the resulting channel capacity is less than that without fading, as long as the same average SNR is maintained.

2.3

Diversity and Parallel Channels: MIMO

We emphasise that we are discussing direct point-to-point communication between a transmitter and a receiver. We have already seen that the signal from the transmitter can reach the receiver over multiple paths. Since it can be expected that random fading along these paths will be independent, combining the signals from these paths in some manner might lead to better performance than working with the aggregate signal. Such diversity can be obtained in various ways. If the receiver has multiple antennas (see Figure 2.8), and if the antennas are spaced sufﬁciently far apart (at least half the carrier wavelength) then, for the same transmitted signal,

G1 G2

receiver

X

^ X

GK

Figure 2.8 A single-input-multiple-output (SIMO) system comprising one transmit antenna and K receive antennas.

2.3

Diversity and Parallel Channels: MIMO

37

the signals received at the different antennas fade approximately independently.5 To see how such independently faded copies can be exploited, let us consider the following model for the signal received along each path. Yk = Gk X + Zk

where k, 1 ≤ k ≤ K, indexes the diversity “paths,” and X is the transmitted (complex) symbol. The Zk , 1 ≤ k ≤ K, are zero mean, i.i.d. circularly symmetric normal random variables, with variance σ 2 . Recalling the notation Hk = each 2 jθ k |Gk | , let us write Gk = Hk e , that is, on the k-th path, the transmitted symbol X is scaled by Hk and rotated by θk . Assuming that the receiver knows the values of θk , 1 ≤ k ≤ K, it can be shown that the optimum strategy is to form a linear combination of the K received signals by using complex weights μk e−jθk , to obtain K

Y=

μk e−jθk Yk

k=1

⎛ =⎝

K

⎞

μ k Hk ⎠ X +

k=1

K

μk e−jθk Zk

k=1

Note that rotation by θk does not destroy the circular symmetry of the noise, Zk . Let the transmitted power be P, that is, E(|X|2 ) = P. If the symbol detection is based on the statistic Y , then the performance of this receiver algorithm will be based on the received SNR Ψrcv =

2 Hk P K 2 2 k=1 μk σ K k=1 μk

Now, by the Cauchy-Schwartz inequality, we have ⎛ ⎝

K

k=1

μk

⎞2 Hk ⎠ ≤

K k=1

μ2k

K

Hk

k=1

5 To understand the relationship between antenna spacing and low correlation between received signals, let

us recall the concept of coherence time. Multipath fading observed by a mobile has low correlation between f

time instants separated by Tc , which is roughly the reciprocal of fd = cc v, where fc is the carrier frequency, c is the speed of light, and v is the speed of the mobile. Equivalently, fd = λvc , where λc is the wavelength of the carrier. It follows that fade correlations are weak over a distance equal to the carrier wavelength. A precise analysis of the phenomenon actually shows that the correlations are weak over distances as little as half the wavelength. Note that λc = 30 cm for fc = 1 GHz, and λc = 6 cm for fc = 5 GHz.

38

2 Wireless Communication: Concepts, Techniques, Models

with equality when μk = a H some a (i.e., the vector (μ1 , μ2 , . . . , μK ) k for √ √ √ when is a multiple of the vector H1 , H2 , . . . , HK ). Choosing the weights μk , 1 ≤ k ≤ K, in this way maximizes the predetection SNR, yielding ⎛ Ψrcv = ⎝

K

⎞ Hk ⎠ Ψxmt

k=1

where, as before, Ψxmt = σP2 is the transmit SNR. We now wish to study the bit error probability for this approach. Suppose that the bit error probability with AWGN decreases exponentially with the received SNR (see Problem 2.1). Then, the average bit error rate is proportional to

E e

−Ψrcv

=E e

−

K k=1

Hk Ψxmt

Recall our discussion in Section 2.1.4, and hence, write Hk = πΦk where π is the path loss and shadowing factor from the transmitter to the receiver (taken to be a constant over the time scale to which this analysis applies), and Φk , 1 ≤ k ≤ K, represent Rayleigh fading over the various paths. This yields K − k=1 πΦk Ψxmt E e−Ψrcv = E e

Assuming that the fading at the different antennas are independent and identically distributed, we take the Φk , 1 ≤ k ≤ K, to be i.i.d. exponentially distributed with mean, say, φ. We then have K E e−Ψrcv = E e−Φ1 πΨxmt =

1 1 + φπΨxmt

K

−K ≈ Ψrcv

where the approximation holds for large average received SNR Ψrcv = φπΨxmt . Recall that for Rayleigh fading the probability of error decreased only as the reciprocal of Ψrcv . Thus, by combining the received signals over multiple paths, the bit error probability performance has been substantially improved. From the form for the decay of the bit error probability with Ψrcv , we say that we have a diversity gain of K. The transmitter could also just repeat the signal over time, and if the repetitions are spaced apart by more than the coherence time (see Section 2.1.4)

2.3

Diversity and Parallel Channels: MIMO

39

then the received signals fade independently. It turns out that commonly used channel codes provide a better chance of successful decoding if the channel error process is uncorrelated over the code symbols. We saw earlier that the channel fade process, Gk , is correlated over periods called the channel coherence time, which depends on the speed of movement of the mobile device. Interleaving is a way to obtain an uncorrelated fade process from a correlated one. Basically the transmitter does not send successive symbols of a codeword over contiguous channel symbols, but successive symbols are separated out so that they see uncorrelated fading. In between, other codewords are interleaved. We say that interleaving exploits time diversity, that is, the fact that channel times separated by more than the coherence time fade independently. Observe that interleaving introduces interleaving delay, which adds to the link delay, and hence to the end-to-end delay over the wireless network. Also, interleaving fails if the fading is very slow, for example if the relative motion stops, and the transmitter-receiver pair are caught in a bad fade. In the discussion earlier in this section, we considered the case in which multiple independently faded copies of a transmitted symbol arrive at the receiver. By appropriate combining of these received symbols, the probability of error is reduced. Suppose that the channel is such that the transmitter can, in parallel, transmit several symbols, each of which is then independently faded and received. Then the available power P can be distributed over the parallel channels to obtain a higher bit rate than if all the power was used on a single channel; see Problem 2.7, and, for more details, Chapter 6. Physically, parallel channels between a transmitter-receiver pair can arise if the system bandwidth is partitioned into several orthogonal channels (e.g., by partitioning in frequency and time), and then several of these channels are simultaneously available for communication between the transmitter-receiver pair. Even for narrow-band systems, multiple parallel channels can arise if the transmitter and receiver use multiple antennas (see Figure 2.9). As before, let the system bandwidth be W Hz. Suppose antennas that there are N transmit and M receive antennas. Let Gk, i, j 1 ≤ i ≤ M, 1 ≤ j ≤ N denote the channel gain between the transmit antenna j and the receive antenna i, at the k-th symbol. As we know, these channel gains will capture the path loss, the shadowing, and the multipath fading, and will be modeled as complex valued random variables. Let Gk denote the M × N channel gain matrix at symbol k. T Let Xk = Xk,1 , Xk,2 , . . ., Xk,N denote the input symbols into the N transmit antennas at the k-th symbol time. These too are complex valued, as would be the T case for general two-dimensional constellations. Let Yk = Yk,1 , Yk,2 , . . ., Yk,M denote the corresponding complex valued channel outputs. Hence, we can write Yk = Gk Xk + Zk

40

2 Wireless Communication: Concepts, Techniques, Models G1,1 G2,1

1

1

GM,1 2

G2,N

N GM,N

M

Figure 2.9 A multiple-input-multiple-output (MIMO) system comprising N transmit antennas and M receive antennas.

where Zk is the M × 1 additive noise process. The components of Zk are zero mean i.i.d. circularly symmetric Gaussian random variables, each with variance σ 2 ; also the Zk sequence is i.i.d. over k. This also means that Zk,i , 1 ≤ i ≤ M, are complex with their real and imaginary parts being zero mean independent Gaussian random 2 variables, each with variance σ2 . There is a total transmit power constraint of P: 1 |Xk,j |2 < P n→∞ n n

N

lim

k=1 j=1

Deﬁne, as before, Ψxmt = σP2 . Let us also assume i.i.d. Rayleigh fading between each transmit-receive antenna pair. Then Hk,i,j = |Gk,i,j |2 are i.i.d. exponentially distributed with a common mean over the antennas, say, φ. If the distance between the transmit antennas and the receive antennas is large, then the path losses between the antenna pairs would be the same. Let us denote this common path loss by π, as in the diversity analysis shown earlier. We “pull” the average path

2.3

Diversity and Parallel Channels: MIMO

41

loss, and the mean of the Rayleigh fading out of the channel gain matrix, leaving the mean power gain of the elements in the channel gain matrix to be 1. Then the received SNR, averaged over Rayleigh fading, is (as before) written as Ψrcv = φπΨxmt

The transmitter does not know the channel gains, and it can be shown that the best strategy is for the transmitter to split its power equally over the N transmit antennas. Then, given a sample of the gain matrix, say, G, it can be shown that the capacity of this channel is given by Ψrcv † C = W log2 det IM + (2.21) bits/second G·G N where det(·) denotes the “matrix determinant,” IM denotes the M × M identity matrix, and G† denotes “conjugate-transpose.” Now G·G† is an M×M Hermitian matrix (i.e., its conjugate-transpose is the same as itself). The theory of matrices provides the following facts: 1. The eigenvalues of G · G† are real and nonnegative. 2. The number of positive eigenvalues is no more than min{M, N}. Let us index the eigenvalues in decreasing order of magnitude and denote them by λ1 ≥ λ2 ≥ . . . ≥ λmin{M,N} . Then using the fact that the determinant of a square matrix is equal to the product of its eigenvalues, and that the eigenvalues of IM + ΨNrcv G · G† are of the form 1 + λj ΨNrcv , we obtain the following simpliﬁcation: min{M,N} Ψrcv C=W log2 1 + λj (2.22) bits/second N j=1

We see that, under the assumptions we have made, the multiple transmit antenna and multiple receive antenna system (also called a multiple-input-multiple-output (MIMO) system) is equivalent to several parallel channels. Note that, for different realizations of the channel gain matrix, the gains of the parallel channels (the eigenvalues λj , 1 ≤ j ≤ min{M, N}) will be different. In effect, we have parallel channels with random gains. Let us consider the situation in which M = N , and all the eigenvalues are equal, say, λ. Then λΨrcv C = WM log2 1 + bits/second M

42

2 Wireless Communication: Concepts, Techniques, Models

We see that for a single transmit and receive antenna system the capacity (i.e., W log2 (1 + Ψrcv )) scales as log Ψrcv for large Ψrcv , whereas for an M × M MIMO system (with equal eigenvalues) the capacity scales as M log Ψrcv . This is called multiplexing gain. Thus, we ﬁnd that a multiple antenna system can be used to obtain diversity gain (as explained above for one transmit antenna and M receive antennas), or can be used to increase the channel capacity by the creation of parallel spatial channels between the transmit and receive antenna groups. For an N transmit antenna, and M receive antenna system, the diversity gain is bounded by M × N , whereas the multiplexing gain is limited to min{M, N}. We note that the above discussion assumed that the channel gains are unknown at the transmitter. If channel gain estimates could be provided to the transmitter, then it could judiciously choose the transmitted symbols and their powers so that the better of the parallel spatial channels are assigned the larger transmit powers. We will study such optimal power allocation problems in the OFDMA context in Chapter 6.

2.4 Wideband Systems Unlike the narrow-band digital modulation used in FDM-TDMA systems, in CDMA and OFDMA the available spectrum is not partitioned, but all of it is dynamically shared among all the users. The simplest viewpoint is to think of CDMA in the time domain and OFDMA in the frequency domain. In a wideband system, a user’s symbol rate is much smaller than the symbol rate that the channel 1 can carry (i.e., W ). 2.4.1 CDMA In CDMA a user’s symbol, which is of duration L channel symbols (also called chips), is multiplied by a spreading code of length L chips. This is called direct sequence spread spectrum (DSSS), since this multiplication by the high rate spreading code results in the signal spectrum being spread out to cover the system bandwidth. If the user’s bit rate is R and the chip rate is Rc (> R), then L = RRc (> 1) and is called the spreading factor. The spreading codes take values in the set {−1, +1}L and are chosen so that each code is approximately orthogonal to all the time shifts of the other codes, and also to its own time shifts. Then the spread symbols are transmitted. All the signals interfere because they occupy the same radio bandwidth. We provide a simple analysis of such a system, with reference to the depiction in Figure 2.10. There are M users. The symbol duration is R1 , during which there are L chips. Denote the chip time by τc = R1c . We can write the transmitted signal from User 1 (see Figure 2.10) as x1

L−1 j=0

S1,j p(t − jτc )

2.4 Wideband Systems

43

3 (2)

(2)

S2,0

S2,(L 2 1)

2 x1

1 S1,0

S1,(L 2 1)

Figure 2.10 Depiction of the superposition of CDMA symbols. The transmissions of three users are shown. The tall ticks denote symbol boundaries and theshort ticks denote chip boundaries. A symbol of User 1 that has the value x1 ∈ {+ E1 , − E1 } has been shown. It has been spread by the code S1,j , 0 ≤ j ≤ L − 1. Interfering symbols of the other users are also shown. The interfering users are assumed to be chip synchronous but their symbols are randomly offset from that of the symbols of User 1.

where x1 is the user’s information carrying symbol, S1,j , 0 ≤ j ≤ L − 1,is User 1’s W spreading code, and p(t) is the baseband pulse that is bandlimited to − W 2 , 2 , and has the property ∞ p2 (u) du = 1 −∞

√ √ Let xi ∈ {+ Ei , − Ei }, where Ei corresponds to the transmit power used by User i. Let hi,1 denote the magnitude of the channel attenuation from the transmitter of User i to the receiver of User 1. For simplicity, let us work at the baseband, and then we can write the received signal at the receiver of User 1, over the duration of one symbol, 0 ≤ t ≤ R1 , as

y(t) =

L−1 j=0

h1,1 x1 S1,j p(t − jτc ) +

M L−1

(i)

hi,1 xi,j Si,j p(t − jτc ) + N(t)

i=2 j=0

where xi,j denotes the value of the symbol of User i that interferes with User 1 at (i) the j-th chip in User 1’s symbol (see Figure 2.10), Si,j denotes that a shifted version (denoted by the superscript (i)) of the spreading code of User i interferes with the chips of User 1, and N(t) is additive white Gaussian noise with power spectral

44

2 Wireless Communication: Concepts, Techniques, Models

density N0 , bandlimited to − W , W . The receiver of User 1 now performs the 2 2 following operation:

+∞

−∞

y(u)

L−1

S1,j p(u − jτc ) du

j=0

yielding the following statistic,6 based on which the transmitted symbol from User 1 has to be detected:

h1,1 x1 L +

M L−1

(i)

hi,1 xi,j Si,j S1,j + Z

i=2 j=0

where Z is zero mean Gaussian with variance N0 L (to see how this is obtained, see the derivation in the appendix of this chapter). The spreading codes are pseudo(i) random sequences taking values in {−1, +1}, and hence we model xi, j Si,j S1,j , 0 ≤ √ √ j ≤ L − 1, as i.i.d. random variables taking values in {+ Ei , − Ei }, each with equal probability. Thus we obtain the following symbol-by-symbol model for the CDMA channel: Yk = L h1,1 Xk + Ik + Zk (2.23) where Ik is the interference, Zk is additive noise (which is an i.i.d. Gaussian sequence with zero mean and variance N0 L), and we have assumed that the channel gains are not varying with time. Since the interference is the sum of contributions from many independent random variables, we model it also as having a Gaussian distribution. Note, from this calculation, that Ik has zero mean, and variance M

L hi,1 Ei

i=2

Hence, the detection performance will depend on (see Section 2.1.1)

M

L2 h1,1 E1

i=2 L

hi,1 Ei + N0 L

=

M

L h1,1 P1

i=2 hi,1

Pi + N0 W

6 The integration over (−∞, +∞) will actually cover neighboring symbols as well. But, because the pulses p(t − jτc ) are orthogonal, the terms that we display are all that we will get.

2.4 Wideband Systems

45

where we have taken Rc = W , and Pi = Ei ×Rc as the transmit power of User i (the power is the energy per chip times the chip rate). Thus the detection performance depends on the ratio M

L h1,1 P1

i=2 hi,1

Pi + N0 W

(2.24)

Now we can see why L is also called the processing gain. The effective predetection signal-to-interference-plus-noise ratio (SINR) for a user is the received SINR h P (i.e., M 1,1 1 ) multiplied by the processing gain L. i=2

hi,1 Pi +N0 W

Recalling the notation Td for the delay spread of the channel, let us write Ld = Tτcd : Ld is the number of chip-times that correspond to the delay spread. Now consider the signal arriving over paths that have delays that are multiple of the chiptimes. If the receiver can lock into any of these paths, then the transmitted symbol can be decoded as described earlier. If the paths fade independently, however, then we can exploit multipath diversity in much the same way as explained in Section 2.3. Because of the orthogonality property mentioned earlier, at the receiver, multiplication of the received signal by various shifts of the spreading code, and appropriate linear combination of the results, yields a detection statistic that is the sum of several faded copies of the user symbol. Since these shifts correspond to as many paths from the transmitter to the receiver, this is called multipath resolution. In the context of CDMA systems this is achieved by the Rake receiver. We note that this is exactly the same procedure as explained for receive antenna diversity in Section 2.3. Thus the Rake receiver permits a desired bit error rate to be achieved with a smaller SINR. Advanced receiver techniques such as interference cancellation now also are employed in CDMA systems. Scheduling transmissions in a CDMA system involves a decision as to the spreading codes and the power levels to be allocated to the users. These determine the rate at which a particular bit ﬂow can be transmitted. Of course, this decision will have to be made jointly for all users, since the decision for one user impacts every other user. We turn to such resource allocation problems in Chapter 5. 2.4.2 OFDMA We begin by recalling some notation. The system bandwidth is denoted by W , and the delay spread by Td . In a wideband system we are dealing with a situation 1 in which Td >> W , so that intersymbol interference has to be dealt with if we 1 . For example, we may have directly do digital modulation at a symbol rate of W W = 5 MHz, and Td = 5 μsec. OFDMA is based on OFDM (orthogonal frequency division multiplexing) (see [43]), which can be viewed as statically partitioning the available spectrum into several (e.g., 128 or 512) subchannels, each of bandwidth B, such that B > Td ); see Figure 2.11. The term orthogonal in OFDM refers to the fact that the center frequencies of the subchannels are separated by the reciprocal of the OFDM block time, T (see Figure 2.12). This makes the carriers approximately orthogonal over the block time. The subchannels can then be overlapping (i.e., B > T1 ), while the orthogonality between the subcarriers facilitates demodulation at the receiver. Let Xj,k , 1 ≤ j ≤ n, denote the j-th symbol in the k-th OFDM block (see Figure 2.11). The batch of n symbols, which are transmitted in parallel, is also called an OFDM symbol. Then the earlier discussion suggests that the predetection channel output can be written as Yj,k = Gj,k Xj,k + Zj,k

(2.25)

X1,k

X1,k11

X2,k

X2,k11

X3,k

X3,k11

X4,k

X4,k11

X5,k

X5,k11

{ {

User bit stream

{

11010011100111 0110 011011

OFDM Carriers

} } }

} } } }

where j, 1 ≤ j ≤ n, indexes the subcarrier and k ≥ 1 indexes the successive OFDM symbols. Gj,k denotes the fading on the j-th subcarrier during the k-th OFDM symbol. Zj,k denotes an additive noise sequence, which is taken to be i.i.d. zero mean, Gaussian.

T T T Successive OFDM blocks

Figure 2.11 Depiction of the mapping of user bits into OFDM symbols. Here there are ﬁve OFDM carriers. Serially arriving user bits are split into pairs that are mapped successively into ﬁve parallel channel symbols (X1, k , X2, k ,. . ., X5,k ), k ≥ 1 (for example, the 4-QPSK constellation could be used).These ﬁve channel symbols comprise an OFDM block, which is transmitted over the block time T.

2.4 Wideband Systems

47 1 T

B W

Figure 2.12 In OFDMA, the system bandwidth, W, is partitioned into overlapping subchannels, each of bandwidth B, with their center frequencies spaced apart by 1 , where T is the OFDMA symbol duration.

T

Let us see how this model can be justiﬁed. By the orthogonality requirement, the carrier spacing is the reciprocal of the OFDM block time, T1 . Then the number of carriers, n, is related to the system bandwidth, W , by 1 ×n=W T

As an example, consider T = 100 μsec, so that the carrier spacing is 10 KHz and, for W = 5 MHz, n = 500. Suppose the channel delay spread, Td , is such that 1 × n >> Td W

even though

1 W

< Td . Then, combining the previous two equations we ﬁnd that T >> Td

that is, the delay spread is much smaller than the OFDM symbol duration. We see that this is true in our numerical example, where T = 100 μsec and Td = 5 μsec. 1 Thus, the model in (2.25) is justiﬁed if the condition W × n >> Td holds. We see 1 that a frequency selective channel (for which W < Td ) gets converted to n parallel channels, each of which is frequency nonselective. If it is further true that T × N > W ), and we are left with ∞

1 Ck p(t − kT) + √ U(t) 2 k=−∞

(2.27)

U(t) is white Gaussian with power spectral density N20 Watts/Hz. W , + Since the signal is now bandlimited to the interval − W 2 2 , the average noise

The noise

√1 2

power is W ×

N0 2

=

WN0 2

Watts; this means that 1 t→∞ t

t

lim

0

U(x) √ 2

2 dx =

WN0 2

where the integrand on the left is the power dissipation if the noise was put across a 1 ohm resistor; the integration yields energy over (0, t), and the division by time yields the average power. The receiver also needs to synchronize to the pulse boundaries. Once this is done the demodulator then needs to look at each received pulse and determine which symbol it is carrying. This step is called detection. Let us now see √ how the k-th√symbol is detected, that is, how it is determined whether Ck = + Es , or Ck = − Es . The received signal is multiplied by the pulse p(t − kT) and integrated (−∞, +∞), the pulse p(t) being assumed to be known at the receiver.7 Since over +∞ 2 −∞ p (t) dt = 1, and the shifted pulses are orthogonal, this yields Ck +

+∞

−∞

U(t) √ p(t − kT) dt 2

Now U(t) is a zero mean Gaussian process; hence, using the fact that a linear combination of Gaussian random variables is again Gaussian, we conclude that 7 Since the pulses are practically time limited to some small multiple of T, such an integration can be

performed by storing the received signal for some multiple of T, before starting the integration.

50

+∞ −∞

2 Wireless Communication: Concepts, Techniques, Models U(t) √ p(t 2

− kT) dt is a zero mean Gaussian random variable, which we denote

by Zk . Thus, E(Zk ) = 0, and the variance of Zk is obtained as E

+∞

−∞

U(t) √ p(t − kT) dt 2

1 = E 2

2

+∞ +∞ −∞

−∞

U(t)p(t − kT)U(x)p(x − kT) dt dx

Since U(t) is a white Gaussian noise process, with power spectral density N0 , W W bandlimited to − 2 , + 2 , it can be shown that the covariance function of U(t) is given by E(U(t)U(x)) = N0

sin πW(x − t) π(x − t)

It then follows that +∞ +∞ 1 E U(t)p(t − kT)U(x)p(x − kT) dt dx 2 −∞ −∞ +∞ N0 +∞ sin πW(x − t) = p(t − kT) dt p(x − kT) dx 2 −∞ π(x − t) −∞ However,

sin πWx is just the πx W − 2 ,+ W 2 . Since

pass band frequencies, we have

+∞

−∞

transfer function of an ideal low pass ﬁlter with the pulse p(t) is bandlimited to this same range of

sin πW(x − t) p(t − kT) dt = p(x − kT) π(x − t)

We therefore conclude that E

Zk2

=E

+∞ −∞

U(t) √ p(t − kT) dt 2

2 =

N0 2

In a similar manner it can be shown that E(Zk Zl ) = 0, for k = l . Hence, since they are jointly Gaussian, Zk and Zl are independent for k = l . Thus, we ﬁnd that we have the symbol-by-symbol channel model Yk = Ck + Zk

where Zk is a sequence of i.i.d. zero mean Gaussian random variables with variance N20 .

Problems

51

Problems 2.1

Show that Perror−AWGN decreases exponentially with x, Q(x) ≈

2.2

2

x √1 e− 2 x 2π

Es N0 .

(Hint: for large

.)

Consider a mobile radio environment in which we model only path loss and Rayleigh fading. The path loss exponent is η. The transmit power, averaged over Rayleigh fading, at the reference distance d0 from a transmitter is P. a. Write down an expression for the random received power Prcv (d) at a receiver at a distance d = ad0 , and obtain the distribution of Prcv (d). b. Two cochannel transmitters (indexed 1 and 2) are simultaneously transmitting at distances d1 = a1 d0 and d2 = a2 d0 from the receiver. A transmission can be decoded if its signal to interference ratio exceeds γ. Ignoring the receiver noise, obtain the probability that the transmission from Transmitter 1 is decoded, treating the signal from Transmitter 2 as interference. This is called the capture probability (of Transmitter 1 over Transmitter 2). c. Determine β such that if a2 > (1 + β)a1 then the probability of transmission 1 being decoded is greater than 1− ( > 0 is very small).

2.3

Consider the binary modulation scheme analyzed in Section 2.1.1. Obtain the bit error rates for various SNR values γ = 12 dB, 11 dB, 10 dB, and 9 dB. In each case, calculate the probability of packet error for 1500 byte packets. Hence compare the plots in Figure 2.9 with the AWGN plot in Figure 2.12. Hint: Use the approximation Q(x) ≈

2

x √1 e− 2 x 2π

.

2.4

For the same situation as Problem 2.3 consider Rayleigh fading. For average (Rayleigh-faded) SNRs γ = 12 dB, 24 dB, and 36 dB, obtain the fraction of time that the SNR is less than 9 dB. Hence explain why a very large SNR is required in Figure 2.12 to obtain a high throughput.

2.5

By using the concavity of log(1 + x), show that the capacity in (2.16) is less than that in (2.18). What practical insight do we get from this? xmt Use Jensen’s inequality to show that Cfading − CSIR ≤ W log2 1 + E(H)P . WN

2.6 2.7

0

Consider two AWGN channels with the same (power) fading h, and noise power σ 2 . We have an amount of power P to assign. If the power Pi is hPi assigned to Channel i, the capacity achieved is ln 1 + σ 2 . Is it better to put all the power into one channel or to split the power over the two channels? What is the optimal power assignment, assuming that the transmitter knows that the two channels have the same power gain?

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CHAPTER 3 Application Models and Performance Issues

I

n Chapter 2 we provided an understanding of the issues and techniques involved in carrying bit streams over wireless channels. The resources required to carry a bit stream depend on the characteristics of the stream (e.g., the average rate, peak rate, and rate variability), and the performance required by the application generating the bit stream (e.g., an interactive voice call requires end-to-end delay bounds, but can tolerate some data loss). In this chapter we will discuss the major types of applications that telecommunication networks are used for, and the performance issues related to these applications, particularly when wireless networks are involved.

Overview We begin by providing a “big-picture” view of telecommunication networks as they exist today, showing all the elements, including the phone network, the Internet, and various wireless access networks. Then we outline various application scenarios that can arise in these interconnected networks. We classify network applications into three types according to the trafﬁc they generate for the network to carry, namely, elastic trafﬁc, real-time stream trafﬁc, and store-andforward stream trafﬁc. Taking interactive telephony as the principal example of real-time stream trafﬁc, we point out that the trafﬁc offered to the network can be either constant bit rate (CBR) or variable bit rate (VBR). We provide the quality of service (QoS) objectives for each case. The predominant use of the Internet is for applications such as e-mail and web browsing, which generate elastic trafﬁc. Such applications are also increasingly important for wireless access networks, as users begin to use their handheld devices for Internet access. The remainder of the chapter provides an understanding of this important type of trafﬁc. We show the need for feedback control of elastic trafﬁc sources. In the Internet such feedback control is exercised by the Transmission Control Protocol (TCP), which uses an adaptive window mechanism for managing the rates of elastic trafﬁc sources. The latter part of this chapter will be devoted to a discussion of the performance of TCP over wireless links.

54

3.1

3 Application Models and Performance Issues

Network Architectures and Application Scenarios

In Figure 3.1 we provide a simpliﬁed view of telecommunication networks as they exist today. The public switched telephone network (PSTN) has carried telephone calls for nearly a century. In this network, the calls are multiplexed onto the links by using circuit switching. Packet switched public data networks have evolved from the early X.25 networks to the present ubiquitous Internet. Cellular networks have provided mobile access since the early 1980s, and already have evolved through three generations of commercial deployment. As discussed in Chapter 1, a variety of resource allocation techniques are employed in cellular networks; these techniques will be the subject of Chapters 4, 5, and 6. In campuses and enterprises, mobile devices such as laptops and personal digital assistants (PDAs) obtain access to Internet services via wireless local area networks (WLANs). Figure 3.1 also shows some emerging technologies, namely, wireless metropolitan area networks (WMANs), and ad hoc multihop wireless mesh networks. Figure 3.1 also shows certain devices, generically called gateways (GW), interconnecting the PSTN, the Internet, and cellular networks. Note that we show only bearer gateways that are in the path of the actual application trafﬁc. Signaling gateways are also needed because signaling protocols are different in networks

Ad Hoc Multihop Wireless Network

D WLAN

R TX

GW

Internet

TX

A

WMAN

R

R

PSTN

CO

AP/BS

R R GW

LAN Switch

GW Cellular Network

VoIP Client

B

media server

E

C

Figure 3.1 A simpliﬁed view of the public switched telephone network (PSTN) and the Internet, and how they connect with each other and various wireless networking technologies. GW denotes a gateway; there are gateways for signaling and call control, and also for transferring trafﬁc across network boundaries. Other abbreviations are: CO – Central ofﬁce, TX – Trunk exchange; R – Router.

3.1

Network Architectures and Application Scenarios

55

that have evolved independently. For example, a signaling protocol called Session Initiation Protocol (SIP) is used to set up voice calls in the Internet, whereas the Signaling System No. 7 (SS #7) is used in the PSTN. In this setting we can identify several different instances of point-to-point communication, each of which gives rise to certain resource allocation issues. A common instance is that of a voice call between a ﬁxed line telephone instrument on the PSTN (e.g., A in Figure 3.1) and a cellular phone (e.g., C). We will often use the more generic term mobile station (MS) for a mobile device such as a cell phone or a PDA; in some technologies the terms station (STA) or subscriber station (SS) are used instead of MS. One of the functions of the gateway, GW, between the PSTN and the cellular network is to convert between the constant bit rate (CBR) ﬂow of voice bytes in the PSTN to a lower bit rate voice coding scheme over the resource limited cellular wireless network. In cellular networks, typically, voice is handled as a CBR stream of a lower bit rate than that in the PSTN. In an FDM-TDMA system, such as GSM, there are channels, each of which can carry one call at a ﬁxed rate. An accepted call is assigned to a channel for the entire duration of the call. Thus, this is essentially a circuit multiplexing system. A call for which no free channel is available is blocked and the main performance measure for the system is the blocking probability, deﬁned as follows. Let A(t) be the number of call arrivals until time t , and B(t) be the number of calls blocked in the same time; then the blocking probability Pb is given by B(t) t→∞ A(t)

Pb = lim

whenever the limit exists. In a CDMA cellular system, the voice connection is handled by assigning to it the required coded rate. However, unlike typical FDMTDMA systems, a variety of rates can be assigned. Each accepted call needs to be assigned a transmit power level, and whether or not a call is accepted depends on the rate requirements of the other calls that have already been accepted, and the resulting interference levels once the new call is accepted. Again, the performance measure of interest is the probability of call blocking. Another possible type of connection is a voice call between the PSTN instrument A and a voice over IP (VoIP) endpoint, say, B. The GW between the PSTN and the Internet then would convert between the CBR ﬂow of voice bytes in the PSTN to an asynchronous ﬂow of voice packets in the Internet. Since the packets ﬂow asynchronously in the packet network, and can be queued in buffers in the network routers, certain new issues arise, which will be discussed in Section 3.3. Similarly, there could be a voice call between either A or B and the endpoint D, which accesses the Internet via a WLAN or a WMAN. In such cases, the packet multiple access mechanism over the wireless access network affects the performance of the voice call. Yet another scenario is that of the MS C being used to browse the contents of the web server E which is attached to the Internet. We will see later, in Section 3.4,

56

3 Application Models and Performance Issues

that this kind of application is quite different from voice, as there is no intrinsic rate at which the data should be transferred from the web server to the mobile phone. Feedback-based rate control algorithms are employed to ensure some sort of rate fairness between such connections, and efﬁcient utilization of network resources. In the Internet, such control is exercised by TCP in conjunction with implicit or explicit congestion feedback from the network. In addition, a cellular access network such as a CDMA cellular network, or an OFDMA network, would have its own rate control algorithms. Unlike these centrally controlled cellular systems, a random access WLAN (see Chapter 7) does not have an explicit rate allocation mechanism. Hence, when a device such as D is engaged in web transfers from the Internet, it is of interest to determine what the throughput is, and what kind of fairness is achieved. Finally, the MS C or the device D could be displaying a video that is stored in the server E. Now, once the video starts playing, the network should provide this connection the average rate required to transport the video stream. Variability in the rate at which the network transports the video can be compensated by buffering a sufﬁcient amount of the video in the playout device. Such buffering must be done in such a way that the playout does not starve (i.e., the buffer empties out), nor does the buffer overﬂow. Figure 3.1 also shows an ad hoc multihop wireless mesh network attached to the Internet. Although multihop mobile wireless networks have been studied for more than three decades (in the early years under the name packet radio networks), even today the deployments of such networks are still experimental. It is one of the more active research areas in wireless networking, and we will provide a research oriented discussion in Chapters 8, 9, and 10. The IEEE 802.16 suite of protocols (popularly known as WiMax) now contains the deﬁnition of a mesh networking standard. Under this standard, nodes that cannot directly access a WiMax base station (BS) can form a static mesh network that is connected at some point to the WiMax BS. We can think of these as managed mesh networks. Such networks will be expected to carry all Internet services, respecting the QoS objectives that we will describe. On the other hand, ad hoc wireless mesh networks, such as community networks formed from WiFi access points in homes, cannot be expected to provide any consistent QoS to the applications they carry. We might expect that these would be used primarily for nonreal-time store-and-forward applications, such as e-mail and web browsing.

3.2 Types of Trafﬁc and QoS Requirements Based on the discussion of the various scenarios in Section 3.1, we can infer that applications generate one of the types of trafﬁc in the following list. Some example applications that generate each type of trafﬁc are also listed. • Elastic trafﬁc; e.g., WWW browsing, FTP ﬁle transfers, and electronic

mail

3.2 Types of Trafﬁc and QoS Requirements

57

• Real-time stream trafﬁc; e.g., packet voice telephony • Store-and-forward stream trafﬁc; e.g., streaming movies or music over

the Internet. In the remainder of this section we discuss the characteristics of these trafﬁc types, and also their quality of service (QoS) requirements.

Elastic Trafﬁc Consider a data ﬁle, residing on the disk of the server E (shown in Figure 3.1) that needs to be transferred to the disk of a portable computer attached to the Internet (e.g., the laptop D, which connects via a WLAN), or to the memory of the cell phone C. Although the human (or some machine application) that wishes to achieve this ﬁle transfer would like to have the transfer completed in, say, a second or two, the source of data itself does not demand any speciﬁc transfer rate. If the data transfer does not lose data, no matter how fast or slow it is (but as long as the rate is positive), the ﬁle will sooner or later get transferred to the destination device. We say that, from the point of view of the network, this source of trafﬁc is elastic. Many store-and-forward services (with the exception of media streaming services) are elastic; e.g., ﬁle transfer, WWW download, electronic mail (e-mail). In this list, the ﬁrst two are distinguished by the fact that they are nondeferable (i.e., the network should initiate the transfer immediately), whereas e-mail is deferable. We observe that elastic trafﬁc does not have an intrinsic temporal behavior, and can be transported at arbitrary transfer rates. Thus the following are the QoS requirements of elastic trafﬁc. • Transfer delay and delay variability can be tolerated. An elastic transfer

can be performed over a wide range of transfer rates, and the rate can even vary over the duration of the transfer. • The application cannot tolerate data loss. This does not mean, however,

that the network cannot lose any data. Packets can be lost in the network (owing to uncorrectable transmission errors or buffer overﬂows) provided that the lost packets are recovered by an automatic retransmission procedure. Thus effectively the application would see a lossless transport service. Since elastic sources do not require delay guarantees, the delay involved in recovering lost packets can be tolerated. In practice, of course, users will not tolerate arbitrarily poor throughput, high throughput variability, and large delays. Hence a network carrying elastic trafﬁc will need to manage its resource-sharing mechanisms in a way such that some minimum level of throughput is provided. Further, some sort of fairness must also be ensured between the ongoing elastic transfers. Elastic trafﬁc can also be carried over circuit multiplexed networks (e.g., the PSTN or GSM cellular networks), or over networks that allocate a ﬁxed rate to

58

3 Application Models and Performance Issues

the elastic connection (e.g., a second generation CDMA cellular network). In this case, shaping of the trafﬁc so as to match the allocated rate should be carried out by the source. Obvious examples would be Internet access over a dial-up line in the PSTN, or a ﬁxed rate connection over a cellular access network being used for Internet access.

Real-Time Stream Trafﬁc Consider digitized speech emanating from an end-device involved in interactive telephony. This could be a periodic stream of bytes or packets, or, if silence suppression is employed then it could be an on-off stream of bytes or packets. Obviously, this source of trafﬁc has an intrinsic temporal behavior, and this pattern needs to be preserved for faithful reproduction of the speech at the receiver. The network will introduce delay: ﬁxed propagation delay, and, in packet networks, queuing delay that can vary from packet to packet (see Figure 3.2). Playout delay introduced at the receiver (to mitigate the effect of random packet delay variation) will be larger the more variable the packet delay. Hence, the network cannot serve such a source at arbitrary rates, as it could in the case of elastic trafﬁc. In fact, depending on the adaptability of such a real-time stream source, the network may need to reserve bandwidth and buffers in order to provide an adequate transport service to the source. Applications such as real time interactive speech or video telephony are examples of real-time stream sources.

first packet in a burst

peak rate 5 R bits/sec h

source output

X3 X2

network output

X1 t1

t2

t3 h

b input to playout device playout delay

Figure 3.2 A sequence of packets from a voice talk-spurt being transported across a packet network, and then being played out at the receiver after a playout delay. Each source packet contains h seconds of voice, the packet delays are X1 , X2 , . . ., the packets arrive at the receiver at times t1 , t2 ,. . ., and the playout delay is b. Notice that immediate playout at time t1 would have resulted in the talk-spurt being broken.

3.2 Types of Trafﬁc and QoS Requirements

59

The following are the typical QoS requirements of real-time stream sources. • Delay (average and variation) needs to be controlled. Real-time interactive

trafﬁc such as that from packet telephony would require tight control of source-to-sink delay; for example, for wide area packet telephony the delay may need to be controlled to less than 200 ms with a probability more than 0.99. Packets that do not conform to the delay bound are considered to be lost. • There is tolerance to data loss. Note that, from the point of view of the

receiver, packets can be lost for two reasons: (1) buffer overﬂows, or unrecovered link losses in the network, or (2) late arrivals at the receiver. Owing to the high levels of redundancy in speech and images, a certain amount of data loss is imperceptible. As an example, for packet voice in which each packet carries 20 ms of speech, and the receiver does lost-packet interpolation, 5 to 10% of the packets can be lost without signiﬁcant degradation of the speech quality [81], [54]. Because of the delay constraints, the acceptable data loss target cannot be achieved by ﬁrst losing and then recovering the lost packets; in other words, stream trafﬁc expects the intrinsic loss rate from the packet transport service to be bounded.

Store-and-Forward Stream Trafﬁc We can distinguish what we have just described as real-time stream trafﬁc from the kind of trafﬁc that is generated by applications such as streaming audio and video. Such applications basically involve a one-way transfer of an audio or video ﬁle stored on the disk of a media server. Consider a video stored in a server being played over a network. For example, the computer D or the handheld device C in Figure 3.1 may be used to watch a movie stored in the Server E. In order for the received video to be useful, the playout device should be continuously “fed” with video frames so that it is able to reproduce a smooth video output. This can be achieved by providing a guaranteed rate to the transfer, as would be done, for example, in a CDMA cellular system in the context of the device C. Alternatively, owing to the fact that the transfer is one way, a more economical way is to treat the transfer as elastic, and buffer the video frames as they are received. This would be the approach taken when the video is transferred over the random access WLAN to the computer D. Playout is initiated only after a sufﬁcient number of video frames has been buffered so that a smooth video playout can be achieved in spite of a variable transfer rate across the contention-based WLAN. Note that the same description holds for streaming audio. Thus, the problem of transporting streaming audio or video becomes just another case of transferring elastic trafﬁc, with appropriate receiver adaptation. Note, however, that the elasticity here is constrained since the average rate at

60

3 Application Models and Performance Issues

which the network transports the video bit stream must match the rate at which the video has been coded. Simple interactivity, such as the ability to rewind, can also be supported by the receiver storing frames that have already been played out. This, of course, puts a burden on the amount of storage that the playout device needs to have. An alternative is to trade off sophistication at the receiver with the possibility of interactivity across the network; that is, the press of the rewind button stops the video playout, frames stored in the playout device are used to create a rewind effect, and meanwhile additional past frames are fetched from the server. But this would need some delay and throughput guarantees from the network, requiring a service model somewhere in between the elastic and the real-time stream model that we have described earlier. We conclude that the QoS requirements of a store-and-forward stream transfer would be the following: • The average transfer rate provided in the network should match (in fact,

should be greater than) the average rate at which the stored media has been encoded. • The transfer rate variability should not be too large.

Thus store-and-forward stream trafﬁc is like stream trafﬁc since it has an intrinsic average rate at which it must be transported, but it does not have strict delay bounds, and hence the network can provide it a time varying transfer rate. In fact, TCP can be used to transport store-and-forward streaming media, provided the average TCP throughput does not drop below the average coded rate of the media. The added beneﬁt of TCP is that it recovers lost packets.

Closed and Open Loop Trafﬁc: It is appropriate to refer to real-time stream trafﬁc as open loop as it has an intrinsic temporal behavior. Typically, the rate of ﬂow on a connection is determined by the application, and these sources are not controlled by the network. In some systems, a limited amount of controllability is possible, by the sink alerting the source of poor playout quality, to which the source can respond by using a lower bit rate coder. On the other hand, closed loop controls invariably are used when transporting elastic trafﬁc, and, hence, such trafﬁc can be called closed loop. By means of implicit feedback (packet loss) or explicit feedback (control bits in packet headers) the source of the trafﬁc is made to continually adjust its rate of emitting data.

3.3

Real-Time Stream Sessions: Delay Guarantees

In this section we will discuss trafﬁc modeling and QoS issues for real-time stream sessions in the context of voice telephony. 3.3.1 CBR Speech Consider a voice call between a pair of endpoints in Figure 3.1. For example, the PSTN phone A and the cellular phone C, or between B and D, or between C

3.3

Real-Time Stream Sessions: Delay Guarantees

61

and B. In each end device, electrical signals from a microphone are digitized and coded by a speech codec. A typical approach is to sample the analog signal from the microphone at 8000 samples per second, to quantize the resulting continuous amplitude samples into 256 predetermined levels, and then encode each of these levels into 8 bits (one byte). The output of such a speech coder is called PCM (Pulse Code Modulation) coded speech (ITU’s G.711 standard). The PCM encoder, thus, yields a CBR source that produces 1 byte every 125 μseconds. A PCM source can be compressed to yield CBR sources at various rates. For example, ITU’s G.729 vocoder takes PCM as the input and produces 10 bytes of coded speech every 10 ms, thus yielding a coded bit rate of 1 KBps (kilobytes per second). However, this speech coder has a coding delay of 15 ms and a decoding delay of 7.5 ms. An important measure of the performance of network telephony is the Mouth-to-Ear (MtoE) delay—the delay between a sound being produced at the source device and this being heard at the other end. Thus, if the G.729 speech coder is employed, then there is a minimum MtoE delay of 22.5 ms. In order to carry a CBR voice source, it is necessary for the network to use a service rate greater than or equal to the voice bit rate. Further, if the source is allocated exactly the constant bit rate then there will not be any queuing. Hence, for CBR sources it is sufﬁcient to allocate the CBR rate. Consider a voice call between the PSTN phone A and the cell phone C. If the gateway GW converts PCM speech arriving over the PSTN to CBR speech at rate R, then the cellular network can just allocate resources so that the voice call is provided a service rate of R. This is typically what is done in an FDM-TDMA cellular system (such as GSM), or in a CDMA cellular system. We will discuss resource allocation issues in these two types of systems in Chapters 4 and 5, respectively. 3.3.2 VBR Speech In speech generated by interactive telephony, there are low energy periods that correspond to silences while the speaker listens, or to gaps between words, sentences, and utterances. The coder output corresponding to these inactive periods can be discarded or encoded at a lower rate. This yields a variable bit rate (VBR) coded speech. The VBR speech can be handled as a variable rate byte stream, or can be packetized for transport over a packet network. One approach is to take a certain number of bytes from the source (e.g., 160 bytes or 20 ms of speech from a PCM source) and generate a packet from these. It may happen that a talkspurt ﬁnishes before 160 bytes have been collected; in such a case a short packet is generated. The packetizer must wait to accumulate a packet; thus bytes that arrive early in the packet have to wait for those that arrive later, until the packet is formed. This results in a packetization delay, which can, obviously, be reduced by using shorter packets. Packets cannot be very short either, as there could be a signiﬁcant amount of header overhead in each packet (e.g., in the Internet there would be at least 12 bytes for RTP (Real-time Transport Protocol), 8 bytes for UDP (User Datagram Protocol), and 20 bytes for IP). If the coder output

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during speech inactivity periods is discarded, then the output of a packetizer will comprise bursts of packets (during which packets are generated at a constant rate) and periods during which no packets are generated. Note that although the inactive periods do not have speech information in them, the duration of the gaps is indeed information that needs to be conveyed to the receiver. One of the difﬁculties in the transport of packetized VBR speech is in the retention of such timing information. Since packets are transmitted only during active periods, the inactive periods can only be approximately replicated at the receiver. It has been found that the resulting errors are not noticeable if the inactive periods are long. Thus the voice activity detection function (after the speech encoder) does not discard bytes from short inactive periods. Consider again the voice conversation between the PSTN phone A and the cell phone C in Figure 3.1, with VBR speech being used in the cellular network; that is, the voice arrives over the PSTN as a CBR ﬂow, but is encoded into a VBR ﬂow at the gateway, GW. Suppose the VBR speech source is allocated the service rate C in the cellular network (see Figure 3.3). Let us denote by R the peak rate of the VBR source, and by r¯ the average rate. Thus, for example, if the on-off VBR source has an average on duration of 400 ms and average off duration of 600 ms, then with R = 64 Kbps, we will have r¯ = 400400 + 600 × R = 25.6 Kbps. It is clear that it is a waste of bandwidth to make C > R, and that it is necessary that C ≥ r¯. Now suppose we take C < R. Notice that when the voice source is emitting data at the rate R, then the link buffer builds up at the rate (R − C ) Kbps. Any byte that arrives when the buffer level is, say, B bits will be delayed by CB ms. A priori, we do not know for how long this rate mismatch will last (the average rate r¯ = 25.6 Kbps could have been obtained with a 4 sec on time and a 6 sec off time too!). Hence, if we want to bound the delay of the voice bytes in the link buffer, in the absence of any other information about the source, our only recourse is to use C = R. This approach of peak rate allocation could be one way in which the cellular network manages its resources, and is typically the approach adopted in FDM-TDMA and CDMA cellular systems. In CDMA systems, even though the peak rate is allocated to a call, the on-off nature of VBR speech is exploited because during the voice silence periods a call does not cause interference (see

R r r T then the packet is discarded.

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65

the receiver can stretch out the delay of each arriving packet to T ; thus packets that are delayed more than T are lost and may be interpolated. We are still left with the problem of determining a value for T . There are two alternatives. The packet network may have the ability to provide a delay guarantee at call setup (e.g., Pr(X > T) < where X is the delay of packets in the network; see Figure 3.5). In such a case, the endpoints specify the trafﬁc characteristics of the source they want to be carried, and the values of T and . The network evaluates whether the call can be accepted, and, if the call is accepted, the network sets up the appropriate mechanisms along the path of the call so that the delay objective is met. Now T is known to the receiver at call setup time, and the procedure shown in Figure 3.5 can be performed. If the network cannot provide delay guarantees, then the receiver would need to estimate T as the call progresses. Time stamps carried by the voice packets in their headers would be used to obtain a statistical estimate of T . This estimate can then be used to set the playout delays of arriving packets. Since there is no guarantee, the value of T could be larger than desired and could vary over time as congestion in the network varies. 3.3.4 QoS Objectives We gather that the MtoE delay for a packet voice call is the sum of several terms as shown in the following equation: MtoE Delay = coding delay + packetization delay + network propagation delay + network transmission and queuing delay + receiver playout delay + decoding delay

In this expression, the network propagation delay is just the signal propagation delay over the various media interconnecting the routers and switches in the path of the call. A rule of thumb is to compute this ﬁxed delay as 5 ms per 1000 Km of cabled transmission. Thus, for example, between points in the continental United States and India separated by a distance of about 20,000 Km, the one-way WAN propagation delay would be about 100 ms. For a geostationary satellite link, the one-way propagation delay is computed as the time taken for radio waves to travel from the transmitter up to the satellite and then down to the receiving ground station, or about 250 ms. In addition to the MtoE delay, some voice packets can be lost, either owing to buffer overﬂows in routers, or because they arrive after their scheduled playout time at the receiver. Thus, an example of the QoS expected by a voice call could be: Pr MtoE Delay > 200 ms < 0.02 and

Pr Packet Loss < 0.05

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3 Application Models and Performance Issues

We recall that packet loss includes loss due to late arriving packets, as well as loss due to buffer overﬂows and errors in the network. All such missing packets will need to be interpolated by the voice decoder, which leads to degradation in voice quality. Notice that the MtoE delay has some ﬁxed parts (coding and decoding delay, packetization delay, and propagation delay), and some variable parts (transmission and queueing delay, and the playout delay). It is these delays that are governed by the characteristics of the trafﬁc emitted by the source, and the way the trafﬁc is handled in the network (i.e., the other trafﬁc with which it is multiplexed, and the resource allocation decisions made by the network). Letting X denote the variable (random) delay, and then subtracting the ﬁxed delays from the MtoE delay target, the network performance requirement can be reduced to Pr(X > T) <

where, for example, = 0.02. Now consider a call between the devices B and D. The computer B is attached to the Internet by a high-speed enterprise or campus LAN, whereas D is attached to the Internet by a contention-based WLAN. One approach to the analysis of such a situation is to break down the end-to-end QoS objective into subnetwork-wise objectives. Thus, one could break up the end-toend delay bound T as T = T1 + T2 , and the probability of violating the delay bound can be split up as = 1 + 2 . We can call T1 and T2 the delay budgets in the respective subnetworks. Let the stationary random delay over the Internet segment of the call be denoted by X1 and that over the WLAN be denoted by X2 . Suppose we ensure that, for each i = 1, 2, Pr(Xi > Ti ) < i

It will then follow that Pr(X > T) = Pr(X1 + X2 > T1 + T2 ) ≤ Pr({X1 > T1 } ∪ {X2 > T2 }) < 1 + 2 =

where the ﬁrst inequality follows from the simple observation that if X1 + X2 > T1 + T2 , then it cannot be that X1 ≤ T1 and X2 ≤ T2 ; the last inequality is just the union bound. The resource allocation in the WLAN can then be performed so as to ensure that the voice packet delay exceeds T2 with a probability less than 2 . The same approach can be used if end-device D is attached to the Internet via a WMAN (see Figure 3.1).

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67

3.3.5 Network Service Models In the previous section we showed how to derive, from the end-to-end delay QoS problem for a connection, a QoS objective for the access network. If the access network accepts calls based on peak rate allocation, then the only issue is to design the network for a desired call blocking probability. For this purpose, the Erlang blocking model (see Appendix D) can be used. If the access network assigns a ﬁxed rate less than the peak rate of a VBR call, then the model depicted in Figure 3.3 can be used. The analysis of such models has been discussed at length in [89, Chapter 5]. In some access networks, however, the service rate applied to a connection may not be constant. For example, in OFDMA systems, the number of bytes to be served from a queue can vary from frame to frame depending on the fading in the various carriers, the competing trafﬁc, and the power constraint. Thus, in this case we have a dynamically controlled server; see Chapter 6. Detailed analysis of such systems to obtain buffer occupancy distributions or delay distributions is difﬁcult. Wireless LANs, based on the IEEE 802.11 standard, are contentionbased systems. Hence the service applied to a queue is time varying because the number of contending nodes varies over time, as some queues empty out while others receive new trafﬁc. Some progress has been made on developing analytical models for the performance analysis of wireless LANs; some of these approaches will be discussed in Chapter 7.

3.4

Elastic Transfers: Feedback Control

Elastic trafﬁc is generated by applications whose basic objective is to move chunks of data between the disks of two computers connected to the network. Elastic ﬂows can be speeded up or slowed down depending on the number of ﬂows contending for the capacity of the network. Figure 3.6 shows that, at the most basic level, an elastic session simply involves the transfer of some ﬁles from one host attached to the packet network to another host. For example, the two hosts could be e-mail relays; each ﬁle transfer would then correspond to an e-mail being forwarded toward its destination mail server. Alternatively, the source host could be an FTP archive, and at the destination host, a user is downloading several ﬁles during an FTP session. A similar example would be that of the source being a web server, and the destination being a client with a web browser, using the HTTP (Hyper-Text Transfer Protocol) protocol to browse the ﬁles at the server. In an internet, for example, when a user requests a web page (using an HTTP GET request), ﬁrst, a base ﬁle is downloaded, which in turn may trigger the transfer of several embedded objects, such as images. When there are embedded objects, the exact mechanism for downloading the objects depends on the version of HTTP in use. In HTTP 1.0, for the transfer of the base ﬁle, and for each embedded object ﬁle, a separate TCP connection would be set up between the client and the server.

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source of files Packet Network

destination of files

Figure 3.6 An elastic session simply involves the transfer of one or more ﬁles from one host to another.

In HTTP 1.1, in order to reduce connection set up overheads, a TCP connection once set up would be reused for several ﬁle transfers between the same client and server. In all these cases the basic problem is to transfer each ﬁle in its entirety from the source machine to the destination machine. This is the primary objective. There is no intrinsic rate at which the ﬁles must be transferred. In fact the transfer rate can vary as a ﬁle is transferred. Although the user downloading the ﬁles would want to receive the ﬁles quickly, this requirement is not really a part of the service deﬁnition of an elastic session. Further, we note that there is no intrinsic packet size that the ﬁles need to be segmented into during their transfer. The transfer protocol can just view each ﬁle as a byte stream, and transfer varying amounts of it in each packet. We will deal primarily with point-to-point elastic sessions, and this is what we will mean when we use the term “elastic session.” Thus an elastic session involves a connection between two endpoints. The network determines a route between the endpoints in each direction; if the session lasts long enough, there is a possibility that the route, in either direction, changes during the session. During a session, data transfers may take place in either direction, with possible gaps between the successive transfers. For example, if a user at a computer logs on to an FTP server, then FTP’s get or put commands can be used to download or upload ﬁles. The user may need to do some other activities in between the ﬁle transfers (e.g., read what is downloaded and make notes); in user models, these gaps often are referred to as think times. Similarly, a user browsing a web server would download a web page, and spend some time looking at it, before downloading another web page from the same site. If the user shifts to browsing another web server, then we view this as another session starting, typically over a different pair of network routes.

3.4

Elastic Transfers: Feedback Control

HOST 1

69

bandwidth of this pipe is shared between a varying number of elastic flows

WEB

HOST 2

SERVERS

feedback control necessary to slow down or speed up the traffic sources HOST n

users downloading data from the web servers

Figure 3.7

Several users dynamically share a link to download ﬁles from web servers.

3.4.1 Dynamic Control of Bandwidth Sharing Figure 3.7 shows a very simple “network” comprising a single link over which several users, on their respective hosts, are downloading ﬁles from some web servers. Let us take the link capacity to be C bps, and assume that the local networks attaching the users and the web servers to this link are inﬁnitely fast. We use this simple scenario to illustrate and discuss some basic issues that arise when several elastic sessions share the network bandwidth. In fact, the situation depicted in Figure 3.7 is similar to what happens when several mobile users download ﬁles (text, music, video, etc.) from a server attached to a cellular operator’s own high speed local area network. The important difference is that in cellular systems the system bandwidth is not managed as one “fat pipe.” Suppose, to begin with, a single user initiates a download from a web server over the link. It is reasonable to expect that an ideal data transfer protocol will (and should) provide this ﬁle transfer with a throughput of C bps. This much bandwidth is available, and if all of it is provided to the transfer, the session will be out of the system as early as possible. Now suppose another user starts a session, while the ﬁrst ﬁle transfer is still progressing. When the corresponding web server starts transferring data toward the user, the total input rate into the link (from the web server’s direction) will exceed C bps. If the ﬁrst ﬁle transfer is proceeding at C bps, then the link’s service rate will be exceeded no matter how slowly the second server sends its data. This will lead to link congestion. The network device that interconnects the server’s LAN to the backbone link will have buffers “behind” this link. These buffers can absorb excess data that accumulate during this overload, provided the situation does not sustain for long. In addition,

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this situation is clearly unfair, with one user getting the full link rate, and the other user getting practically no throughput; this situation should not be allowed to persist. Both of these issues (congestion and unfairness) require that there be some kind of feedback (explicit or implicit) to the data sources so that the rate of the ﬁrst transfer is reduced, and that of the second transfer is increased, so that ultimately each transfer obtains a rate of C2 bps. Now suppose the ﬁrst ﬁle transfer completes; then, if the second transfer continues to proceed at C2 bps, the link’s bandwidth is wasted, and the second session is unnecessarily prolonged. Hence, when the ﬁrst session departs, the source of the second session should increase its transfer rate so that a throughput of C bps is obtained. In summary, from this discussion we conclude that an explicit or implicit feedback control mechanism needs to be in place so that as the number of sessions varies, the transfer rate provided to each session varies accordingly. By an explicit feedback we mean that control packets ﬂow between the trafﬁc sources, sinks, and the network, and these packets carry information (e.g., an explicit rate or a rate reduction signal) that is used by the sources to adapt their sending rates. On the other hand, implicit feedback can be provided by packet loss or increase in network delay; that is, a source can reduce its rate on sensing that one of the packets it sent may not have reached the destination. For example, in much of the current Internet, TCP (Transmission Control Protocol) uses an implicit feedbackbased congestion control mechanism. Explicit rate control was proposed for the ABR (available bit rate) service in ATM (asynchronous transfer mode) networks. The idea was to associate with each ABR session a ﬂow of control cells (called Resource Management (RM) cells) generated by the source. As the RM cells of a session ﬂow through the network, the ATM switches in their path could set an explicit rate value in these cells. The sink would return the RM cells back to the source, and the source could use the explicit rate in the returned RM cells to adjust its cell emission rate. On the other hand, TCP uses a windowbased transmission protocol. For a ﬁxed round trip delay, the TCP throughput is proportional to the average window size. Thus, the window can be adapted to vary the TCP transfer rate. Window adaptation works by a TCP source detecting lost packets, taking these as indications of rate mismatch and network congestion, and voluntarily reducing its transmission rate by reducing the transmission window. 3.4.2 Control Mechanisms: MAC and TCP As mentioned in the previous section, and as is clear from our discussions in Chapter 2, in wideband cellular wireless networks, the entire system bandwidth is not used as one fat pipe. Instead, radio resource allocation is done on an MS by MS basis, depending on the channel conditions to the mobiles. Thus, even at the medium access layer it is possible to implement control strategies that achieve some sort of a rate allocation objective over the MSs. For example, the objective could be

3.4

Elastic Transfers: Feedback Control

71

equal rate allocation (this is an example of a more general fairness objective called max-min fairness); such an approach might be very inefﬁcient as the MS with the weakest link will determine the rate that all MSs obtain. Another objective could be to allocate rates so as to maximize the total rate over the MSs; such an approach might be very unfair as MSs with poor connectivity might obtain no throughput. We will examine MAC level rate allocation for elastic trafﬁc in CDMA cellular systems in Chapter 5. In CSMA/CA based wireless LANs, the medium access control protocol results in some default bandwidth sharing. If only downlink ﬁle transfers are considered then it is found that the IEEE 802.11b standard results in equal rate sharing, irrespective of the physical rate at which MSs are connected. We will provide an analytical model for understanding this in Chapter 7. Bandwidth sharing in the wide area Internet is controlled by the Transmission Control Protocol (TCP) that resides in all end-systems attached to the Internet, including Internet-enabled cellular phones. In OSI terminology, TCP is a Transport Layer protocol. Thus TCP sits between the applications and IP, the Internet’s packet routing and forwarding protocol. TCP is connection oriented, which means that a connection has to be established between the endpoints before data transfer can start, and this connection is taken down when the data transfer completes. TCP enhances the unreliable, nonsequential packet transport service provided by IP to a reliable and sequential packet transport service. It uses a window-based packet loss recovery mechanism to achieve this function. In addition, the windowbased mechanism is employed for two other major functions that TCP provides: (1) sender-receiver ﬂow control, which prevents a fast source of packets (at the application level) from overwhelming a slow sink, and (2) adaptive bandwidth sharing in the network. The TCP transmitter maintains a congestion window that increases if packets are acknowledged in sequence. On the other hand if the desired acknowledgment (ACK) fails to show up then the transmitter takes this as an indication of congestion, and reduces the transmission window. The transmission window can also be controlled by the receiver, by a window advertisement in the ACK packets. By the latter means, the receiver can exercise ﬂow control over the transmitter. For a connection, the number of packets in the network is roughly related to the TCP window, and the average window divided by the mean round trip packet delay is an estimate of the TCP throughput. The adaptive window-based congestion control mechanism of TCP has evolved over several versions. In the earliest version, any packet loss resulted in a transmitter time-out, and the reduction of the congestion window to one. A TCP receiver continues to accept packets even if previous packets are missing. For all such out-of-order packets, the transmitter returns an ACK packet “asking” for the ﬁrst missing packet. These ACKs are called duplicate ACKs. Thus, duplicate ACKs are an indication of out-of-order packets at the receiver, and multiple duplicate ACKs are indicative of packet loss. In a later version, called TCP Tahoe, loss recovery was initiated at the transmitter by the receipt of three duplicate ACKs; this was called fast retransmit. However, the transmitter dropped the congestion

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window to one. Thus, although loss recovery started earlier than the older version (say, OldTahoe), the window had again to be built up from one. TCP Tahoe was followed by TCP Reno in which, on the receipt of three duplicate ACKs, the congestion window was cut by half, and loss recovery was initiated; this was called fast retransmit and fast recovery. A more aggressive loss recovery than TCP Reno was implemented in TCP NewReno. This version was followed by another improvement in TCP SACK, in which the TCP transmitter uses the ACK packets to send the pattern of missing TCP packets back to the TCP transmitter. From the foregoing, we see that the rate allocation achieved by elastic transfers to or from devices that attach to the Internet via wireless access networks will be governed by the interaction between the rate allocation strategies in the wireless MAC, the behavior of the wireless channel, and TCP’s window-based end-to-end control mechanisms. 3.4.3 TCP Performance over Wireless Links In [89, Chapter 7] we have discussed the TCP protocol at length and have studied models for evaluating the performance of TCP-controlled ﬁle transfers in several situations. We studied a model that can be used to obtain the performance of TCP controlled ﬁle transfers with random packet loss. We saw that the performance of TCP can be signiﬁcantly affected by packet loss. In these discussions, the concern was with congestion-related loss; that is, either a packet was lost owing to buffer overﬂow, or a packet was deliberately dropped at a router queue owing to imminent congestion. We were not concerned with the possibility of packet loss in the physical bit carriers. In a sense, we were assuming a wired physical infrastructure. Wired links can be properly established so that they have small BERs. On the other hand, mobile wireless links can have high packet loss rates, and are subject to random variations in their quality. Also, in CSMA/CA based wireless LANs, it is unrealistic to model the service provided to a ﬂow as being at a constant bit rate. It is therefore of interest to study the performance of TCP transfers over wireless access networks, particularly in light of the growing importance of mobile wireless access to the Internet. It is well known that the bandwidth delay product (BDP) (normalized to the packet length) along a path in a network is deﬁned as 2Cδ L

where C is the bottleneck link rate along the path of the TCP connection, 2δ is the round-trip propagation delay (RTPD), and L is the packet length. If the TCP window grows to the BDP and stays at that value, then the bottleneck link can be kept fully occupied. This yields the highest possible TCP throughput on that path. With this in mind, let us consider elastic transfers from the server E to an MS associated with the cellular network in Figure 3.1. Let us suppose that the cellular system assigns a ﬁxed service rate to each transfer, where the rate

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Elastic Transfers: Feedback Control

73

depends on the condition of the channel to the MS and on the other MSs that are being served by the system. Typical service rates would be in 100s of Kbps, and hence the cellular network would be the bottleneck in the path of the TCP connection. Even if the server E is 20,000 Km away (halfway around the earth), thus yielding an RTPD of about 200 ms, we have a BDP of about four packets for a bottleneck rate of 250 Kbps and L = 1500 bytes. The TCP maximum window size implemented in various operating systems is 20 packets or more. Hence, assuming that the wide area Internet has negligible packet loss, the TCP window is well above the minimum to keep the bottleneck link busy.1 This observation permits us to make the simpliﬁcation that we may ignore the wide area packet network, and study only the interaction between TCP and the behavior of the wireless link. It is as if the server E was attached to the local area network of the cellular operator.

Independent Packet Losses With this discussion in mind, Figure 3.8 shows a simple scenario in which a mobile host is doing a TCP controlled ﬁle transfer from a ﬁle server on a wired LAN. The LAN wireless router network would be located at the base station. The propagation delay between the base station and the mobile host is negligible. The BER on the wireless link is such that packets are lost with probability p. The packets are lost independently; correlated losses owing to channel fading are not modeled here. Only ACKs are sent from the mobile host to the LAN, and since these are small (40 bytes), their loss probability is ignored; recall that TCP uses cumulative ACKs, which further limits the effect of ACK loss. The link layer random packet losses

Server

TCP IP

Mobile Station

Base Station LAN – Wireless Router

TCP IP

IP link

link modem

modem

local area network

Figure 3.8 A mobile station transferring data over a wireless link from a server on the LAN attached to the base station.

1 A simple way to quantify the effect of random losses in the wide area Internet is to use the square root formula 1.5 p where p is the packet loss probability. This formula gives an approximation to the mean

window size of a TCP connection over a wide area network, if the loss probability is p and the connection stays in congestion avoidance. Typical values of p in a well-engineered ISP network would be 0.001 or 0.005. The resulting average window is well above the 4 required to keep the bottleneck link busy.

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3 Application Models and Performance Issues

protocol is unable to recover all the wireless packet losses; hence any residual packet losses have to be recovered by TCP. As the TCP transmitter on the ﬁle server grows its window, the wireless link buffer builds up. The buffer can hold as many packets as needed; that is, there is no buffer loss. Eventually, a loss occurs in the wireless link, and one of the loss recovery mechanisms is invoked. The throughput of a large ﬁle transfer can be analyzed via a stochastic model of the TCP protocol with random packet losses. A sample of results obtained from this analysis is shown in Figure 3.9. The parameters and the results are normalized. We plot the ﬁle transfer throughput versus the packet loss probability. The throughput is normalized to the bit rate of the wireless link. One set of parameters that would correspond to the results is LAN speed; 10 Mbps wireless link bit rate: 2 Mbps; TCP packet length: 1500 bytes (hence the packet transmission time is 6 ms); time-out granularity: 420 ms; minimum time-out: 600 ms; and Wmax = 24 packets, where Wmax is the maximum TCP window. The performance of four versions of TCP is compared: OldTahoe (which is the name we give to the version of TCP that predates Tahoe and always requires time-outs to recover losses), Tahoe, Reno, and NewReno. We observe that even with a packet loss probability of 0.001, the throughput with OldTahoe is less than the full link rate,

Packet throughput, normalized to speed of lossy link

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1023

NewReno: K = 3 Tahoe: K = 3 Reno: K = 3 OldTahoe 1022 1021 Packet Error Probability

100

Figure 3.9 File transfer throughput (normalized to the link’s bit rate) vs. packet loss probability for various versions of TCP; OldTahoe refers to a version that recovers losses only by timeout. K is the duplicate ACK threshold for fast-retransmit in the TCP loss recovery protocol. Adapted from Kumar [85].

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Elastic Transfers: Feedback Control

75

and drops to just over 50 percent of the link rate for a packet loss probability of 0.01. The other three TCP versions implement fast-retransmit and they yield 100 percent throughput at p = 0.001, and better than 95 percent throughput up to p = 0.01. Beyond 1 percent packet loss, the performance of these versions too begins to drop, and is not much better than OldTahoe for a 10 percent packet loss rate. Reno is slightly better than Tahoe up to p = 0.02, but becomes worse for large loss rates since multiple losses cause it to waste more time than Tahoe. The more aggressive fast-recovery of NewReno results in this version yielding almost 90 percent throughput up to p = 0.03. We can make a broad observation that random packet loss probabilities larger than 1 percent can signiﬁcantly affect the performance of TCP, with these parameters. Note that the coarse time-out and minimum time-out values are large in this example. Smaller values for these parameters will yield better performance, as losses will then result in less wastage of link capacity. Thus, we see that there is a maximum packet loss probability below which the TCP throughput is just the bit rate of the wireless link. If the packet loss probability is ensured to be less than this maximum then the effect of TCP can be ignored, and we can just take the bit rate provided by the MAC mechanisms as the transfer rate obtained by the elastic application. If a packet loss probability of p is desired, and the packet length is L bits, then the BER on the wireless link should satisfy the requirement p = 1 − (1 − )L . An upper bound on p thus yields an upper bound on the BER. Hence we see that the performance of the application we wish to carry on the wireless link puts a requirement on the performance of the link. We saw in Chapter 2 that the BER on a wireless link is a function of the SNR. Hence, for a given modulation and coding scheme, the desired BER places a requirement on the minimum SNR at which the link can operate. We also note that a desired packet loss probability can be obtained by using an ARQ protocol over a physical link with a higher BER than calculated from the formula above. Since the propagation delay on cellular links is negligible (i.e., the number of bits “in ﬂight” is much smaller than the packet length), a stop-and-wait ARQ sufﬁces. The overall effect of using an ARQ protocol is that we have a lower bit rate link (due to ARQ overheads and retransmissions) with the desired packet loss rate.

Correlated Packet Losses We now turn to the performance of TCP controlled ﬁle transfers over a fading channel. In Section 2.1.4 we discussed models for channel fading. We pointed out that the fading is correlated in time. Thus for a given average BER there would be periods when the BER is greater than the average, and periods during which the BER is less than the average. A similar statement can be made for the packet error rate if ﬁxed length packets are being used, as is typically the case with large ﬁle transfers over TCP. A simple approach is to model the channel as being in one of two states: a Good state (during which a packet transmission is

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3 Application Models and Performance Issues (12g)

g

Good

Bad

b

(12b)

Figure 3.10 Transition structure of the two-state Markov model for a fading channel.

successful), and a Bad state (during which a packet transmission is unsuccessful). A further simpliﬁcation is to model the state process as a two-state Markov process, on the state space {Good , Bad } (see Figure 3.10). The durations in each state are taken to be multiples of the packet transmission time. The transition probabilities of the Markov chain are obtained by specifying the amount of fading that leads to a bad transmission (at other times a good transmission is assumed). The marginal distribution of the fading process and results about correlations in the fading process can be used to obtain the transition probabilities. For a packet length L, channel bit rate C , and Doppler frequency fd , the parameter fd CL is a measure of the fade durations relative to the packet transmission time; thus L fd C = 0.01 means that channel coherence time is roughly 100 packet transmission times. Using this Markov model for the channel state, the analysis of the throughput of a long ﬁle transfer under TCP can be performed by developing a certain stochastic model. In performing this analysis, in addition to the state of the TCP window adaptation process, the state of the channel will also need to be maintained. Figure 3.11 shows some typical numerical results with Rayleigh fading. The normalized throughputs with TCP Tahoe and Reno are plotted versus the average packet error probability, with and without fading. For the results with fading, the parameter fd CL = 0.01. The other parameters are the same as in Figure 3.9, except that the local area network is taken to be inﬁnitely fast. Notice that the performance without fading is similar to that depicted in Figure 3.9. With fading, the performance is signiﬁcantly different. For the same probability of error, we ﬁnd that the performance of TCP Tahoe increases substantially, whereas that of TCP Reno drops for p < 3 × 10−2 , and improves for large packet loss probabilities. This can be understood as follows. With independent losses, the repeated reductions in the window lead to a small effective window; hence when a loss occurs there are not enough packets in circulation to generate the number of duplicate ACKs required for a fast retransmit. Thus with uncorrelated losses, time-outs are more frequent. When packet errors are clustered (as in the case of fading), the durations between packet loss events are larger. Hence with correlated packet losses, the TCP transmitter is able to grow its window to larger values than

3.4

Elastic Transfers: Feedback Control

77

packet throughput, normalized to link bit rate

1.0

0.8

0.6

0.4

0.2

0.0 1023

Tahoe analysis (i.i.d.) Tahoe analysis (fading) Reno analysis (i.i.d.) Reno analysis (fading) Tahoe simulation (i.i.d.) Tahoe simulation (fading) Reno simulation (i.i.d.) Reno simulation (fading)

1021 1022 packet error probability

100

Figure 3.11 File transfer throughput (normalized to the link’s bit rate) vs. packet loss probability for TCP Tahoe and Reno; with independent losses (denoted as i.i.d.), and with Rayleigh fading with fd L = 0.01. Adapted from Zorzi et al. [144]. C

in the independent packet loss case (for the same average packet error probability). When a loss does occur, it is more likely that there are enough successful packets sent subsequently in the window to trigger a fast retransmit. Even if a time-out does occur, it is long enough to last out the fade, so that when transmission resumes, the channel is likely to be in the Good state. For small values of p, the performance of Reno is worse since Reno requires additional duplicate ACKs for recovering each lost packet. With correlated losses, multiple losses are more likely and this results in Reno wasting more time than Tahoe. Reno attempts to perform a fast-retransmit for each lost packet, spends time in this process waiting for duplicate ACKs, and then times out anyway. For large values of p, the two protocols have similar behavior since with the high loss rate the window grows to small values, the number of duplicate ACKs are insufﬁcient to trigger a fast-retransmit, and hence it is very likely that both protocols recover with a time-out. Although this discussion illustrates the effect of correlated errors on TCP controlled ﬁle transfer performance, it is important to make a comparison by ﬁxing the average SNR. The same two-state Markov model can be used. The SNR that corresponds to a Bad state is ﬁrst ﬁxed. Then for each SNR and Doppler frequency, the two-state Markov model can be parametrized. Sample results

3 Application Models and Performance Issues Packet throughput, normalized to speed of wireless link

78 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3

TCP Tahoe; no fading TCP Tahoe; fade = 1 pkts TCP Tahoe; fade = 2 pkts limit, speed –>0

0.2 0.1 0 10

15

20 25 30 Mean Signal to Noise Ratio (dB)

35

40

Figure 3.12 File transfer throughput (normalized to the link’s bit rate), withTCP-Tahoe, vs. SNR in dB, with no fading (AWGN only) and with Rayleigh fading. The legend fade = n pkts means that the mean Bad state duration is n packets, where a Bad state occurs if the SNR < 10 dB. Adapted from Kumar and Holtzman [87].

for TCP Tahoe are shown in Figure 3.12. In order to compare with the results presented earlier, no channel coding or link level retransmissions are taken into account. The normalized throughput is plotted against the average SNR in dB. We observe that without fading, an SNR of about 12 dB sufﬁces to obtain a TCP throughput of over 90 percent of the link rate. This is because the packet error probability itself is very small without fading. With fading, however, much larger Rayleigh faded SNRs are required; between 25 to 30 dB for a throughput of 90 percent of link rate. We notice that slower fading and hence more correlated errors improves the TCP throughput, but the throughput even with this improvement is much worse than that without fading. We also show the case of speed → 0. This corresponds to the fade level being constant during the entire TCP transfer; either the channel is good throughout the transfer, or is bad throughout. This is a bound on the achievable throughput with fading. As the faded SNR decreases, the probability of the Good state reduces and hence the bound rapidly decreases for decreasing average faded SNR.

3.5

Notes on the Literature

Extensive analytical treatments of QoS issues and models is provided in [89], [133], and [119]. References [81] and [54] provide a discussion of issues in transporting

3.5

Notes on the Literature

79

voice over packet networks. The material on analysis of TCP controlled ﬁle transfer throughput over lossy wireless links has been taken from the papers by Kumar ([85], which assumes i.i.d. packet loss) and by Zorzi et al. ([144], which accounts for correlated packet losses). An approach for two-state Markov modeling of a fading channel is provided by Zorzi et al. in [146]. Additional references on TCP throughput analysis with correlated packet losses in the wireless setting are Kumar and Holtzman [87], Zorzi and Rao [145], and Anjum and Tassiulas [3].

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CHAPTER 4 Cellular FDM-TDMA

T

he FDM-TDM technique for allocating spectrum and time resources to calls is the most classical one, and systems based on this technique carry a substantial majority of the cellular telephony trafﬁc around the world. Many of the basic techniques of cellular telephony emerged from the design of such systems; for example, techniques such as frequency reuse management, cell sectorization, power control, and handover management.

Overview One of the main ideas developed in this chapter is that of spatial reuse by partitioning the available FDM carriers into reuse groups, and then allocating these reuse groups to cells in such a way that cochannel interference is within acceptable limits. It is shown that the cochannel interference constraint places a constraint on the D R ratio, the ratio between the shortest distance between cochannel cells and the cell coverage radius. This analysis is based on signal-to-interference ratio (SIR) modeling, where we use a power attenuation model that includes path loss and shadowing. Assuming that the cells form a hexagonal tessellation of the plane, the D R ratio is related to the number of cochannel reuse groups into which the cells must be partitioned. It is shown how the partitioning of the channels and other system parameters affect the spectrum efﬁciency. Once channel allocation constraints are understood, various channel allocation strategies are considered and a call blocking analysis is developed. Finally, we consider intercell handovers. We show how signal strength measurements from neighboring BSs are used to determine that a call needs to be handed over between cells. An approximate handover blocking analysis is also shown. The chapter ends with an overview of call handling in GSM, the most widely deployed FDM-TDMA cellular system.

4.1

Principles of FDM-TDMA Cellular Systems

Suppose that a system bandwidth of Wsystem is to be used for providing FDMTDMA based telephony services in a certain coverage area, say a town in some country. The operator would have to pay a substantial fee to the authority managing the spectrum in that country, and hence it is in the operator’s interest to maximize the revenue from operating the service while keeping costs down. We begin by providing an understanding of the issues involved in the efﬁcient design

82

4

Cellular FDM-TDMA

of an FDM-TDMA cellular telephony system. We will refer to a mobile handset as an MS (mobile station) and to the ﬁxed stations that are connected to the wire-line network as base stations (BSs). There are several commercial implementations of FDM-TDMA technology for mobile telecommunication, the one with the most widespread deployment being the GSM system (see Section 4.7). In any of these implementations, the system bandwidth, Wsystem , is partitioned into several nonoverlapping FDM channels, each of which is then digitally modulated, and then time slotted to yield FDM-TDM channels. A guard band is left vacant at either end of an operator’s spectrum allocation to prevent power from one operator’s system from interfering with another system. For example, in the GSM system the FDM channel spacing is 200 KHz. After digital modulation, each such channel is time slotted to provide eight TDM channels, each of which can carry one direction of a digitized voice call. Each voice call has two directions, and hence for each call we need two links to be established, one from the MS to the BS, and one from the BS to the MS. The way these two links are established is called the duplexing technique. In FDMTDMA systems the common duplexing mechanism employed is frequency division duplexing (FDD); that is, two separate FDM carriers are used to carry the two directions of a call. Thus, the operator actually gets two nonoverlapping segments of the radio spectrum, each of bandwidth Wsystem (see Figure 4.1). One of these is the uplink band and the other is the downlink band. Each band is partitioned into an equal number of nonoverlapping FDM channels. The FDM channels in the uplink and downlink bands are then paired, as shown in Figure 4.1, for two FDM channels j and k. Thus, when we say that FDM channel j is assigned to an MS, for the purpose of making a telephone call, then actually two TDM slots, one in each of the two FDM channels with center frequencies fju and fjd , are assigned to the call, for the entire duration of the call. For example, if the operator leases a Wsystem of 5 MHz, then (allowing for a total guard band equal to the bandwidth of one FDM channel), the system can be used to carry 24 × 8 = 192 simultaneous calls. Let us denote the FDM channel

fj u

fku

fj d

fkd

frequency uplink band

downlink band

Frequency division duplexing in FDM systems: The FDM channels fju and fjd are paired, as are the channels fku and fkd . Figure 4.1

4.1

Principles of FDM-TDMA Cellular Systems

83

bandwidth by W , the number of FDM channels in the system bandwidth by C (24 in the preceding example), and the number of trafﬁc carrying TDM slots per FDM channel by s (8 in the preceding example). Let N = C × s be the number of calls that can be carried simultaneously. We will further denote the set of FDM carriers by {f1 , f2 , . . . , fC }. In view of our earlier discussion on duplexing, each element of the set of carriers {f1 , f2 , . . . , fC } actually denotes an uplink-downlink pair. The system can now be set up as shown in the left panel of Figure 4.2. We observe that a large power will need to be used in order to serve the MSs at the periphery of the coverage area, in order to ensure that the power received at either end of an MS-BS link is such that the SNR exceeds the minimum required for the desired bit error rate for the voice coder being used. Such MSs will quickly drain their batteries, and will also cause interference to systems in neighboring coverage areas. Also, the maximum number of users that can be simultaneously served in this simple system is N . Let B(ρ, n) denote the blocking in an Erlang blocking system with a load of ρ Erlangs and n servers (see Appendix D, Section D.5.1). If the arrival rate of new calls into the system is λ per second, and the mean holding time of a call is h seconds, then for this system ρ = λh, and the blocking probability becomes B(ρ, N). Table 4.1 shows a sample Erlang table from which the number of servers that would be required to obtain a speciﬁed blocking probability for a given trafﬁc intensity can be obtained.

BS

BS

BS

BS

BS

BS

Figure 4.2 Spatial reuse: In the left panel, all the MSs associate with one BS, and the entire band is used to serve all the calls in the desired coverage area. In the right panel the band is reused at multiple BSs.

84 n↓

15 16 17 18 19 20 21 22 23 24 25

n↓

Loss Probability 0.0001

0.001

0.002

0.005

0.01

0.02

0.05

0.1

0.2

4.7812 5.3390 5.9110 6.4959 7.0927 7.7005 8.3186 8.9462 9.5826 10.227 10.880

6.0772 6.7215 7.3781 8.0459 8.7239 9.4115 10.108 10.812 11.524 12.243 12.969

6.5822 7.2582 7.9457 8.6437 9.3515 10.068 10.793 11.525 12.265 13.011 13.763

7.3755 8.0995 8.8340 9.5780 10.331 11.092 11.860 12.635 13.416 14.204 14.997

8.1080 8.8750 9.6516 10.437 11.230 12.031 12.838 13.651 14.470 15.295 16.125

9.0096 9.8284 10.656 11.491 12.333 13.182 14.036 14.896 15.761 16.631 17.505

10.633 11.544 12.461 13.385 14.315 15.249 16.189 17.132 18.080 19.031 19.985

12.484 13.500 14.522 15.548 16.579 17.613 18.651 19.692 20.737 21.784 22.833

15.608 16.807 18.010 19.216 20.424 21.635 22.848 24.064 25.281 26.499 27.720

Table 4.1 Part of the Erlang table showing the trafﬁc intensity that can be offered to a link of capacity n (rows) circuits for speciﬁed blocking probabilities (columns).

15 16 17 18 19 20 21 22 23 24 25

4.1

Principles of FDM-TDMA Cellular Systems

85

Telephony systems typically are designed for blocking probabilities such as 0.01 or 0.02. If a single phone is expected to provide a load of 0.1 Erlangs (e.g., two calls per hour, with an average holding time of 3 min), then for a coverage area with 2500 MSs the Erlang load is 250, and we see that this proposed system (with N = 192) will yield an unacceptable probability of call blocking B(250, 192).1 The right panel of Figure 4.2 shows the idea of spatial reuse. The MS-BS communication is done using smaller powers. The same channel can be used in several places in the system, provided that the cochannel interference is such that the signal-to-interference plus noise ratio (SINR) of any MS-BS link is maintained above a threshold. Suppose that, by exploiting spatial reuse, each channel could be reused, say, ﬁve times in the same coverage area; then we would have effectively multiplied the number of calls that can be simultaneously handled by a factor of 5. On further thought, however, a problem becomes evident with the simple arrangement in Figure 4.2. If an MS is not close to any BS, then in order to serve it from some BS, on any channel, a large transmission power will need to be used. This will cause cochannel interference if the same channel is reused elsewhere in the coverage area, thus rendering spatial reuse less effective. In order to address this problem, the cellular FDM-TDMA approach is to tessellate the coverage area into cells, each of which has a BS. The set of FDM carriers is partitioned into subsets called reuse groups. These channel groups are then assigned to the cells in such a way that cells with the same group of channels (called cochannel cells) are not close together. How close cochannel cells can be depends on the SINR required for reliable communication. Each cell then acts as an Erlang blocking system for the calls that require a channel in it. We observe that, if the SINR required is large, then the cochannel cells will need to be kept far apart. This will require more channel reuse groups, and hence fewer channels per reuse group. This brings us to another issue. If the reuse groups have only a small number of carriers in them, then trunking efﬁciency is lost. By this we mean the following. For a ﬁxed probability of blocking, , let ρ (n) denote the Erlangs that can be carried when the number of servers is n, B(ρ (n), n) =

(4.1)

To understand this, notice that each column of Table 4.1 corresponds to a value of , and each element in that column gives the value of ρ (n) for the corresponding n in the ﬁrst column. Now deﬁne g (n) = ρn(n) , that is, g (n) is the Erlangs per server that can be offered, when the number of servers is n and the target blocking 1 Note that the maximum load that 192 servers can carry is just 192, so a load of 250 Erlangs will give a

blocking probability close to 1. As a rule of thumb, for B(ρ, n) to be as small as 0.01 or 0.02, it is necessary that ρ < n. This follows because, with a blocking probability of , the rate of calls that are carried is (1−)λ, and time average number of busy servers is (1 − )λh = (1 − )ρ (by Little’s Theorem; see Appendix D). For low blocking, however, the average number of busy servers will be substantially less than n (see Table 4.1).

86

4

Cellular FDM-TDMA

ρ probability is . For n = 1, note that B(ρ, n) = 1+ρ , which yields g (1) = 1− . Thus, a very small load (per server) can be handled if there is just one server. However, g (n) increases monotonically to 1 as n increases. Thus, we see that it is beneﬁcial to not partition the set of carriers into small groups, as this reduces trunking efﬁciency. We conclude that a larger target SINR results in smaller reuse groups, which results in lower trunking efﬁciency. It follows that there is a trade-off between keeping the SINR above a required threshold and keeping the trunking efﬁciency high. The number of reuse groups we use in a system will be denoted by Nreuse . We observe from this discussion that the SINR requirements, the spatial reuse, and the system efﬁciency are intimately linked, and some analysis is required to evaluate the trade-offs.

4.2

SIR Analysis: Keeping Cochannel Cells Apart

In Figure 4.3, we depict uplink and downlink cochannel interference in a conﬁguration in which a channel is reused at the ﬁve BSs shown. The circular boundaries indicate the coverage of each BS; these are assumed to be of radius R. The distance between the centers of each of the outer BSs and the one in the middle is D. It is intuitively clear that a large D R ratio will be required if the cochannel interference has to be kept very small. In this section we will study how to carry out the cochannel interference analysis with a target SINR, in order to determine the required D R ratio.

BS1 MS1 D 2R R BS4

MS0 BS2

D 2R MS4

R

D 2R MS2 MS0

D 2R

D MS3 BS3

Figure 4.3 Depiction of downlink (left panel) and uplink (right panel) cochannel interference. In each case the MS0 is taken to be in the most unfavorable position, such that the desired signal will suffer the maximum attenuation and the interference will suffer the least attenuation. D is the shortest distance between BSs of cochannel cells, and R is the coverage radius of each BS.

4.2

SIR Analysis: Keeping Cochannel Cells Apart

87

For the purpose of studying the cochannel interference, the MS whose signal performance is being analyzed is considered to be in the most unfavorable position (at the periphery of its BS’s coverage area), and the interferers are also assumed to be in the most unfavorable position, as close as possible to the receiver of the desired transmission. For example, in the right panel of the ﬁgure, the uplink is being considered, and therefore the interferers are cochannel MSs in the other cells. Notice that these are being assumed to be at the peripheries of their own cells, and placed so that they are as close as possible to BS0. Such worst case conﬁgurations are used to determine how far away cochannel cells need to be kept. At this point we recall the material in Section 2.1.4. Let H denote the channel power gain (actually, an attenuation) between the transmitter of the desired signal and its receiver, and let Hi denote the power gain from the i-th cochannel interferer to the receiver of the desired signal. Let there be NI interferers. We will view all transmitter powers as being the Rayleigh faded mean values at the reference distance d0 (see Section 2.1.4). With this convention, let P be the power used by the transmitter of the desired signal, and Pi , 1 ≤ i ≤ NI be the powers of the interfering transmitters. It then follows that the SINR at the receiver is given by Ψ=

PH I N0 W + N i=1 Pi Hi

We note that these FDM-TDMA systems use narrowband modulation, and hence the SINR requirements are in the range of 10 dB to 20 dB. Also the noise power, N0 W , is very small; approximately −120 dBmW (i.e., 10−12 mW). It is, therefore, assumed that the noise power is much less than the received signal power, and we neglect this term in the denominator. Let d be the distance between the transmitter and its receiver, and di the distance between the i-th interfering transmitter and the receiver. We can then write (see Section 2.1.4) H= Hi =

d d0 di d0

−η −η

10−

(ξ+ξ0 ) 10

10−

(ξi +ξ0 ) 10

where ξ, ξ0 , and ξi , 1 ≤ i ≤ NI , normally are distributed and correspond, respectively, to the shadowing at the transmitter of the desired signal, at the receiver, and at the NI interferers. Here the ξ, ξ0 , ξi , 1 ≤ i ≤ NI , are i.i.d. normally 2 distributed, 0 mean, and with variances σ2 ; thus, the lognormal shadowing standard deviation on any path is σ dB. This form of the lognormal shadowing is used since shadow fading comprises a part due to the shadowing near the receiver (which is common to all paths to the receiver), and a part near the transmitters

88

4

Cellular FDM-TDMA

(which is assumed to be independent for widely separated transmitters). Hence, we can write the SINR (or, simply, the SIR) Ψ as

−η

(ξ+ξ0 )

10− 10 Ψ= −η (ξi +ξ0 ) NI di P 10− 10 i i=1 d P

d d0

0

Notice that the terms ξ0 all cancel. We can then rewrite the SIR expression in the following form: Ψ=

1 − 10 −10 log10 P+10η log10

10 NI

10

i=1

d +ξ d0

1 − 10 (−10 log10 Pi +10η log10

di +ξi ) d0

(4.2)

Notice that in the numerator we have a log-normally distributed random variable 1 of the form 10− 10 Q , where Q has units of dB, and is normally distributed with d E Q = m := −10 log10 P + 10η log10 dB d0 VAR(Q) = υ :=

σ2 2

and in the denominator we have a sum of NI log-normally distributed random 1 variables of the form 10− 10 Qi , where Qi also has units of dB, and is normally distributed with di E Qi = −10 log10 Pi + 10η log10 dB d0 VAR(Qi ) =

σ2 (= υ) 2

Thus, we have 1

10− 10 Q

Ψ= NI

i=1

1

10− 10 Qi

where Q, Qi , 1 ≤ i ≤ NI , are independent normally distributed random variables that essentially model the shadowing. Since shadow fading is correlated over distances of several 10s of meters, we assume that the shadowing is “sampled” once during a call, and independent samples of the shadow fading random variables are taken from call to call. We also assume that a call, during its holding time, samples the entire distribution of the Rayleigh fading; it does not get “stuck” in

4.2

SIR Analysis: Keeping Cochannel Cells Apart

89

a deep fade. This corresponds to our use of mean transmit powers averaged over Rayleigh fading. We are now interested in ensuring that the SIR exceeds a threshold γ with a high probability, say, 1 − . Note that, given a target BER, γ will be obtained from an analysis of the underlying modulation scheme under Rayleigh distributed ﬂat fading and additive white Gaussian noise; see Section 2.1.4. Then would be the outage probability. What does this probability mean? Consider instances of calls from or to MSs at the boundaries of the coverage areas of the cells in which they are handled. Then the fraction of such calls that will experience a BER higher than the target will be less than . This is because for such calls, we have assumed that the cochannel interferers are placed at the most unfavorable locations; refer back to Figure 4.3. One approach to carrying the analysis forward is to approximate the I 1 − 10 Qi distribution of N by a log-normal distribution. Such an approximation i=1 10 is known to work well. Also, the resulting SIR threshold analysis becomes simple, since the ratio of two independent log-normal random variables is obviously log-normal. So, let us write NI

1

1

10− 10 Qi ≈ 10− 10 QI

i=1

The approximation is performed by matching the mean and second moment of the random variables on the two sides. This is called the Fenton-Wilkinson method, the details of which can be found in standard texts on wireless digital communication (see, for example, [123]). Let us suppose that this procedure yields QI as normally distributed with mean mI and variance vI . Then, we have 1

Ψ = 10− 10 (Q−QI )

or, equivalently, (Ψ)dB = QI − Q

where QI − Q is in dB, and is normally distributed with mean mI − m and variance v + vI . Thus, mI − m is the mean SIR in dB, and the SIR variance is v + vI . We need that QI − Q > (γ)dB with a large probability, where, as usual, (γ)dB = 10 log10 γ. From Figure 4.3, we notice that in the downlink worst case situation (left panel of the ﬁgure), the BSs are all taken as transmitting to MSs at the peripheries of their coverage areas. Similarly, in the uplink worst case situation (right panel) the BSs are all receiving from MSs at the peripheries. We thus assume that the transmission powers P, Pi , 1 ≤ i ≤ NI , are all equal. It follows from (4.2) that the transmitter powers cancel in the SIR expression. The mean value, m, corresponds to the path loss from the transmitter of the desired signal to its receiver. Further, mI depends on the path losses from the interferers to the receiver, and is the effective path loss of the interfering transmitters, in dB. We can thus require that mI − m > 0,

90

4

Cellular FDM-TDMA

that is, the interferers have a larger effective power attenuation to the receiver than does the desired transmitter. Figure 4.4 depicts a typical situation, when the target probability of exceeding γ is being met. The normal density of (Ψ)dB has been plotted in this ﬁgure. It follows that for an outage probability (i.e., to ensure that Pr (QI − Q) < γ < ,) there is a τ , such that we need to ensure that √ mI − m > γ + τ v + v I Such a τ will be obtained from a table of the tail of the normal distribution. For example, τ0.01072 = 2.3, as can be seen from Table 4.2. This inequality provides the insight that shadowing variance of the signal and of the interference add up, and a larger value of this total variance requires the cochannel reuse to be designed so that there is a larger difference between the mean interference attenuation and the signal attenuation, mI − m. As an application of this analysis, consider the uplink conﬁguration shown in the right panel of Figure 4.3. Since, in this case, the interferers are as close as possible to the receiver of the desired signal, this situation is worse than the downlink situation shown in the same ﬁgure. It can be shown that the FentonWilkinson analysis yields m = 10η log10 R

2 ea v NI3 1 mI = 10η log10 (D − R) − ln 2 2a ea v + (NI − 1) 2 1 ea v − 1 vI = 2 ln +1 NI a

␥

mI 2 m

c in dB

Figure 4.4 A sketch of the normal probability density of the SIR, Ψ, in dB. target SIR, and mI − m is the mean SIR.

γ is the

4.2

SIR Analysis: Keeping Cochannel Cells Apart

91

z

Q(z)

z

Q(z)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

0.50000 0.46017 0.42074 0.38209 0.34458 0.30854 0.27425 0.24196 0.21186 0.18406 0.15866 0.13567 0.11507 0.09680 0.08076 0.06681 0.05480 0.04457 0.03593 0.02872

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

0.02275 0.01786 0.01390 0.01072 0.00820 0.00621 0.00466 0.00347 0.00256 0.00187 0.00135 0.00097 0.00069 0.00048 0.00034 0.00023 0.00016 0.00011 0.00007 0.00005

Table 4.2 The probability under the right “tail” of the ∞ −x2 normal (Gaussian) distribution: Q(z) = √1 e 2 dx. 2π z

where a :=

ln 10 10

≈ 0.23026. We thus get the requirement

10η log10

2 a v−1 D 1 e − 1 > γ + τ v + 2 ln +1 R NI a 2 ea v NI3 1 + ln 2 2a ea v + (NI − 1)

(4.3)

We notice that this inequality places a constraint on the ratio between D, the distance between cochannel cells, and R, the cell radius. Let us look at two numerical examples, both with η = 4. Consider ﬁrst v = 0, that is, there is no log-normal shadowing, just path loss. The D R constraint reduces to 40 log10

D − 1 > γ + 10 log10 NI R

(4.4)

92

4

Cellular FDM-TDMA

Exercise 4.1 Obtain the expression in (4.4) directly from the SIR expression in (4.2). For NI = 6, we ﬁnd 40 log10

D − 1 > γ + 7.78 R

On the other hand, if the shadow fading standard deviation is 8 dB, then 2 v = σ2 = 32. For η = 4, NI = 6, and outage probability = 0.01, (4.3) yields 40 log10

D − 1 > γ + 25.25 R

We conclude that shadow fading has a signiﬁcant effect on the

D R

ratio.

Discussion a. We observe, from the preceding analysis, that as long as the transmitter powers are all assumed to be equal, the required D R ratio does not depend on the actual values of the transmit powers. b. We notice also that only the ratio D R is determined, but not the absolute values of D and R. This provides the important insight that the cell sizes can be shrunk while retaining the D R ratio. This increases the system call handling capacity, since the channel groups in each cell are used to serve a smaller cell area. This approach to increasing the system capacity has its limitation, however. As the cell size decreases the MSs tend to more frequently require intercell handovers. Since the blocking of handover requests leads to the dropping of ongoing calls, an increase in handover rates needs more channels to be reserved for handover handling (see Section 4.6), thus leading to a possible reduction in call handling capacity. In addition, the higher frequency of handovers results in more signaling load, thus possibly overloading the call handling processors in the system.

4.3

Channel Reuse Analysis: Hexagonal Cell Layout

It is evidently not practical to use the cell conﬁguration shown in Figure 4.3, as this leaves large portions of the service area uncovered. Hence, as explained in Section 4.1, the service area is tessellated with cells. The set of FDM carriers is partitioned into disjoint sets, which are assigned to subsets of the cells, in such a way that cochannel cells respect the D R ratio. In order to analyze such a system, it is convenient to take the cells to be hexagons of equal size. This permits an easy visualization of the tessellation in the two-dimensional plane. It is then useful to recall the simple geometrical concepts shown in Figure 4.5.

4.3

Channel Reuse Analysis: Hexagonal Cell Layout

93

C5 3 R Area 5 3 C 2 5 3 3 R 2 2 2

C R

Figure 4.5 Hexagon geometry: relations between the cell width, C, the cell radius, R, and the area of the hexagon.

4.3.1 Cochannel Cell Groups In Figure 4.6 we show a tessellation of the plane with hexagonal cells. The FDM channels are partitioned into reuse groups. One of these groups is assigned to Cell 0, shown at the center of the cell layout. This will be our reference cell in the following discussion. We next wish to determine which other cells in the layout should use the same group of carriers. For this purpose it is convenient to work with a coordinate system with axes inclined at 60◦ to each other, as shown by the axes u and v in the ﬁgure. For simplicity in our description, we draw a third “axis,” w. The axes pass through the center of the reference cell. There is an angular separation of 60◦ between u and v, and the same between v and w. Notice that moving a cell width, C , along any of the axes takes us to the center of a neighboring cell. Thus, let C be unit length along the axes. Now, starting from the origin of this system (the center of Cell 0), move i units along the u axis and then j units along the v axis. Observe that for i = 3 and j = 2 this brings us to the cell labeled 1. Let the Euclidean distance between the centers of Cell 0 and Cell 1 be D(i, j), the distance between two cells whose relative positions depend on (i, j) in the manner just explained. The following calculation follows from simple geometry. √ 2 3 1 2 D(i, j) = j + i+j 2 2 = =

j2

3 1 + i2 + ij + j2 4 4

i2 + ij + j2

In a similar manner, ﬁxing i = 3 and j = 2, we can identify Cells 2, 3, 4, 5, and 6, as shown in Figure 4.6. These will be the cochannel cells in relation to Cell 0. Observe that if we carry out the same procedure, for i = 3 and j = 2, for Cell 1 in the ﬁgure, then we will obtain Cells 2, 0, and 6, and three other cells, above and to the right of Cell 1; these cells are not shown in the ﬁgure. Thus, this process yields

94

4

Cellular FDM-TDMA

v 2

u

1

w

j D 3

i 0

4

6

5

Cells 0,1,2,3,4,5, and 6 are cochannel cells to locate a cochannel cell w.r.t. to a cell: move i cells along an axis, then turn clockwise and move j cells

Figure 4.6 Tessellation of the coverage area by hexagonal cells. Cells 0, 1, 2, 3, 4, 5, 6 are cochannel cells for (i, j) = (3, 2).

a subset of the hexagons that tessellate the plane. Applying the procedure starting from any element of this subset yields the same set of cells. Notice, however, that if we start from one of the cells adjacent to Cell 0, and use the same (i, j), then we will get a subset of cells that is disjoint from the previous one. In fact, looking at Figure 4.7, for each cell in the large dashed hexagon with Cell 0 at its center, we will obtain a different subset of hexagons, and all these subsets (19 for (i, j) = (3, 2)) are mutually disjoint and together they form a partition of the tessellation. We can call each of these subsets of cells a cochannel group. 4.3.2 Calculating Nreuse The number of cochannel groups (which we had denoted earlier by Nreuse ) thus depends on the choice of (i, j). For example, with (i, j) = (1, 0) there is only one cochannel group. What is the general relation between (i, j) and the number of cochannel groups? This can √ be worked out as follows. In Figure 4.7 the area of the large dashed hexagon is 23 D2 (where we recall that the unit of length is the cell width, C ). There are as many cochannel groups as the number of cells in this large hexagon. Exactly one cell from any of the cochannel groups lies in this large hexagon. Hence, given a large coverage area A, the number of cells in a cochannel group is √ A 2 (see Figure 4.5). The total number of cells is √A . Thus, the ( 3/2)D

3/2

4.3

Channel Reuse Analysis: Hexagonal Cell Layout

2

95

1

A2

D

A1

3

0

6

A3 A0 A6 4

5

A4 A5 The large dashed hexagons are centred at cochannel cells

Figure 4.7 Tessellation of the plane by hexagonal cells. There is one cell from each cochannel group in each of the large dashed hexagonal areas. Notice that Cells A0, A1, A2, A3, A4, A5, and A6 belong to a different cochannel group than Cells 0, 1, 2, 3, 4, 5, and 6.

number of cochannel groups is D2 . Obviously, the number of cochannel groups has to be the same as the number of groups into which we partition the set of FDM carriers (i.e., Nreuse (as deﬁned earlier)). Thus Nreuse = D2 = i2 + ij + j2

We also observe that for a given (i, j), following this procedure for ﬁxing the cochannel cells, we have also ﬁxed the D R ratio to the value (see Figure 4.5): D(i, j) = R =

3(i2 + ij + j2 ) 3Nreuse

Table 4.3 shows the values of Nreuse and D(i,j) R that are obtained for various values of (i, j). Recall that the SIR analysis in Section 4.2 yielded a constraint on the D R ratio. For example, (4.3) provided a constraint on the D ratio when a reference R

96

4

Table 4.3

Cellular FDM-TDMA

i

j

N reuse

D(i,j) R

1 1 2 2 3 2 3 4 3 4 4

0 1 0 1 0 2 1 0 2 1 2

1 3 4 7 9 12 13 16 19 21 28

1.73 3.00 3.46 4.58 5.20 6.00 6.24 6.93 7.55 7.94 9.17

Nreuse, and

D(i,j) ratio, for relative locations of cochannel cells, (i, j ). R

cell is surrounded by NI cochannel cells whose centers are all at distance D from the center of the reference cell. We say that the analysis considered only the ﬁrst tier interferers, the nearest cochannel cells. In general, in large cellular networks, there will be second and third tier interferers and even more beyond. Of course, the interference from second, third, and higher tiers is substantially lower than that from the ﬁrst tier, especially when the path loss exponent, η, is large. In any case, the SIR analysis yields a D R ratio, and then Table 4.3 can be used to determine the value of Nreuse that provides this D R ratio, and the corresponding (i, j) to be used to lay out the cells. For example, if the required D R ratio is 7, then we must take Nreuse = 19, which is achieved with (i, j) = (3, 2). 4.3.3 D R Ratio: Simple Analysis, Cell Sectorization It is instructive to compare various cases of ﬁrst tier cochannel interference while ignoring shadowing, and accounting only for path loss. Such analysis provides quick insight into the comparisons between the various cases. Thus, accounting only for path loss, and taking the transmitter powers, in the worst-case transmitterreceiver conﬁgurations, to be equal (see the discussion in Section 4.2), the following is the general expression for the SIR R−η Ψ = N −η I i=1 Di 1 = −η NI Di i=1

R

where R is the cell radius, NI is the number of ﬁrst tier interferers, and Di , 1 ≤ i ≤ NI , is the distance of the i-th interferer from the receiver in the reference cell.

4.3

Channel Reuse Analysis: Hexagonal Cell Layout

97

Figure 4.8 shows the forward channel (downlink) worst-case situation, where approximations have been made for the various distances between the interferers and the receiver. We see that Ψ= 2

D R

−η

−1

+

1 −η D R

+

D R

−η

+1

Suppose we take Nreuse = 9; then Table 4.3 provides D R = 5.20, from which we ﬁnd that, for η = 4, Ψ = 95.09 = 19.78 dB. In Figure 4.9 we show the reverse channel (uplink) worst case situation. Here we see that Ψ= 6

1 D R

−η

−1

D D1R D2R R D2R D1R D

Figure 4.8 Seven cochannel cells, showing the worst-case conﬁguration of ﬁrst tier downlink interferers. The arrows point at the receiver, and show the direction of the desired signal and the interference. The distances are approximations, and, in general, the cochannel cells may have a different relative orientation from the one shown.

98

4

Cellular FDM-TDMA

D2R D2R D2R

R D2R D2R D2R

Figure 4.9 Seven cochannel cells, showing the worst case conﬁguration of ﬁrst tier uplink interferers. The arrows point at the receiver, and show the direction of the desired signal and the interference. The value D – R is an approximation, and, in general, the cochannel cells may have a different relative orientation from the one shown.

For Nreuse = 9 (i.e., D R = 5.20) we have, for η = 4, Ψ = 51.86 = 17.14 dB. Thus we see that the uplink provides a more than 2.5 dB worse performance for the same D R ratio. In each of these cases, there are six ﬁrst tier interferers at a receiver. If directional antennas are used in the BSs then the number of interferers can be reduced. This is achieved by a technique called sectorization, which is depicted in Figure 4.10. Each cell is shown divided into three 120◦ sectors, each with a directional antenna whose angular coverage is designed to coincide with the angular spread of the sector. Thus, an MS in a given sector of a cell is served by the antenna in that sector. Further, the channels are reused only in the corresponding sectors of the cell reuse groups. This is shown in Figure 4.10, where the carrier f is shown being reused in a particular sector of all the cells in a reuse group of seven cells. To see the advantage of doing sectorization, consider the downlink worstcase situation depicted in the left panel of Figure 4.10. Notice that the MS in Cell 0 sees only two ﬁrst tier interferers, the two BSs in Cells 2 and 3. The corresponding

4.4

Spectrum Efﬁciency

99

1

1 f

f 6

2 f

f

f D1R

6

2

0

f 0 R

Rf 3

5

3 f

D

f

f

D D

5 f

4

4 f

f

Figure 4.10 Seven cochannel cells, with 120◦ sectorization, showing the worst case conﬁguration of ﬁrst tier downlink interferers (left) and uplink interferers (right). The distances shown are approximations.

sectors in Cells 4, 5, and 6 could be using the same channel, but their antenna main lobe is not “visible” to the MS in Cell 0. Making suitable approximations for the distances, the following is the forward channel SIR, ignoring the shadowing. Ψ=

D R

1 −η

+1

+

−η D R

Taking η = 4, Nreuse = 9 (i.e., D R = 5.20) we ﬁnd that Ψ = 489.13 = 26.89 dB, a 7.1 dB improvement over the case without sectorization. The right side of Figure 4.10 shows the worst-case uplink interferers with sectorization. The SIR is given by Ψ= 2

1 −η D R

With Nreuse = 9 (i.e., D R = 5.20), taking η = 4, we ﬁnd that Ψ = 365.58 = 25.63 dB, which is 8.5 dB better than without sectorization. Note, however, that sectorization implies smaller sets of channels in each sector, thus reducing the trunking efﬁciency.

4.4

Spectrum Efﬁciency

Let us recall the following system parameters. The RF spectrum allocated to the system is Wsystem , the number of FDM carriers the system bandwidth is partitioned into is C , the number of TDM slots per carrier is s. Assuming equal cell sizes, let

100

4

Cellular FDM-TDMA

a denote the area of each cell. Further, let K denote the number of sectors in each cell (e.g., K = 3 for 120◦ sectors). Recall the deﬁnition of Nreuse , and the function g (n) (see Section 4.1). Let us consider the simplest approach of partitioning the C carriers into Nreuse subsets. Each subset of carriers is then further partitioned into K sets, each

of which is allocated to the same sector in all the cells in a reuse group of cells. Each slot in each carrier in a sector can carry one call. For the present we assume that a call that is initiated in a sector stays in the same sector for its entire duration; sC that is, there are no handovers in the system. Thus, the Nreuse K slots in a sector, along with the call arrivals to or from MSs in that sector, constitute an Erlang blocking model. It follows that, for a target blocking probability of , the number of Erlangs that can be offered to a cell is given by g

sC

×

Nreuse K

sC Nreuse K

×K

where the ﬁrst term is the number of Erlangs per slot in a sector. Let A denote the coverage area of the system. Then the Erlang capacity of the system, denoted by Λ, is given by A Λ = × g a

sC Nreuse K

×

sC Nreuse

Let us deﬁne the spectrum efﬁciency of the system as the Erlang capacity per unit area per Hz of system bandwidth, and denote this by ν. We then have ν := =

Λ A Wsystem 1 sC × g × a Wsystem

Wsystem Wsystem Nreuse K sC

1 Nreuse

(4.5)

sC Notice that Wsystem is ﬁxed for a given system bandwidth, and depends on the FDM-TDM modulation scheme being used. For example, in the GSM system, the FDM carrier spacing is 200 Khz, and there are eight TDM slots per FDM carrier. Thus, given Wsystem , and allowing for some guard bandwidth on either side, the value of C is determined. In Section 4.3 we saw how Nreuse and K can sC Wsystem 1 be chosen to achieve the required SIR. Notice that the term g Wsystem Nreuse K Nreuse decreases with increasing Nreuse or K, but we need to set Nreuse and K so that the SIR requirements are met while keeping this trunking efﬁciency term as large as sC Wsystem 1 possible. Note that g Wsystem Nreuse K Nreuse also increases with Wsystem , but having leased a certain amount of the spectrum, the operator will want to work within this leased amount. Finally, the Erlang capacity of the system can be increased by

4.5

Channel Allocation and Multicell Erlang Models

101

decreasing a; that is, by reducing the cell size. Of course, there are limits to this scaling. As the cell size decreases, there are three issues: a. As the cell size decreases, we need to consider handovers, and the handover rate increases with decreasing cell size. This will impact the Erlang capacity, as resources need to be reserved for handovers. b. The signaling load increases due to the increased handover rate. This means that higher capacity call handling systems need to be installed. c. Reducing cell size requires the installation of more base stations, which can be expensive. Finally, the design of any given system will have to balance these trade-offs.

4.5

Channel Allocation and Multicell Erlang Models

From the expression for spectrum efﬁciency in (4.5), we can infer that, apart from reducing the cell size, another way to increase the efﬁciency is to improve the channel utilization. The earlier analysis assumed a uniform ﬁxed assignment of the FDM carriers to the cells and their sectors. In such an assignment, it is possible that channels are idle in one cell, whereas another cell is overloaded. The trunking efﬁciency can be improved if the channels are viewed as being in various common pools, from which allocations are made as needed. Of course, such dynamic channel allocation must respect the cochannel SIR constraints as the channels are allocated, released, and reallocated to various cells. 4.5.1 Reuse Constraint Graph A simple model that can be used for designing and analyzing dynamic channel allocation strategies is to specify pairwise reuse constraints. Given an array of cells, pairwise reuse constraints specify which pairs of cells cannot use the same FDM carrier at the same time. For example, Figure 4.11 shows a linear array of rectangular cells, such as might be deployed along a highway. The diagrams in the middle and bottom of the ﬁgure depict pairwise reuse constraints as constraint graphs. In a constraint graph, each cell is represented by a vertex. There is an edge between two vertices if an FDM carrier cannot be simultaneously used in both of the corresponding cells. Thus, the constraint graph in the middle of Figure 4.11 only constrains neighboring cells from reusing the same channel; channels can be simultaneously used in alternate cells. The constraint graph at the bottom, however, permits a channel to be reused only in cells that are separated by at least two other cells. We note that, in general, representation of reuse constraints by pairwise constraints is conservative. It is possible that among three cells, any two cells can reuse the same channel, but, if the third cell also uses that channel, then, due to the increased interference, the SIR in all the cells may be at an unacceptable level.

102

4 1

2

3

4

5

6

7

8

9

Cellular FDM-TDMA 10

Figure 4.11 A linear array of 10 cells (top), and two sets of pairwise reuse constraints (middle and bottom), shown as constraint graphs.

The modeling of such, more general, constraints requires hypergraphs, a generalization of graphs in which edges are subsets of nodes with cardinality greater than two. Such models have been studied in the literature, but we will consider only pairwise constraints in this book. Formalizing this discussion, let B = {1, 2, . . . , N} denote the set of cells (or, equivalently, base stations). Let (B, C) denote the constraint graph with C being the edge set; that is, for i ∈ B and j ∈ B, (i, j) ∈ C if the same carrier cannot be used in Cell i and Cell j simultaneously. We note that the constraint graph is undirected; that is, (i, j) ∈ C if and only if (j, i) ∈ C . Let F = {f1 , f2 , . . . , fM } be the set of FDM carriers that need to be assigned to the cells. Suppose that a certain number of calls need to exist in each of the cells. This will require a certain number of carriers xj in each of the cells j, 1 ≤ j ≤ N , in order to be able to carry those calls. For example, if each carrier has eight TDM slots, then in order to carry 9 to 16 calls in a cell, two carriers are needed. Let xj denote the number of carriers required in Cell j, 1 ≤ j ≤ N . We say that the vector x = (x1 , x2 , . . . , xN ) is feasible if there exists an allocation of xk carriers to Cell k such that the reuse constraints are respected. As a simple illustration, if F = {f1 , f2 }, and we have the reuse constraints shown in the middle of Figure 4.11, then x = (2, 1, . . . , 1) is not feasible. Deﬁne X = {x : x feasible}

Recalling some standard concepts from graph theory, we say that a clique of (B, C) is a fully connected subgraph. Thus, a carrier can only be used in exactly one of the cells that form a clique. A maximal clique is one that is not contained in any other clique. We will simply refer to maximal cliques also as cliques. Thus, in the bottom diagram of Figure 4.11, the cliques are {1, 2, 3}, {2, 3, 4}, and so on.

4.5

Channel Allocation and Multicell Erlang Models

103

4.5.2 Feasible Carrier Requirements Let Q be the number of cliques (i.e., maximal cliques) in (B, C). Consider the Q×N matrix A with 1 if cell j is in clique i aij = 0 otherwise We see that a necessary condition for x ∈ X is A·x ≤M 1

where we recall that M is the number of carriers, and 1 is the Q × 1 vector of 1s. Note that this inequality simply says that, for each i, 1 ≤ i ≤ Q, N j=1 aij xj ≤ M, where the expression on the left of this inequality is the number of carriers needed in Clique i in order to achieve the carrier allocation given by x. Let us denote XCPA = {x : A · x ≤ M 1}

where the sufﬁx CPA expands to clique packing allocation. It may appear that XCPA is a convenient characterization of X . Since every carrier allocation must satisfy the clique constraints, we see that X ⊂ XCPA . In general, however, X is a strict subset XCPA ; that is, in general, it can be that x ∈ XCPA , but x ∈ X . An example is shown in Figure 4.12. We can also observe that, if the constraint graph shown in Figure 4.12 is a subgraph of a constraint graph, then X = XCPA .

Exercise 4.2 Consider a linear array of cells (1, 2, . . . , N) (as shown in Figure 4.11) with a constraint graph that has the property that if cells i and j, i ≤ j, are in a clique, then all k such that i < k < j are also in the same clique. Argue that for this situation X = XCPA . Show that if x ∈ XCPA then a feasible carrier assignment is obtained via a greedy algorithm that starts by assigning the required carriers to the clique to which the left-most cell belongs, and then moves across the cells from left to right, reassigning carriers as need arises.

4.5.3 Carrier Allocation Strategies Based on the preceding discussion, we can identify various carrier allocation strategies. We recall that, for a system with N cells, x = (x1 , x2 , . . . , xN ) denotes a vector of carrier requirements. Given a set of reuse constraints, a given x may or may not be feasible. We have deﬁned X as the set of all feasible carrier requirement vectors: X = {x : x is feasible}. a. Fixed Carrier Allocation (FCA). The carriers are allocated statically to the cells in such a way that the reuse constraints are satisﬁed. For example, if F = {f1 , f2 }, and we have the reuse constraints shown in the middle of Figure 4.11, then (f1 , f2 , f1 , f2 , . . .) is a valid allocation. With this allocation,

104

4 f1

1

5

2

4

Cellular FDM-TDMA

3

?

f2

f2

f1

Figure 4.12 A pentagon reuse constraint graph for ﬁve nodes is shown on the left. With M = 2, the vector x = (1,1,1,1,1) satisﬁes the clique constraints, but there is no feasible allocation of carriers to cells, as seen in the diagram on the right.

x = (1, 1, 1, . . .) is feasible, and x = (2, 1, 1, . . .) is not. For a given ﬁxed allocation of carriers, let XFCA denote the set of feasible carrier requirements x. Clearly, XFCA ⊂ X .

b. Maximum Packing Allocation (MPA). By deﬁnition, for every x ∈ X there is a carrier assignment that achieves x. When a call arrives to a cell and is accepted, then this will result in a carrier requirement vector y. Under MPA, if y ∈ X , then the call is accepted, even if this requires a rearrangement of the carriers. This is not a practical approach as the rearrangement requires a lot of signaling, and the forced handovers of calls as carriers are being swapped. Writing the set of feasible carrier requirements under MPA by XMPA , we have XMPA = X . c. Clique Packing Assignment (CPA). Since the characterization of XCPA is simple, for theoretical purposes we may assume that each x ∈ XCPA is acceptable. In general, we have XFCA ⊂ X = XMPA ⊂ XCPA

where, as we have seen, the last containment can be strict. Another channel allocation strategy, which can be viewed as a hybrid of FCA and MPA, is that of channel borrowing. Some channels are statically assigned to cells, whereas others are permitted to be borrowed between cells, in order to accommodate local load variations. 4.5.4 Call Blocking Analysis If a carrier allocation respects the SIR constraints, or if it satisﬁes certain reuse constraints that, in turn, assure the SIR constraints, then, with a high probability,

4.5

Channel Allocation and Multicell Erlang Models

105

the accepted calls will experience an acceptable voice quality. This was the purpose of the analysis that we discussed in Section 4.2. Once a particular carrier allocation strategy (denoted CA, generically) is chosen, then the carrier requirement vector x will remain in XCA . Calls will need to be blocked for this to happen; if acceptance of a new call results in a carrier requirement vector x ∈ XCA , then the arriving call is blocked. In addition to a good voice quality during a call, users also are concerned about the probability of their requests being blocked, or accepted requests being dropped because of handover blocking. In this section we show how blocking probabilities can be obtained for carrier allocation strategies. Consider any carrier assignment strategy, and let XCA denote the set of feasible carrier requirements, x, as discussed earlier. We will assume, for simplicity, that each carrier can carry just one call (rather than, for example, eight in the GSM system). In this section, we also assume that calls stay in the cells into which they arrive, that is, that there are no handovers between cells. Let the arrival rate of calls into Cell j be λj , 1 ≤ j ≤ N . The arrival processes are assumed to be Poisson processes that are independent from cell to cell. We assume that the time duration for which a call holds a carrier has mean μ1 , and that the holding times from call to call are i.i.d. We also assume that the carrier holding times are exponentially distributed; this assumption can be relaxed, but we will not dwell on that aspect in this discussion (see, however, Appendix D, Section D.5.1). In this setting, let Xj (t), 1 ≤ j ≤ N , denote the number of carriers utilized in Cell j (equivalently, the number of calls in Cell j) at time t . Then consider the vector random process X(t) = (X1 (t), X2 (t), . . . , XN (t)). If the chosen carrier assignment strategy is used then, for all t , X(t) ∈ XCA . With the assumptions we have made on the arrival processes and carrier holding time distributions, it can easily be seen that the process X(t) is a continuous time Markov chain (CTMC; see Appendix D, Section D.2). For ﬁnite and positive arrival rates and mean holding times, this CTMC is positive recurrent, since it has a ﬁnite number of states. In order to obtain the blocking probabilities we need the stationary distribution π(x), x ∈ XCA . Then the blocking probability of calls arriving into Cell j, denoted by Pb, j is given by Pb,j = π(x) (4.6) {x∈XCA : x+ej ∈XCA }

where ej is the unit vector with a 1 in the j-th position. Note that Pb,j is the fraction of time during which an arrival into Cell j will be blocked. The fact that this is the same as the fraction of calls arriving into Cell j that are blocked (the quantity on the right-hand side of (4.6)) is a consequence of the Poisson Arrivals See Time Averages theorem (PASTA) (see Appendix D, Section D.4.2). The average blocking over all the cells is then given by Pb =

N j=1

λj N

i=1 λi

Pb, j

106

4

Cellular FDM-TDMA

which can be understood by observing that the probability that a call arrival is for λ Cell j is N j . i=1

λi

It remains to determine the stationary distribution π(x), x ∈ XCA . Notice that only the following state transitions are possible in the CTMC X(t). For x ∈ XCA , we can have x → x + ej for some j, 1 ≤ j ≤ N (due to an arrival into Cell j), or x → x − ej (due to a call completion in Cell j; here we require xj > 0 in x). Let λ ρj = μj , 1 ≤ j ≤ N , the Erlang load on Cell j. Deﬁne π(x) ˆ =

xj ρj N Πj=1 xj !

Now consider the transition x → x + ej , and notice that π(x) ˆ × λj = π(x ˆ + ej ) × (xj + 1)μj

Also, for the transition x → x − ej , where xj > 0, we have π(x) ˆ × xj μj = π(x ˆ − ej ) × λ j

With these observations, and deﬁning GCA =

π(x) ˆ

(4.7)

{x:x∈XCA }

it can be shown that (see Exercise 4.3) the stationary distribution is given by xj

ρj 1 π(x) = ΠN j=1 GCA xj !

(4.8)

Exercise 4.3 Use Theorem D.8 in Appendix D to prove that what is being claimed in (4.8) is correct. 4.5.5 Comparison of FCA and MPA Consider the three-cell example, and the corresponding pair-wise constraint graph shown in Figure 4.13. There are M carriers, each of which can handle one call. If we partition the set of carriers into two equal parts, and assign one set to Cells 1 and 3, and the other set to Cell 2, then the reuse constraints are met, and we get the XFCA shown by the dashed box in Figure 4.13. On the other hand, in MPA, any carrier not used in Cell 2 can be used in both Cells 1 and 3; XMPA is also shown in the ﬁgure. Suppose that the arrival rate of calls is the same in all the cells. Let us ﬁrst numerically investigate the blocking probabilities for M = 2. Figure 4.14 shows the set of states in XMPA . The set of states in which calls to

4.5

Channel Allocation and Multicell Erlang Models

107

x3 1

M

2

3

XFCA

M/2

M

M

M/2

XMPA

x2 M/2 M

x1

Figure 4.13 On the top right is shown a 3-cell example and the corresponding reuse constraint graph. There are M carriers. The sets XFCA and XMPA are the points with integer coordinates inside the regions shown.

000

010

001

101

002

102

011

100

020

111

200

110

201

202

Figure 4.14 The set of states for the three-cell network using maximum packing channel allocation with two channels. The downward transitions occur at rate λ and the upward transitions are at rates that are multiples of μ.

108

4

Cellular FDM-TDMA

Cell j are blocked are as follows. For Cell 1: (020), (110), (200), (111), (201), and (202); For Cell 2: (002), (011), (020), (110), (200), (102), (111), (201), and (202); For Cell 3: (020), (011), (002), (111), (102), and (202). Let λj = λ for j = 1, 2, 3. The following blocking probabilities are easy to obtain. GMPA = Pb,1 = Pb,3 = Pb,2 = Pb =

1 4 9 ρ + 2ρ3 + ρ2 + 3ρ + 1 4 2 1 4 4ρ

+ 32 ρ3 + 2ρ2 GMPA

1 4 4ρ

+ 2ρ3 + 32 ρ2 GMPA

2 1 Pb,1 + Pb,2 3 3

Figure 4.15 shows a plot of blocking probability in each of the cells and the overall blocking probability. For comparison, the blocking probability from a ﬁxed channel allocation is also shown; one channel is allocated to Cell 2, and the 1 0.9 0.8 0.7 P1 P2 P Pf

0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5

6

7

8

9

10

Figure 4.15 Plot of the blocking probability for the 3-cell network shown in Figure 4.14, with M = 2, under MPA, in Cell 1 (P1), Cell 2 (P2), and the overall blocking probability (P). Pf is the blocking probability for FCA, with 1 channel allocated to Cell 2, and the other to both Cells 1 and 3.

4.5

Channel Allocation and Multicell Erlang Models

109

other to both Cells 1 and 3. Note that in the middle cell, the blocking probability is worse with MPA than with FCA for large ρ. As can be seen from the set of states that block a call to Cell 2, and also the expression for Pb,2 , there are many more states that affect the blocking in the middle cell. Let us now consider a linear array of N > 3 cells, again with the constraint that neighboring cells cannot reuse the same channel. Extending the two-cell analysis via enumeration for N > 3 is clearly tedious. We therefore do an asymptotic analysis as N → ∞. Before describing the result, let us see what to expect. With increasing N , the number of middle cells increases as N → ∞, and the blocking probability behavior that we saw for Cell 2 in the numerical example earlier should become typical. We will see that this indeed is what happens. Let us consider the blocking probability in an interior Cell i. For the same reuse constraints, we see that the set of states XMPA , is deﬁned by (see Exercise 4.2) XMP = {x : xi + xi+1 ≤ M

for i = 1, . . . , N − 1}

and the blocking states for Cell i are deﬁned by {x : xi−1 + xi = M or xi + xi+1 = M}

or, equivalently, the set of blocking states for Cell i are {x : xi = M, or, (xi−1 + xi = M, 0 ≤ xi < M), or (xi + xi+1 = M, 0 ≤ xi < M)}

Thus, using the union bound, the blocking probability at Cell i with MPA is bounded as follows ⎛ ⎞ M−1 M−1 M k M−k k M−k 1 ⎝ ρ ρ ρ ρ ρ MPA ⎠ Pb,i ≤ G2,i (k) G3,i (k) G1 + + GMPA M! k! (M − k)! k! (M − k)! k=0

k=0

Here G1 , G2,i (k) and G3,i (k) are given by G1 :=

N ρ xj xj ! n=1

x∈X1

G2,i (k) :=

n=i

x∈X2, i (k)

G3,i (k) :=

x∈X3,i (k)

N ρ xj xj ! n=1

n=i−1,i

N ρxj xj ! n=1

n=i,i+1

where X1 is the set of states in which Cell i has M calls, X2,i (k) is the set of states in which Cell i has k calls and Cell (i − 1) has (M − k) calls, and X3,i (k) is the set of states in which Cell i has k calls and Cell (i + 1) has (M − k) calls.

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First, let us see what happens for low values of ρ. For low values of ρ, the G (k) 1 , G2,iMPA , and higher powers of ρ will be insigniﬁcant and we can argue that GGMPA G3,i (k) GMPA

all approach 1 as ρ → 0. Then, as ρ → 0, we can write MPA Pb,i ≤

M−1 ρk ρM−k ρM +2 M! k! (M − k)! k=0

⎛ ⎞ M ρ M ⎝ M! ρM +2 − 1⎠ = M! M! k! (M − k)! k=0

=

ρM M 2M+1 − 1 M ρM +2 (2 − 1) = ρ M! M! M!

For ﬁxed channel allocation, each cell would be allocated M 2 channels and the blocking probability would be (as before, see Appendix D, Section D.5.1) FCA Pb,i

=

ρM/2 (M/2)! M/2 ρk k=0 k!

≈

ρM/2 (M/2)!

where, since ρ is small, in the denominator we just retain the unit term. We can now FCA decreases as ρM/2 , whereas PMPA decreases see that, for large M, and small ρ, Pb,i b,i faster than ρM (for a precise calculation we can use Stirling’s approximation for the factorials). Hence, MPA would perform better at low loads. Let us now see what happens when ρ is large. The stationary probability of there being x active calls in Cell i can be written as πi (x) =

1 GMPA

x∈X3 (x)

M ρ x ρ xj x! j=1 xj !

1

=

GMPA

ρx φ(i, x) x!

j =i

where X3 (x) is the subset of XMPA in which there are x calls active in Cell i and φ(i, x) :=

M ρx j x∈X3 (x)

j=1 j=i

xj !

We can see that the carried load in Cell i is the average number of active calls in Cell i and is given by M x=1 xπi (x). Subtracting the carried load from the offered load (ρ to each cell) and expressing it as a fraction of the offered load, the loss MPA , is probability, Pb,i MPA Pb,i

=

ρ−

1 GMPA

M

ρx x=1 x x! φ(i, x)

ρ

= 1−

1

M−1

GMPA

x=0

ρx φ(i, x + 1) x!

4.5

Channel Allocation and Multicell Erlang Models

111

Obtaining φ(i, x) is involved and we will omit that here. For M = 2, and MPA can be shown to be given by N → ∞, Pb,i MPA Pb,i =

p2 (14 − 10p − 5p2 + 3p3 ) 2(2 + p2 − 2p3 )

Loss probability

where p is the solution in (0, 1) to the cubic equation ρ(1 − p)(2 − p2 ) = 2p. MPA and PFCA as a function of the offered load ρ. Notice Figure 4.16 shows Pb,i b,i at about ρ = 2.6 the ﬁxed channel assignment outperforms the maximum packing dynamic channel assignment! What is more interesting is that it can be shown that as M increases, the crossover happens at lower values of ρ and the crossover point is asymptotically 0! This indicates that for high capacity cellular networks, the ﬁxed channel allocation scheme will outperform the dynamic channel scheme when the load is time and space homogeneous. This result is deﬁnitely counterintuitive; we expect dynamic schemes to be better than static schemes. A heuristic explanation for the effect just described is that the MPA allocation scheme can upset the tight packing of the channels and calls at high loads and spends more time in the many Bad states that are possible with dynamic allocation. A conclusion that we may draw from this analysis is that it might be better to reject some calls, especially at high loads, to be able to improve the overall system performance. The MPA scheme will accept a call if the channels can be rearranged

Crossover probability

Pb,iFCA

MPA

Pb,i

1.0

Figure 4.16 from [73].

MPA and P FCA as a function of Pb,i b,i

2.0

2.6

offered load

ρ for M = 2, and N → ∞. Adapted

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to accommodate it while the FCA will reject a call if all the channels allocated to the cell are busy; that is, it will not borrow channels from other cells to fulﬁll a request. Thus, it is not automatic that a dynamic channel allocation performs better than a ﬁxed allocation scheme. However, we cannot conclude that dynamic channels do not have advantages. Rather that the advantages are realized if the offered load is nonhomogeneous in space and is time varying, in which case the dynamic schemes adapt to the changing load.

4.6

Handovers: Techniques, Models, Analysis

In our discussions thus far, we have essentially assumed that mobiles are conﬁned to the cells in which they initiate their calls. Since typically, neighboring cells do not reuse a carrier, when a mobile moves to a neighboring cell, it must switch over to a different carrier in that cell. This is called a handover. Naturally, for cellular mobile telephony to be a useful service, a handover should be transparent to the user. This imposes two requirements: a. An ongoing call should not experience degradation in service when it is at the fringes of the cell that is handling it. b. A handover should rarely fail due to a channel not being available in the cell into which a mobile call moves. Such a handover failure leads to call dropping, the constraints on which are more stringent than on call blocking. Handovers are performed by the MS making signal strength measurements to neighboring BSs, and conveying this information to the handover management system, which then decides on the need for a handover and the channel to be assigned to the new cell. The transfer of such measurements from the MS to the call handling system became possible in the second generation cellular systems. Note that the handover strategy basically deﬁnes what is meant by a cell’s coverage area. 4.6.1 Analysis of Signal Strength Based Handovers We consider an MS located on the line joining two BSs, BS 0 and BS 1, as shown in Figure 4.17. Let Si (x) = Received signal power from BS i, i ∈ {0, 1}, when the MS is at the distance x from BS 0. Then, recalling the path loss and shadowing model from Section 2.1.4, we have [S0 (x)]dB = S0 (d0 )

dB

− 10η log

x − ξ0 d0

where η is the path loss exponent, and the shadow fading, ξi , i ∈ {0, 1}, is normally distributed with mean 0, and variance σ 2 . Also, 2R − x [S1 (x)]dB = S1 (d0 ) dB − 10η log − ξ1 d0

4.6

Handovers: Techniques, Models, Analysis

113

BS 0

BS 1

MS 0

d 0

2R⫺d 0

R x

2R

2R⫺x

Figure 4.17 Handover: An MS located on the line joining two neighboring BSs that are at the distance 2R. The MS is located at distance x from BS 0. The MS provides signal strength measurements from each of the BSs.

Let us assume that S0 (d0 ) = S1 (d0 ). Then, for d0 ≤ x ≤ 2R − d0 ,

[S0 (x) − S1 (x)]dB

2R − x = 10η log x

+ (ξ1 − ξ0 )

where ξ1 −ξ0 is normally distributed with 0 mean and variance 2σ 2 . In Figure 4.18, we show the variation of [S0 (x) − S1 (x)]dB as the MS moves from BS 0 to BS 1. 2R−x . The two dashed curves above The solid curve shows the mean 10η log x and below the solid curve represent the variability due to shadowing, and can be viewed as the bounds within which [S0 (x) − S1 (x)]dB stays, with a high probability. The√half-width of the curved strip deﬁned by the two dashed curves is proportional to 2σ . Suppose the MS is being served by BS 0. A simple handover approach is to hand over the MS to BS 1 when [S0 (x) − S1 (x)]dB R). There are two issues here: a. If the coverage of either cell extends only up to a distance R, then once the MS is beyond R, the handover should occur with a high probability. b. With this design, if the MS is moving about in the region around the middle of the line joining the two BSs, then it will be repeatedly handed over between the two BSs, thus increasing the chance of the call being dropped, and also increasing the load on the call management processors.

114

4

0

Cellular FDM-TDMA

2R2d0

d0 R

+H 2R 2H

h

Figure 4.18 Handover: The difference in signal strengths, S0 (x) − S1 ( x) (in dB) at an MS that is at position x on the line joining BS 0 and BS 1. For an explanation of h, see the text.

These two issues can be addressed by extending the coverage of each BS beyond R, to an additional distance, say, h. Suppose h is chosen so that 10η log

√ 2R − (R + h) + a 2σ < −H R+h

or 10η log

√ R−h + a 2σ < −H R+h

√ where a 2σ is the half width of the dashed strip, and a is chosen from the standard normal tables so that the tail probability of the random variable ξ1 − ξ0 beyond √ a 2σ is small. This choice of h is shown in Figure 4.18, since at x = R + h the upper dashed curve falls below −H . Now, when deciding to hand over from BS 0 to BS 1, we check if both of the following tests are true: [S0 (x)]dB < Sthreshold [S0 (x) − S1 (x)]dB < −H

for a suitably chosen Sthreshold . Both these tests will succeed beyond R + h with a high probability, and, thus, the handover will take place with a high probability. Further, the reverse handover will take place with a very small probability. Thus, this handover strategy has a hysteresis built into it. Although this design solves the problem of repeated handovers from one cell to the other, the extension of the cell coverage into the neighboring cell impacts the earlier SIR analysis. Let h =b R

4.6

Handovers: Techniques, Models, Analysis

115

so that R + h = (1 + b)R

Thus, in the cochannel interference calculations, we now need to use D (1 + b)R

It follows that a larger D R value will need to be used for a given SIR constraint, thus requiring a larger value of Nreuse , and lowering the spectrum efﬁciency. It is thus important to design handover schemes that can reduce the cell expansion factor b. 4.6.2 Handover Blocking, Call Dropping: Channel Reservation Let us consider a cell in an FDM-TDMA cellular system with new call arrival rate λ0 , and handover call arrival rate (from neighboring cells) λh . Deﬁne Pb = new call blocking probability Ph = handover blocking probability Pd = call dropping probability

Note that a call may undergo several handovers, and the call gets dropped at the ﬁrst of its handovers that is blocked. The preceding deﬁnitions can be formally expressed as Ph = lim

t→∞

number of handovers lost in [0, t] number of handovers in [0, t]

and Pd = lim

t→∞

number of accepted calls dropped in [0, t] number of calls accepted in [0, t]

Note that Pb and Pd are user perceived performance measures, whereas Ph is a measure internal to the system. We need Pd to be very small (e.g., 0.1%), whereas Pb is typically 1 to 2 percent. Let us assume that the time that a call spends in a cell is exponentially distributed with mean 1ν . The duration of a call is exponentially distributed with mean μ1 . Then, assuming that whether or not a handover is blocked is independent from handover to handover, we can write Pd =

ν (P + (1 − Ph ) · Pd ) ν+μ h

ν is the probability that a call leaves the cell This can be understood as follows. ν+μ it is in before it ﬁnishes conversation. If it does leave the cell, then the handover

116

4

Cellular FDM-TDMA

attempt is blocked with probability Ph , or if the handover is not blocked (with probability 1 − Ph ), then we have a renewal point (see Appendix D) and the remaining call experiences dropping with probability Pd . This expression yields Pd =

=

ν ν+μ Ph νPh μ ν+μ + ν+μ ν μ Ph 1 + μν Ph

The second expression on the right may be approximated by μν Ph when μν Ph is much smaller than 1. Where the approximation works, its interpretation is the mean number of handovers per call multiplied by the handover blocking probability. This calculation yields a target value of Ph , given a target for Pd . We had deﬁned λh as the rate of arrival of handovers into a cell, and we observe that this is not a given. But with the exponential distribution assumptions we have made, we can write the following: ν 1 λh = (λ0 (1 − Pb ) + λh (1 − Ph )) (4.9) · ·6 μ+ν 6 This is obtained as follows. λ0 (1 − Pb ) + λh (1 − Ph ) is the rate of accepted calls into a cell. Each accepted call causes a handover to a neighboring cell with probability ν μ+ν . Each cell is surrounded by six cells, and one-sixth of the handovers of each of its neighbors enters the cell. However, Ph and Pb depend on λh . The approach is to iterate, starting with λ(0) . This will yield Pb(0) and Ph(0) . Given Pb(k−1) and Ph(k−1) h we can obtain λ(k) by using (4.9), and thus the iterations can continue. How Pb(k−1) h

and Ph(k−1) are obtained from λ(k) depends on the way channels are assigned to new h calls and handover calls in a cell, and is the next topic of discussion. The remaining question is whether there is a need to discriminate between new calls and handover calls when assigning channels. If they are handled in the same way, then they will get the same blocking probability (i.e., Pb = Ph ). Since the target value of Ph is much smaller than that of Pb , we will be forced to operate with much too small a value of new call blocking, which will result in a very low Erlang capacity. Hence channel reservation is done for handover calls. The common approach is dynamic channel reservation, which means the following. If there are m carriers in a cell, then a number mh < m is chosen; typically, mh is just 1 or 2. When a call arrives, if the number of busy carriers is less than m − mh , then every call is accepted. However, if the number of busy carriers is ≥ m − mh then only handover calls are accepted. If we assume that the arrival process of new calls and handover calls into a cell are independent Poisson processes, then the number of busy carriers becomes a positive recurrent CTMC. Returning to the iterative calculation, earlier, the analysis of this CTMC will provide Pb(k) and Ph(k) ,

4.7 The GSM System for Mobile Telephony

117

given λ0 and λ(k) . At the k-th iteration, let π(k) (i), 0 ≤ i ≤ m, denote the stationary h probability distribution of the CTMC. Then (k)

Pb =

m

π(k) (i)

i = m−mh

(k)

Ph = π(k) (m)

where, again, the PASTA theorem is used (see Appendix D, Section D.4.2).

Exercise 4.4 a. Show the transition rate diagram of the CTMC with dynamic channel reservation for handovers, and obtain the stationary distribution π(k) (·). b. Write a computer program to carry out the proposed iteration and obtain the new call arrival rate, λ0 , that can be offered when m = 16, and mh = 1, for a target Pd = 0.01. Take the mean call duration to be 100 seconds, and the mean time a call stays in a cell to be 50 seconds. Obtain the new call blocking probability, Pb , that is obtained with this value of λ0 . c. What is the new call arrival rate that can be handled if no special treatment is provided to handovers, but we still require that Pd = 0.01?

4.7 The GSM System for Mobile Telephony After about 15 years of deployment, the FDM-TDMA-based GSM system (Global System for Mobile communications) is the most popular cellular system for mobile telephony and related services. Figure 4.19 shows the components of a GSM cellular network. The wireless links are only between the mobile stations (MSs; shown as cellular phone handsets in Figure 4.19) and the Base Transceiver Stations (BTSs). An MS can be in the vicinity of several BTSs, but at any point in time, an active MS is associated with one BTS, the one with which it is determined that it has the highest probability of reliable communication. Several BTSs are linked to Base Station Controllers (BSCs) by wired links. Together, the BTSs and the associated BSC is called a BSS (Base Station Subsystem). The BTSs provide the ﬁxed ends of the radio links to the MSs; it is the BSC that has the intelligence to participate in the signaling involved in connection handovers. In turn, the BSCs are connected to the Mobile Switching Center (MSC), which connects to the ﬁxed network infrastructure. Worldwide, several bands have been used for the operation of GSM networks. The 900 MHz or 1800 MHz bands are the ones commonly used in most countries. In the 900 MHz band the uplink carriers are in the 890–915 MHz frequency band, and the downlink carriers are in the 935–960 MHz frequency

118

4

Cellular FDM-TDMA

NSS: Network and Switching Subsystem VLR

SS7 Network

HLR GMSC BSC

MSC

fixed network infrastructure BSC

mobile stations

BTS

Figure 4.19 The components of a GSM cellular network.

band. As explained earlier in this chapter, if an operator obtains W MHz of spectrum, actually W MHz is provided from the uplink band and another W MHz is provided from the downlink band. This is for the purpose of frequency division duplexing of bidirectional calls. The bandwidth of a GSM operator, in each direction, is then divided into FDM carriers with a spacing of 200 kHz. These FDM carriers are digitally modulated to create a hierarchical TDM carrier. The basic frame time in this TDM carrier is 4.615 ms, which contains eight slots, each of which can be assigned to a different voice call. Each TDM slot can carry 114 bits of payload. Notice that the coded payload bit rate on each carrier is about 200 Kbps. For one standard GSM voice codec, after channel coding, blocks of 456 bits are emitted, which are accommodated in four TDM slots. Since the resources (i.e., the spectrum) of a cellular wireless network are limited, an MS cannot have permanent access to the network, but has to make a request for a connection. Thus, since an MS is not always connected to the network, there are two problems that need to be addressed: a. Between the time that an MS last accessed the network and the time that it next needs to access, the MS may have moved; hence, it is ﬁrst necessary to locate the MS and associate it with one of the cells of the network. b. Since the MS initially does not have any access bandwidth assigned to it, some mechanism is needed for it to initiate a call or to respond to an incoming call.

4.8

Notes on the Literature

119

Location management and call set up are the major activities that need to be overlaid on the basic cellular wireless infrastructure in order to address the ﬁrst problem. In Figure 4.19 we show the additional components that are needed. Together these are called the Network and Switching Subsystem (NSS), and comprise the MSC, the GMSC (Gateway MSC), the HLR (Home Location Register), the VLR (Visitor Location Register), and the signaling network (standardized as Signaling System 7 (SS7), by the ITU). The SS7 signaling network already exists where there is a modern circuit switched phone network. As their names suggest, the HLR carries the registration of an MS at its home location, and a VLR in an area enters the picture when the MS is roaming in that area. Each operator has a GMSC at which all calls to MSs that are handled by the operator must ﬁrst arrive. The GMSC, HLR, and VLR exchange signaling messages over the SS7 network, and together help in setting up a call to a roaming user. Location management is done as follows. An MS will be registered with an operator in its home area. A roaming mobile that is turned on brieﬂy associates itself with a nearby BTS and provides the network the information that it is now in the area. If this happens to be an area other than where the MS normally is registered, then the MS’s identity is used to determine its home location, and the HLR at this location is informed of the whereabouts of the MS. The VLR at the location that the MS is visiting then receives conﬁrmation from the MS’s HLR that this MS is a valid user. Suppose now that someone somewhere in the world calls this MS. The MS’s number is used to determine the GMSC of its home operator. A signaling message is sent over the SS7 network to this GMSC, which determines the HLR where the MS is registered, and sends a message to this HLR. The HLR then, knowing that the MS is roaming, queries the VLR in the area where the MS is roaming. The VLR knows which local MSC the MS is in the control of, and provides this information to the HLR. The HLR forwards this information to the GMSC, which then directly establishes the call to the MS. Let us now turn to the second of the two problems enumerated. In the GSM system there are several permanent channels deﬁned in each cell. Whenever an MS enters a cell it locks into these channels. One of these channels is called the paging and access grant channel (PAGCH). If a call arrives for an MS, and it is determined that the MS may be in a cell, or in a group of cells, then the MS is paged in all these cells. Another such common channel is basically a slotted Aloha random access channel (RACH) (see Chapter 7), and is shared by all the MSs in the cell. When an MS has to respond to an incoming call (i.e., it is paged on the PAGCH) or has to initiate a call, it contends on the RACH in the cell, and conveys a short message to the network. Subsequently, the network allocates a channel to the MS and call set up signaling starts.

4.8

Notes on the Literature

In this chapter we have discussed concepts and techniques that were researched in the 1970s and 1980s, at a time when the ﬁrst analog cellular telephony systems

120

4

Cellular FDM-TDMA

were being experimented with. Bell System’s Advanced Mobile Phone Service (AMPS) and the cellular concept are described in a seminal article by MacDonald in Bell Systems Technical Journal [96]. There are several textbooks devoted to extensive treatments of cellular telephony, including the classic by Lee, and the more recent book by Garg and Wilkes [39]. The widely adopted text by Rappaport [116] discusses cellular mobile systems in conjunction with a detailed coverage of propagation phenomena in cellular mobile communication systems, physical layer techniques, and speech coding. A rigorous derivation of the formula relating the cochannel cell distance D(i, j) and Nreuse was carried out by Gamst [38] using group and ring theory. The Fenton-Wilkinson approximation, and other similar techniques have been derived in the text on mobile communications by Stuber [123]. The comparison of ﬁxed channel allocation and maximum packing allocation has been adapted from Kelly [73], which also provides some very useful insights into several problems in networking. A very accessible and extensive coverage of the GSM standard has been provided by Mouly and Pautet [104].

Problems 4.1.

The coverage of a cell is ﬁrst obtained by ignoring shadow fading (Rayleigh fading can be assumed to be averaged over). If the shadow fading standard deviation is 8 dB, roughly how much additional power is required so that the outage probability for the same coverage is less than 2%?

4.2.

A fade margin of 20 dB is required to combat shadowing and achieve adequate coverage in a cell. a. If the shadowing standard deviation is 8 dB, what was the target outage probability? b. If the path loss exponent is 4, how much additional coverage would be obtained if there is no shadowing?

4.3.

A GSM operator leases 7 MHz of spectrum (i.e., 7 MHz each in the uplink and the downlink), and estimates that a D R of at least 4 is required. If the cell radius, R, is 2 km (assume hexagonal cells), determine the Erlangs per square kilometer for the network, for a target blocking probability of 1%.

4.4.

A GSM operator leases 7 MHz of spectrum. Assuming that the uplink constrains performance, a path loss exponent of 4, and ignoring shadowing and additive noise, and given that an SIR of 14 dB is required, determine the Erlang capacity per cell for a blocking probability of 1%. Do not consider sectorization. Assume a hexagonal cell geometry.

4.5.

Consider a highway cellular system. Assume that the highway is exactly linear, the cells are of length 2R, and the cell width (i.e., the width of

Problems

121

the highway) can be ignored. Frequencies can be reused in cells whose centers are D units apart. The base station in each cell is at its center, and has two directional antennas, one covering each half of the cell (i.e., the cells are “sectorized” into two sectors). a. Relate D, R, and the number of reuse groups N . b. Accounting only for ﬁrst tier interferers, assuming that Rayleigh fading is averaged out, assuming independent log-normal shadowing for all the received signals, determine the minimum D R value so that the SIR falls below 12 dB with a probability of 1%. You must analyze both the forward and reverse channels. The standard deviation of log-normal shadowing is 8 dB. Take the power law path loss exponent to be 4. c. Explain why the SIR analysis is greatly simpliﬁed in this problem by assuming directional antennas; that is, by sectorization. d. Given that there are 200 trafﬁc channels available (assume single channel per carrier) determine the maximum number of Erlangs that each cell can be offered. 4.6.

Consider a channelized cellular system with a total of 320 trafﬁc channels. Denote the cell radius (center to apex) by R, and the minimum distance between cochannel cells by D. Assume that we can average over Rayleigh fading. Take the lognormal shadowing to have a standard deviation of σ = 8 dB, and the path loss component to be 4. Considering only the uplink channel answer the following. a. Obtain the channel reuse ratio for an uplink channel target SIR of 6 dB and an outage probability of 10%. Use the Fenton-Wilkinson method, and a table of the normal distribution. You may assume that the worst case interferer distance is D − R. b. List two assumptions that this analysis makes. In your solution in (a), where is Rayleigh fading being accounted for (even though it is being averaged over)? c. For this reuse ratio and the given number of channels, obtain the Erlang capacity per cell assuming a ﬁxed channel allocation, and a call blocking probability of 2%. Use an Erlang blocking table.

4.7.

Consider a TDM/TDMA cellular system in which each carrier handles eight calls. Voice activity detection (VAD) is used to reduce cochannel interference; an MS does not transmit when there is no speech activity. The probability of an MS being active is p. Consider a hexagonal cell layout; ignore shadowing and Rayleigh fading; take the path loss exponent to be η. In the following, use the standard approximations for the hexagonal geometry. Use tables of the standard normal distribution and Erlang blocking tables.

122

4

Cellular FDM-TDMA

a. Considering only the uplink, and accounting for voice activity, determine the minimum D/R ratio required for a SIR γ, if the probability of SIR falling below γ is allowed to be 2.3%. (Hint: consider the total power at the reference BS, and individual powers from each of the interfering MSs.) b. For γ = 20 dB, η = 4 and p = 0.4 determine the reuse ratio without and with VAD. Show that the effect of VAD is equivalent roughly to reducing the value of γ by 3 dB. 4.8.

In the ﬁgure are shown ﬁve cochannel cells each with four 90◦ sectors, oriented as shown. cell radius 5 R

D

5 cochannel cells, showing the 90 degree vectors

a. Copy the diagram and mark the cochannel sectors with g1 , g2 , g3 , and g4 . b. Ignoring Rayleigh fading and log-normal shadowing, obtain the value of D/R for a reverse channel worst-case S/I of 20 dB. Take the path loss exponent to be 4. 4.9.

a. The ﬁgure shows seven cells and pairwise reuse constraints between them. Show that for these constraints, and two channels, XCPA is strictly larger than XMPA . 1

7

2

6

3

5

4

Problems

123

b. Consider three cells with a triangular pairwise reuse constraint graph. There are N channels and calls arrive to each cell at rate λ. The calls have a mean channel holding time of b. i. Sketch the set of possible vectors of the numbers of calls that can be present in each of the cells (i.e., X ). ii. Find the probability that a call is blocked. 4.10.

Consider a linear array of K cells, with reuse constraint graph given. Let the J × K clique incidence matrix be denoted by A. Assume that the maximum number of frequency channels that can be used simultaneously in the j−th maximum clique is given to be nj , 1 ≤ j ≤ J (nj ≤ M, where M is the total number of frequency channels in the cellular system). Let N = (n1 , n2 , . . . , nJ ). a. Find the set S(N) of feasible cell occupancy vectors x = (x1 , x2 , . . . , xK ). b. Ignoring mobility, assume Poisson call arrivals with trafﬁc intensity ρk in cell k, 1 ≤ k ≤ K, and assume that the cell occupancy vector has the steady-state distribution x

π(x) = G(N)ΠK k=1

ρk k xk !

, x ∈ S(N),

where ⎛ G(N) = ⎝

x∈S(N)

x

ΠK k=1

ρk k xk !

⎞−1 ⎠

.

Show that the steady-state probability that a call arrival in cell k is blocked is Bk = 1 −

G(N) G N − AekT

where ek is the length-K vector (0, . . . , 0, 1, 0, . . . , 0) with the only non-zero element 1 appearing at the k-th position. 4.11.

Consider two neighboring base stations BS 1 and BS 2, a distance 2R apart (where R = 10d0 ), and an MS on the line joining them. Assuming that Rayleigh fading is averaged over, the minimum SINR required for acceptable communication is 10 dB. Let Ψ0 denote the average SNR at

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a distance d0 from a transmitter. Assume that the power law path loss exponent is 4; the shadowing standard deviation is σ = 8 dB. a. With hand-off (at the cell boundary) obtain the (Ψ0 )dB required for an outage probability of 10%. b. With hand-off at 10% beyond the half-way point between the BSs (i.e., 1.1R), repeat part (a).

CHAPTER 5 Cellular CDMA

W

e discussed the CDMA concept in Chapter 2, Section 2.4.1. One of the two major technologies for second-generation cellular systems is based on CDMA. Third-generation (3G) cellular access systems that provide high speed data and multimedia access are also based on CDMA. In this chapter we will study various resource allocation problems in cellular CDMA systems, basing our discussion mainly on SINR (signal-to-interference plus noise ratio) analysis.

Overview Unlike the FDM-TDMA cellular systems discussed in Chapter 4, CDMA cellular systems are based on the principle of universal frequency reuse; that is, the same portion of the spectrum is reused at every BS. These systems employ frequency division duplexing; each system is assigned a pair of bands, one for the uplink and the other for the downlink. These two bands then are used at every BS. Thus, every uplink transmission interferes, in principle, with every other uplink transmission in the system; the same holds for the downlink. As discussed in Chapter 2, Section 2.4.1, the performance of an instance of communication between an MS and a BS depends on the SINR achieved at the receiver (see (2.24)). Second-generation CDMA systems were designed mainly for carrying telephone quality voice. CDMA systems have been evolving so as to be able to efﬁciently carry other guaranteed QoS services, such as interactive video, and also elastic services, such as ﬁle transfer and web access. We consider resource allocation for both these types of services. Each guaranteed QoS connection needs to achieve an SINR target. In Section 5.1, we write down general inequalities that need to be satisﬁed by the transmission powers used at all the uplink transmitters in the system. An important question that we then ask is about the existence of a set of transmit power levels at all the MSs so that the inequalities are satisﬁed. This leads to the concept of admission control; arbitrary collections of MSs, each with its SINR target cannot be handled by the system. Hence, some call requests need to be blocked. Focusing on the uplink problem, we begin by assuming a spatially homogeneous system in which the interference at a BS, from MSs associated with other BSs, can be taken to be just a multiple of the total received power at a BS. We develop the case of a single call class (say, voice) in Section 5.2. We ﬁnd that each call can be characterized by a resource requirement expressed in terms of the

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5

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target SINR. We ﬁnd that the resource requirements of the calls just add up, and the admission control ensures that a certain measure of total system resource is not exceeded. This measure of system resource depends on the other-cell interference. We show how this is calculated, for hard handover and soft handover of calls between BSs. In Section 5.3, we expand our discussion to multiclass calls. The Chernoff bound is used to develop an admission control that again treats each call as having a resource requirement, and the requirement due to a set of calls is the sum of the individual resource requirements. In Section 5.4 we abandon the spatial homogeneity assumption and consider a general conﬁguration of MSs scattered among several BSs. For a given association of MSs and BSs, we develop a necessary and sufﬁcient condition for there to exist a feasible transmit power allocation. The condition is in terms of the PerronFrobenius eigenvalue of a matrix derived from the channel gains. This also leads to an iterative power control algorithm. Finally, in Section 5.5, we consider the scheduling of downlink elastic transfers. Depending on the channel power gains between the BSs and the MSs, there is a convex set of transfer rates that can be achieved to the MSs. There is a trade-off between maximizing the total transfer rate over all the MSs (which leads to maximization of operator revenue), and fairness between the rates assigned to the MSs. We use the sum-utility maximization formulation to compare various approaches. One such formulation leads to the idea of proportional fairness, for which we then show how the ﬁle transfer delay can be analyzed in terms of the M/G/1 processor sharing model. A brief overview of 2G and 3G CDMA cellular standards is then provided in Section 5.6.

5.1 The Uplink SINR Inequalities In CDMA cellular systems, each active mobile station (MS) is associated with one of the base stations (BSs) in its vicinity. When an MS is involved in a conversation, then it is assigned a power level with which it should transmit. As explained in Section 2.4.1, in CDMA access networks the link performance obtained by each mobile station (MS) is governed by the strength of its signal and the interference experienced by the MS’s signal at the intended receiver. For each radio link between an MS and a BS, a SINR target needs to be met. Hence it is important to associate MSs with BSs, and to assign them transmit powers in such a way that signal strengths of intended signals are high and interference from unintended signals is low. It is evident that increasing the transmit power to help one MS may not solve the overall problem, as this increase may cause unacceptably high interference at the intended receiver (i.e., a BS) of another MS. We will say that an association of MSs with BSs, and an allocation of transmit powers, is feasible if all SINR targets are achieved. In some situations there may be no feasible power allocation. The analysis of CDMA systems is performed via certain SINR inequalities. We will begin our discussion by setting up these inequalities in general.

5.1 The Uplink SINR Inequalities

127

Consider a CDMA system with multiple interfering cells (see Figure 5.1). The system bandwidth is W (e.g., 1.25 MHz in the IS-95 standard), and the chip rate is Rc ≤ W (e.g., 1.2288 Mcps (Mega chips per second)) in IS-95. There are M MSs and N BSs, with B = {1, 2, 3, . . . , N} denoting the set of BSs. Let hi, j , 1 ≤ i ≤ M, 1 ≤ j ≤ N , denote the power “gains” (i.e., attenuations) from MS i to BS j. Let A = (a1 , a2 , . . . , aM ), ai ∈ B, denote an association of MSs with the BSs; thus, in the association A, MS i is associated with BS ai . Let pi be the transmit signal power used by MS i, 1 ≤ i ≤ M. For the most part of the following discussion, we will assume that the power gains and the association are ﬁxed. With these deﬁnitions we can write the uplink received signal power to interference plus noise ratio for MS k as (SINR)k =

hk, ak pk {i : 1≤i≤M, i=k} hi, ak pi + N0 W

BS j

hk,j MS k

BS 1

BSN

hi,j

MS i

Figure 5.1 A depiction of the power allocation problem for several MSs in the vicinity of some BSs. The solid lines indicate signals from MSs to the BSs with which they are associated. MS k is associated with BS j, and its signal (solid line labeled hk,j ) is interfered with by all the other MSs associated with the other BSs (dashed lines), and also by the other MSs associated with BS j. The signal from MS k has a channel “gain” of hk,j to BS j.

128

5

Cellular CDMA

where N0 is the power spectral density of the additive noise, and W is the radio spectrum bandwidth. Assume that the interference plus noise is well modeled by a white Gaussian noise process. Various types of calls may be carried on the system; for example, there could be different types of voice telephony calls that use various codecs. Suppose that a call requires a bit rate Rk . In order to ensure a target bit-error-rate (BER) (which is governed by the required QoS for the application being carried; see the discussion to follow later in Section 5.2.2), we need to lower bound the product of the SINRk and the processing gain Lk := RRc (see (2.24)). For example, with Rk = 9.6 Kbps k and Rc = 1.2288 Mcps, we obtain L = 128. If the desired lower bound is γk , then, deﬁning Γk := γk RRk , we obtain (see (2.24)), for MS k, c

hk, ak pk

hi, a pi + N0 W k

≥ Γk

(5.1)

i : 1 ≤ i ≤ M, i = k

For a given association and given channel gains, we thus obtain M linear inequalities in the M uplink powers of the M MSs. Suppose ak = j; then if we deﬁne

Ij :=

hi, j pi

{i : 1 ≤ i ≤ M, ai =j}

the total power received at BS j from MSs associated with it, and deﬁne Io, j :=

hi, j pi

{i : 1 ≤ i ≤ M, ai =j}

the total interference power at BS j from MSs associated with other BSs, then we can write the SINR inequalities as hk, ak pk ≥ Γk (Iak − hk, ak pk ) + Io, ak + N0 W

(5.2)

for each k, 1 ≤ k ≤ M. Let us understand this derivation by looking at the geometry of the twouser case. Both users are associated with the same BS and there is no interference from any other cell. In Figure 5.2 we depict the analysis for two users. The SINR inequalities are (since there is only one BS we write hi,1 as hi ): h1 p1 − Γ1 h2 p2 ≥ Γ1 N0 W −Γ2 h1 p1 + h2 p2 ≥ Γ2 N0 W

5.1 The Uplink SINR Inequalities p2

129

1

p2

2

feasible power controls 2 1 p1

p1

p*

Figure 5.2 Power control feasibility for two users. The left panel shows the situation in which there are feasible power controls; then there is a power control that achieves the SINR targets with equality. The right panel shows a situation in which there is no feasible power control.

with p1 ≥ 0, p2 ≥ 0. These inequalities are depicted in Figure 5.2 by the lines labeled 1 and 2, for MS 1 and MS 2, respectively. The region to the right of, and below, the line labeled 1 is feasible for MS 1, and the region to the left of, and above, the line labeled 2 is feasible for MS 2. It is easy to see that there is a nonempty feasible region if Γ2 h1 h1 > Γ1 h2 h2

equivalently, if Γ1 Γ2 < 1. It can easily be checked that this is equivalent to Γ1 Γ2 + 0:

BS 1

BS 2 h

h h9

h9

Figure 5.3 The uplink power control problem for two cells with each of which there are M MSs associated. The MSs associated with each BS are collocated, and h and h are the channel power gains, as shown.

1 Note that, for simplicity, we are only considering path loss, and not shadowing, so that, for the geometry shown in the picture, it is plausible that the two groups of MSs have the same gains to the BSs.

5.2 A Simple Case: One Call Class

Q≥M >M

131

Γ ((1 + ν)Q + N0 W) 1+Γ Γ (1 + ν)Q 1+Γ

where the strict inequality arises because N0 W > 0. Thus, we ﬁnd that a necessary condition is M

Γ 1 < 1+Γ 1+ν

(5.5)

or, in other words, the number of admitted calls M should satisfy M

0. For the power allocation obtained, the value of Q is given by Q=

Γ (N0 W) M 1+Γ

Γ 1 − M(1 + ν) 1+Γ

and the interference at a BS from MSs associated with the other BS is given by Io = νQ. 5.2.2 Multiple BSs and Uniformly Distributed MSs We now assume that the MSs are uniformly distributed, and the radio propagation is spatially homogeneous, and, thus, that the BSs are uniformly loaded; that is, each BS receives the same total power Q from the MSs associated with it. Then we can continue to assume that at each BS the interference received from MSs not

132

5

Cellular CDMA

associated with it is some factor ν times Q; that is, Io = νQ for every BS. The SINR inequalities become hk pk ≥

Γ ((1 + ν)Q + N0 W) (1 + Γ)

(5.6)

for each k, 1 ≤ k ≤ M, where hk is the channel gain of MS k to the BS with which it is associated. As before, we sum these inequalities over the MSs associated with a BS to obtain the following necessary condition for a set of powers pk , 1 ≤ k ≤ M, to exist: Γ Q ≥M ((1 + ν)Q + N0 W) 1+Γ But then, noting that all the terms on the right are positive, and hence lower bounding this expression, we see that it is necessary that Γ (1 + ν)Q Q >M 1+Γ It follows that a necessary condition for the existence of a set of powers pk , 1 ≤ k ≤ M, that satisfy the SINR inequalities (5.6) is 1 Γ < M (5.7) 1+Γ 1+ν Now suppose that this condition holds, by associating new calls with a BS in such a way as to ensure that the condition is not violated. Taking equalities in (5.6) and summing, we obtain the following power allocation. For each k, 1 ≤ k ≤ M, ⎞ ⎛ Γ N0 W ⎝ 1+Γ ⎠ pk = (5.8) hk 1 − M(1 + ν) Γ 1+Γ

These powers are all positive when (5.7) holds, and, hence, we have a feasible power allocation (which meets the SINR constraints with equality). For this power allocation, by setting Q = M k=1 hk pk , we see that Γ M 1+Γ (N0 W) Q= Γ 1 − M (1 + ν) 1+Γ Thus, the condition expressed by (5.7) is found to be necessary and sufﬁcient for the existence of a feasible power control, in the present setting (i.e., at a BS, the uplink interference from MSs associated with other BSs can be modeled as a factor ν times the total power received at the BS from the MSs associated with it).

5.2 A Simple Case: One Call Class

133

Discussion a. From the previous derivation, we conclude that, in the single class case (with the spatial homogeneity assumptions we made), the following admission control will permit a feasible power allocation. A connection request is characterized by its “effective” resource requirement 1 +Γ Γ . The connection is added to the existing calls at a BS if and only if the following inequality is satisﬁed: Number of existing connections ×

Γ Γ 1 + < 1+Γ 1+Γ 1+ν

(5.9)

where ν is a spatial parameter that captures other cell interference. We will discuss how ν can be derived later in this chapter. b. We notice that a large value of Γ reduces the number of calls we can carry. How is the value of Γ determined? Suppose we wish to carry a new enhanced quality voice call, streaming audio call, or streaming video call using the CDMA access system just described. The source coding scheme that is used will determine the aggregate bit rate R that needs to carried. Also, sophisticated source coders will encode the source into bit streams of varying degrees of importance (called Class A, B, and C bits in some speech coders). When bit errors occur, a radio link layer protocol can recover the CDMA bursts containing the errored bits, but this recovery takes time, which adds to the end-to-end delay for the connection. After some number of attempts, bits may need to be discarded, in the hope that the decoder can reconstruct the speech or audio with some desirable quality using the received bits. It is thus clear that, for each coder, there will be a threshold bit error rate above which the speech (or audio or video) quality will not be acceptable. Finally, the physical layer (PHY) techniques employed (e.g., exploitation of multipath diversity (via a Rake receiver), interference cancellation, multiuser Eb detection) will determine the N , γ, required to provide the desired bit error 0 rate to the connection (see the discussions in Chapter 2). More sophisticated PHY techniques will result in a lower value of γ, hence a lower value of Γ Γ = γ RRc , and thus a lower resource requirement 1 + Γ for the connection. c. To get a feel for the numbers, let us consider telephone quality voice over the IS 95 CDMA system. A commonly used speech coder has R = 9.6 Kbps. The system bandwidth is 1.25 MHz, and the chip rate is 1.2288 Mcps. Thus 6 6 the processing gain is 1.2288×10 = 128 ≈ 21 dB (i.e., 10 log 1.2288×10 ≈ 21). 9.6×103 9.6×103 It turns out that, for the PHY techniques employed in the IS 95 standard, Eb for this speech coder is 6 dB. It follows that the target SINR, the target N 0 1 Γ = Rγc , is 6 − 21 = −15 dB (in fact, Γ = 32 ). The target SINR of −15 dB R

should be contrasted with narrowband systems such as FDM-TDMA (see Chapter 4) where the target SINR could be as high as 8 to 10 dB.

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5

Cellular CDMA

d. We can see that the interference Io = νQ can be reduced by exploiting voice activity detection; the voice call transmits only when carrying actual speech, and turns off during silence periods, thereby reducing the other cell interference for a given number of accepted calls. Note that, roughly, this will result in the factor 1 + ν getting multiplied by the voice activity factor (note that even the intracell interference reduces by the voice activity factor, hence we multiply 1 + ν by this factor). The voice activity factor is typically 0.4 to 0.5, and thus this technique results in the capacity being increased by a multiplicative factor of 2 to 2.5. 5.2.3 Other Cell Interference: Hard and Soft Handover Let us examine the form of the power allocation proposed in (5.8). We will now refer to a BS and the region in which MSs will normally associate with this BS as a cell. Notice that, in the one class case, with homogeneous interference at each cell, the received powers hk pk are all equal at every BS. Thus, when the entire system carries just one type of call (as is the case in the early deployment of all cellular telephony systems), then the powers of all MSs, in any cell, need to be controlled in such a way that the received powers at their respective BSs are all equal. If the power to be received at each BS from any MS has to be the same, then, in order that an MS uses the least transmit power it should associate with the geographically nearest BS (assuming only deterministic path loss proportional to an inverse power of the distance). For a location with coordinates (x, y) let rj (x, y) denote the distance of BS j from the location (x, y). We can say that the default coverage area of BS j is all (x, y) such that rj (x, y) < rk (x, y) for every other BS k. If this is done, then the coverage areas are actually so-called Voronoi cells, which are uniquely determined by the BS locations. We obtained the power allocation shown in (5.8), assuming that the other cell interference factor ν was somehow given. The power allocation actions in one cell, however, affect the other-cell interference seen by other cells. For example, if MS k is at the fringe of the coverage area of the BS with which it is associated then the value of hk will be small, thus requiring a large value of pk (see (5.8)). But this large value of pk will result in a higher level of other-cell interference at neighboring BSs. In fact, it is possible that the MS may have a better channel to a neighboring BS than to the one with which it is associated. If on the basis of this better channel to the neighboring BS the MS is handed over to that BS, then we say that we are performing soft handovers. On the other hand if the region is demarcated into coverage areas on the basis of path loss measurements, and MSs are associated with a BS so long as they are in its coverage area, then we say that we are performing hard handovers. We will carry out an interference analysis, assuming that all calls are of the same type, and hence (for a spatially homogeneous system, as assumed in our simple analysis earlier) the target received power from an MS is the same at every BS. This analysis will yield the value of ν for hard hand-off and for soft hand-off.

5.2 A Simple Case: One Call Class

135

With this we will have all the ingredients to perform a quantitative evaluation of the system capacity as given by (5.7). Let us ﬁrst consider hard hand-off. Let Sr denote the target uplink received power at a BS from any MS associated with it. In Figure 5.4 we show an MS at the location (x, y) in the coverage area of BS 1. The distance of the MS (located at (x, y)) to BS 1 is r1 (x, y), and to BS 0 is r0 (x, y). Modeling the power law path loss and shadowing, it can be seen that the interference power, say, S0 , at BS 0 due to the MS at location (x, y) is given by S0 = Sr

r1 (x, y) r0 (x, y)

η

10(ξ1 (x, y)+ζ(x, y))/10 10(ξ0 (x, y)+ζ(x, y))/10

where η is the path loss exponent, ξ1 (x, y), ξ0 (x, y), and ζ(x, y) are i.i.d. normally 2 distributed random variables with mean 0, and variance σ2 . Here, (ξ1 (x, y)+ζ(x, y)) correspond to the log-normal shadowing on the path to BS 1, and (ξ0 (x, y)+ζ(x, y)) to the log-normal shadowing on the path to BS 2. The shadowing is modeled as being composed of local shadowing around the MS, ζ(x, y), and the shadowing on the two different paths, ξ1 (x, y), ξ2 (x, y). The total shadowing standard deviation over each path is σ .

2

(x, y ) r1(x, y ) 1

3 r0(x, y)

0

6

4

5

Figure 5.4 Other-cell interference with hard hand-off. An MS at the location (x, y) is power controlled by BS 1, and the power it radiates causes uplink interference at BS 0.

136

5

Cellular CDMA

The previous expression can be understood as follows. Starting with the target power Sr at BS 1, we trace back to the MS to obtain its transmission power. This gives the numerator of the expression multiplying Sr . Then we obtain the interference power seen at BS 0. This is obtained by dividing by the channel attenuation along the path from (x, y) to BS 0. Note that the local shadowing terms cancel out, and, further, we assume that the distributions of ξ1 (x, y) and ξ0 (x, y) do not depend on the MS location (x, y). Then denoting these generic random variables by ξ1 and ξ0 , we get S0 = Sr

r1 (x, y) r0 (x, y)

η

10ξ1 /10 10ξ0 /10

Then the total expected other-cell interference at BS 0 is obtained by adding up the interference from all the other cell MSs and taking the expectation of this sum. This computation is done by assuming a uniform distribution of MSs over the coverage area, with density d MSs per unit area, and then integrating over the area outside of the cell covered by BS 0. This yields

ξ1 −ξ0 Io = Sr E 10 10

{(x, y)∈Cell / 0}

rBS (x, y) r0 (x, y)

η d dxdy

(5.10)

where rBS (x, y) denotes the distance of the location (x, y) from the BS in whose cell (x, y) lies. Clearly, the total power received at BS 0 from MSs associated with it is Q = Sr dA, where A is the area covered by a BS. It follows that ν=

ξ1 −ξ0 I0 rBS (x, y) η 1 dxdy = E 10 10 Q r0 (x, y) A {(x, y)∈Cell / 0}

It can be seen that the integral in the right-hand side of this expression does not vary with the cell radius, R. This integral can be numerically evaluated to approximately 0.44 for η = 4. Further, we observe that ln 10 ξ1 −ξ0 = E e 10 (ξ1 −ξ0 ) E 10 10

=e

σ2 2

ln 10 10

2

where we use the fact that ξ1 − ξ0 is normally distributed with mean 0 and variance σ 2 . For σ = 8 dB and η = 4, we then ﬁnd that ν = e(

σ 2 ln 10 2 2 ( 10 ) )

× 0.44 = 2.38. Thus, with an 8 dB standard deviation for the shadowing, and a path loss exponent of 4, the other-cell interference is 2.38 times the power received from MSs within the cell. We notice that with σ = 0 we have ν = 0.44, for η = 4.

5.2 A Simple Case: One Call Class

137

Let us now turn to the same analysis with soft handovers. Figure 5.5 depicts the concept. An MS at location (x, y) is power controlled by either BS 1 or BS 0. What this means is that the MS will use a transmit power that is the smaller of the two values required to achieve a received signal power of Sr at either of the two BSs. In the situation of random shadowing, this will result in the MS causing less interference than if it was dedicated to the more proximate of the two BSs. Thus, with random shadowing, an MS may get power controlled by a geographically farther away BS. For two neighboring BSs i and j (e.g., BS 1 and BS 0), and for a location (x, y) in the region where an MS chooses between either of them (e.g., (x, y) in Figure 5.5 is power controlled by BS 1 or BS 0), deﬁne αi, j (x, y) =

(ri (x, y))η 10ξi (x, y)/10 (rj (x, y))η 10ξj (x, y)/10

r1(x, y) 2

3

1 (x, y)

0 6

r0(x, y)

4

5

Figure 5.5 In soft hand-off, an MS is power controlled by the best of two or more BSs. This diagram shows an MS located at position (x, y) being power controlled by the best of BS 1 or BS 0. Each diamond shaped area, with a BS at each end of its long diagonal, shows the area in which an MS would be power controlled by either of those two BSs. By ♦i,j we will mean the diamond between BS i and BS j; as an illustration, ♦0,3 is shown shaded.

138

5

Cellular CDMA

where ri (x, y) and rj (x, y) are the distances of (x, y) from BS i and BS j, respectively, and ξi (x, y) (resp. ξj (x, y)) corresponds to log-normal shadowing near BS i (resp. BS j). As before, the distributions of these shadowing random variables will be taken to be independent of the location (x, y). Also ξi (x, y) and ξj (x, y) are assumed statistically independent in the following analysis. We can see that αi,j (x, y) is the relative attenuation from (x, y) to the BSs i and j; αi,j (x, y) > 1 implies that the power attenuation from the location (x, y) to BS i is larger (than that to BS j) and hence an MS located at the position (x, y) should be power controlled by j, since this will require the MS to use less transmission power. As before, let d be the density of mobiles per unit of the system coverage area. It can then be seen that the total power received at BS 0 (i.e., intracell power and other-cell interference) is given by 6

♦0, 1

Sr 1{α0, 1 (x,y)≤1} + Sr α1, 0 (x, y) 1{α1, 0 (x, y)

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Policy-Based Network Management: Solutions for the Next Generation John Strassner

Smart Phone and Next-Generation Mobile Computing Pei Zheng and Lionel Ni

MPLS Network Management: MIBs, Tools, and Techniques Thomas D. Nadeau

Developing IP-Based Services: Solutions for Service Providers and Vendors Monique Morrow and Kateel Vijayananda Telecommunications Law in the Internet Age Sharon K. Black

Internetworking Multimedia Jon Crowcroft, Mark Handley, and Ian Wakeman Understanding Networked Applications: A First Course David G. Messerschmitt

Internet QoS: Architectures and Mechanisms Zheng Wang

Integrated Management of Networked Systems: Concepts, Architectures, and Their Operational Application Heinz-Gerd Hegering, Sebastian Abeck, and Bernhard Neumair

TCP/IP Sockets in Java: Practical Guide for Programmers Michael J. Donahoo and Kenneth L. Calvert

Virtual Private Networks: Making the Right Connection Dennis Fowler

TCP/IP Sockets in C: Practical Guide for Programmers Kenneth L. Calvert and Michael J. Donahoo

Networked Applications: A Guide to the New Computing Infrastructure David G. Messerschmitt

Multicast Communication: Protocols, Programming, and Applications Ralph Wittmann and Martina Zitterbart

Wide Area Network Design: Concepts and Tools for Optimization Robert S. Cahn

Optical Networks: A Practical Perspective, 2e Rajiv Ramaswami and Kumar N. Sivarajan

MPLS: Technology and Applications Bruce Davie and Yakov Rekhter High-Performance Communication Networks, 2e Jean Walrand and Pravin Varaiya

For further information on these books and for a list of forthcoming titles, please visit our website at http://www.mkp.com

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WIRELESS NETWORKING Anurag Kumar D. Manjunath Joy Kuri

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Morgan Kaufmann Publishers is an imprint of Elsevier

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Copyright © 2008, Anurag Kumar, D. Manjunath and Joy Kuri. Published by Elsevier Inc. All rights reserved. The right of author names to be identiﬁed as the authors of this work have been asserted in accordance with the copyright, Designs and Patents Act 1988. Designations used by companies to distinguish their products are often claimed as trademarks or registered trademarks. In all instances in which Morgan Kaufmann Publishers is aware of a claim, the product names appear in initial capital or all capital letters. Readers, however, should contact the appropriate companies for more complete information regarding trademarks and registration. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means—electronic, mechanical, photocopying, scanning, or otherwise—without prior written permission of the publisher. Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (+44) 1865 843830, fax: (+44) 1865 853333, E-mail: [email protected] You may also complete your request online via the Elsevier homepage (http://elsevier.com), by selecting “Support & Contact” then “Copyright and Permission” and then “Obtaining Permissions.” Library of Congress Cataloging-in-Publication Data Kumar, Anurag. Wireless networking / Anurag Kumar, D. Manjunath, Joy Kuri. p. cm. – (The Morgan Kaufmann series in networking) Includes bibliographical references. ISBN 0-12-374254-4 1. Wireless LANs. 2. Wireless communication systems. 3. Sensor networks. I. Manjunath, D. II. Kuri, Joy. III. Title. TK5105.78.K86 2008 621.384–dc22 2007053011 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-374254-4 For information on all Morgan Kaufmann publications, visit our website at www.mkp.com or www.books.elsevier.com Printed and bound in the United States of America 08 09 10 11 12

5 4 3 2 1

Contents Preface

xiii

1 Introduction

1

1.1

Networking as Resource Allocation

1

1.2

A Taxonomy of Current Practice

3

1.3

Technical Elements

9

1.4

Summary and Our Way Forward

2 Wireless Communication: Concepts, Techniques, Models 2.1

2.2

12 15

Digital Communication over Radio Channels

16

2.1.1

Simple Binary Modulation and Detection

17

2.1.2

Getting Higher Bit Rates

20

2.1.3

Channel Coding

23

2.1.4

Delay, Path Loss, Shadowing, and Fading

25

Channel Capacity

32

2.2.1

Channel Capacity without Fading

32

2.2.2

Channel Capacity with Fading

35

2.3

Diversity and Parallel Channels: MIMO

36

2.4

Wideband Systems

42

2.4.1

CDMA

42

2.4.2

OFDMA

45

2.5 Additional Reading 3 Application Models and Performance Issues

48 53

3.1

Network Architectures and Application Scenarios

54

3.2

Types of Trafﬁc and QoS Requirements

56

3.3

Real-Time Stream Sessions: Delay Guarantees

60

3.3.1

CBR Speech

60

3.3.2

VBR Speech

61

3.3.3

Speech Playout

63

viii

Contents

3.4

3.5

3.3.4

QoS Objectives

65

3.3.5

Network Service Models

67

Elastic Transfers: Feedback Control

67

3.4.1

Dynamic Control of Bandwidth Sharing

69

3.4.2

Control Mechanisms: MAC and TCP

70

3.4.3

TCP Performance over Wireless Links

72

Notes on the Literature

4 Cellular FDM-TDMA

78 81

4.1

Principles of FDM-TDMA Cellular Systems

81

4.2

SIR Analysis: Keeping Cochannel Cells Apart

86

4.3

Channel Reuse Analysis: Hexagonal Cell Layout

92

4.3.1

Cochannel Cell Groups

93

4.3.2

Calculating Nreuse

94

4.3.3

D Ratio: Simple Analysis, Cell Sectorization R

96

4.4

Spectrum Efﬁciency

4.5

Channel Allocation and Multicell Erlang Models

4.6

99 101

4.5.1

Reuse Constraint Graph

101

4.5.2

Feasible Carrier Requirements

103

4.5.3

Carrier Allocation Strategies

103

4.5.4

Call Blocking Analysis

104

4.5.5

Comparison of FCA and MPA

106

Handovers: Techniques, Models, Analysis

112

4.6.1

Analysis of Signal Strength Based Handovers

112

4.6.2

Handover Blocking, Call Dropping: Channel Reservation

115

4.7

The GSM System for Mobile Telephony

117

4.8

Notes on the Literature

119

5 Cellular CDMA

125

5.1 The Uplink SINR Inequalities

126

5.2 A Simple Case: One Call Class

130

5.2.1

Example: Two BSs and Collocated MSs

130

5.2.2

Multiple BSs and Uniformly Distributed MSs

131

5.2.3

Other Cell Interference: Hard and Soft Handover

134

5.2.4

System Capacity for Voice Calls

139

5.3 Admission Control of Multiclass Calls

140

5.3.1

Hard and Soft Admission Control

141

5.3.2

Soft Admission Control Using Chernoff’s Bound

141

5.4 Association and Power Control for Guaranteed QoS Calls

145

Contents

ix

5.5

Scheduling Elastic Transfers

149

5.6

CDMA-Based 2G and 3G Cellular Systems

154

5.7

Notes on the Literature

155

5.8

Appendix: Perron-Frobenius Theory

156

6 Cellular OFDMA-TDMA

161

6.1

The General Model

162

6.2

Resource Allocation over a Single Carrier

163

6.2.1

Power Control for Optimal Service Rate

165

6.2.2

Power Control for Optimal Power Constrained Delay

171

6.3

Multicarrier Resource Allocation: Downlink

178

6.3.1

Single MS Case

178

6.3.2

Multiple MSs

181

6.4

WiMAX: The IEEE 802.16 Broadband Wireless Access Standard

183

6.5

Notes on the Literature

183

7 Random Access and Wireless LANs 7.1 7.2

7.3

187

Preliminaries

188

Random Access: From Aloha to CSMA

189

7.2.1

Protocols without Carrier Sensing: Aloha and Slotted Aloha

190

7.2.2

Carrier Sensing Protocols

199

CSMA/CA and WLAN Protocols

201

7.3.1

Principles of Collision Avoidance

201

7.3.2

The IEEE 802.11 WLAN Standards

204

7.3.3

HIPERLAN

211

7.4

Saturation Throughput of a Colocated IEEE 802.11-DCF Network

213

7.5

Service Differentiation and IEEE 802.11e WLANs

222

7.6

Data and Voice Sessions over 802.11

225

7.6.1

Data over WLAN

226

7.6.2

Voice over WLAN

230

7.7

Association in IEEE 802.11 WLANs

234

7.8

Notes on the Literature

235

8 Mesh Networks: Optimal Routing and Scheduling 8.1 8.2

243

Network Topology and Link Activation Constraints

244

8.1.1

Link Activation Constraints

244

Link Scheduling and Schedulable Region

247

8.2.1

Stability of Queues

250

8.2.2

Link Flows and Link Stability Region

254

x

Contents 8.3

Routing and Scheduling a Given Flow Vector

257

8.4

Maximum Weight Scheduling

264

8.5

Routing and Scheduling for Elastic Trafﬁc

273

8.5.1

Fair Allocation for Single Hop Flows

277

8.5.2

Fair Allocation for Multihop Flows

280

8.6

Notes on the Literature

9 Mesh Networks: Fundamental Limits 9.1

9.2

9.3

287 291

Preliminaries

292

9.1.1

Random Graph Models for Wireless Networks

293

9.1.2

Spatial Reuse, Network Capacity, and Connectivity

296

Connectivity in the Random Geometric Graph Model

297

9.2.1

Finite Networks in One Dimension

298

9.2.2

Networks in Two Dimensions: Asymptotic Results

302

Connectivity in the Interference Model

309

9.4

Capacity and Spatial Reuse Models

315

9.5

Transport Capacity of Arbitrary Networks

318

9.6

9.7

Transport Capacity of Randomly Deployed Networks

322

9.6.1

Protocol Model

322

9.6.2

Discussion

331

Notes on the Literature

10 Ad Hoc Wireless Sensor Networks (WSNs)

333 337

10.1 Communication Coverage

339

10.2 Sensing Coverage

341

10.3 Localization

348

10.4 Routing

353

10.5 Function Computation

359

10.6 Scheduling

368

10.6.1 S-MAC

369

10.6.2 IEEE 802.15.4 (Zigbee)

370

10.7 Notes on the Literature

Appendices A Notation and Terminology A.1 Miscellaneous Operators and Mathematical Notation A.2 Vectors and Matrices A.3

Asymptotics: The O, o, and ∼ Notation

A.4 Probability

372

375 377 377 377 377 379

Contents

xi

B A Review of Some Mathematical Concepts

381

B.1 Limits of Real Number Sequences

381

B.2 A Fixed Point Theorem

382

B.3 Probability and Random Processes

382

B.3.1

Useful Inequalities and Bounds

382

B.3.2

Convergence Concepts

384

B.3.3

The Borel-Cantelli Lemma

385

B.3.4

Laws of Large Numbers and Central Limit Theorem

385

B.3.5

Stationarity and Ergodicity

386

B.4 Notes on the Literature C Convex Optimization

387 389

C.1 Convexity

389

C.2 Local and Global Optima

389

C.3 The Karush-Kuhn-Tucker Conditions

390

C.4 Duality

391

D Discrete Event Random Processes

393

D.1 Stability Analysis of Discrete Time Markov Chains (DTMCs)

393

D.2 Continuous Time Markov Chains

394

D.3 Renewal Processes

398

D.3.1 Renewal Reward Processes

398

D.3.2 The Excess Distribution

399

D.3.3 Markov Renewal Processes

399

D.4 Some Topics in Queuing Theory

401

D.4.1 Little’s Theorem

401

D.4.2 Poisson Arrivals See Time Averages (PASTA)

402

D.5 Some Important Queuing Models

403

D.5.1 The M/G/c/c Queue

403

D.5.2 The Processor Sharing Queue

404

D.6 Notes on the Literature

405

Bibliography

407

Index

417

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Preface Another Book on Wireless Networking? The availability of high performance, low power, and low cost digital signal processors, and advances in digital communication techniques over the radio frequency spectrum have resulted in the widespread availability of wireless network technology for mass consumption. Several excellent books are now available that deal with the area of wireless communications, where topics of recent interest include multiple-input-multiple-output (MIMO) systems, space time coding, orthogonal frequency division multiplexing (OFDM), and multiuser detection. Wireless networks are best known in the context of ﬁrst- and secondgeneration mobile telephony (AT&T’s analog AMPS system in the ﬁrst generation, and the GSM and CDMA digital systems in the second generation). There are books that provide coverage of such wireless networks, and also those that combine a comprehensive treatment of physical layer wireless communication with that of cellular networks. In the last decade, however, there has been an explosion in the development and deployment of new wireless network technologies, and in the conceptualization of, and research in, a variety of newer ones. From the ubiquitous WiFi coffee shop and airport networks to the emerging WiMAX systems, which promise broadband wireless access to mobile users, the menu of wireless access networks promises to become so comprehensive that wired access from user devices may soon become a relic of the past. Research on wireless mesh networks (so-called ad hoc wireless networks), which started in the 1970s, is being pursued with renewed vigor due to the availability of inexpensive and interoperable mobile wireless devices. In addition, the widespread use of wireless sensor networks (in conjunction with emerging standards such as Zigbee and IEEE 802.15.4) is a clear and present possibility. Thus the variety and scale of wireless networks is unprecedented, and, in teaching courses in our institutions, we have felt the need for a comprehensive analytical treatment of wireless networking, keeping in mind the technical developments in the past, the present, and the future. This book is the outcome of our efforts to address this need. The foremost aspect of networking, wireline or wireless, is the design of efﬁcient protocols that work. Taking the view that the devil is in the detail, protocols with “working code” often gain widespread acceptance. With the increasing variety in networks and applications, and also in their scale, complex interactions (e.g., between devices using a particular protocol, or between protocols at the various layers) need to be understood. Although computer

xiv

Preface

simulation is a useful vehicle for understanding the performance of protocols, it is not always sufﬁcient, because, once again, the devil is in the detail. The assumptions made in deriving simulation models play an important role in the results that are obtained. If a simulation program simply encodes the standard, then running the simulation only provides a plethora of numbers, with no new insights being gained. Further, large simulation models, although possibly closer to reality, take a lot of effort to develop and debug, and are slow to execute, thus rendering them not very useful in the early stages of experimentation with algorithms. This is where analytical models become very important. First, the process of deriving such models from the standards, or from system descriptions, provides very useful insights. Second, the analytical models can be used to help verify large simulation programs, by providing exact results for subcases of the model being simulated. Third, research in analytical modeling is necessary to develop models that can be programmed into simulators, so as to increase simulation speed. Finally, the analytical approach is very important for the development of new and efﬁcient protocols, and there is a trend toward optimization via reverse engineering of well-accepted protocols. In addition to the variety of networks and protocols that need to be understood, there is a large body of fundamental results on wireless networks that have been developed over the last ﬁfteen years that give important insights into optimal design and the limits of performance. Examples of such results include distributed power control in CDMA networks, optimal scheduling in wireless networks (with a variety of optimization objectives involving issues such as network stability, performance, revenue, and fairness), transmission range thresholds for connectivity in a wireless mesh network, and the transport capacity of these networks. Further, the imminence of sensor networks has generated a large class of fundamental problems in the areas of stochastic networks and distributed algorithms that are intrinsically important and interesting. This book aims (1) to provide an analytical perspective on the design and analysis of the traditional and emerging wireless networks, and (2) to discuss the nature of, and solution methods to, the fundamental problems in wireless networking. For the sake of completeness, traditional voice telephony over GSM and CDMA wireless access networks also is covered. The approach is via various resource allocation models that are based on simple models of the underlying physical wireless communication.

About the Book and the Viewpoint After the speciﬁcation of the protocols and the veriﬁcation of their correctness, we believe that networking is about resource allocation. In wireless networks, the resources are typically spectrum, time, and power. That theme pervades much of this book in our quest for models for performance analysis, for developing design insights, and also for exploring the fundamental limits. Once a problem has been analytically formulated, we draw upon a wide variety of techniques

Preface

xv

to analyze it. In this process we will use techniques drawn from, among others, probability theory, stochastic processes, constrained optimization and duality, and graph theory. We believe it is necessary to make forays into these areas in order to bring their power to bear on the problem at hand. However, we have attempted to make the book as self-contained as possible. Wherever possible, we have used only elementary concepts taught in basic courses in engineering mathematics. A brief overview of most of the advanced mathematical material that we use is provided in the appendix. Also, wherever possible we have avoided the theorem–proof approach. Instead, we have developed the theorems or results and then formally stated them. After the introductory chapter, we begin the presentation of the main material of the book in Chapter 2 by giving an overview of the physical layer issues that are so much more important to understand wireless networks than they are for wireline networks. Wireless networks are viewed as being either access networks or mesh networks. In access networks mobile wireless nodes connect to an infrastructure node, and in mesh networks they form an independent internet and may or may not connect to an infrastructure network. Access networks are covered in Chapters 4 through 7 and mesh networks are covered in Chapters 8 through 10. The wireless networking aspect of the book begins in Chapter 3. Like in our earlier book, Communication Networking: An Analytical Approach, we precede the discussion on access networks by listing the issues and setting the performance objectives of a wireless network in Chapter 3. FDM-TDMA cellular networks (of which GSM networks are a major example) are discussed in Chapter 4, with the focus on signal-to-interference ratio analysis, on channel allocation, and on the call blocking and call dropping performance. Chapter 5 is on CDMA networks where the main emphasis is on interference management via power allocation. Whereas the trafﬁc model in Chapter 4 and in much of Chapter 5 is an arrival process of calls, each with a rate requirement, in Chapter 6, on OFDMA access networks, we consider buffered models, and discuss power allocation over time and over carriers with the objectives of stability and mean delay. In Chapter 7, we discuss the performance of distributed allocation of channel time in wireless LANs. We begin our discussion of mesh networks in Chapter 8 by considering optimal routing and scheduling in a given mesh network. One can view this class of problems as the optimal allocation of time and space in a network. In Chapter 9 we explore fundamental limits of this time and space allocation to the ﬂows. Chapter 10 is on the emerging area of sensor networks, a rich ﬁeld of research issues including connectivity and coverage properties of stochastic networks, and distributed computation. Some of the material in Chapter 5 and most of the material in Chapters 6 through 10 are being covered in a wireless networking textbook for the ﬁrst time. We have not obtained new results for the book but we have trawled the literature to pick out the fundamental results and those that are illustrative of the issues

xvi

Preface

and complexities. Wherever possible, we have simpliﬁed the models for pedagogic convenience.

Using the Book This is a graduate text, though a ﬁnal year undergraduate course could be supplemented with material from this text. Some understanding of networking concepts is assumed. A quick introduction may also be obtained from Chapter 2 of our earlier book, Communication Networking: An Analytical Approach. Most of the chapters are self-contained and we believe that an instructor can pick and choose the chapters. A course that needs to cover voice and data access networks (including cellular networks and wireless LANs) could be based on Chapters 4 through 7. One can say that these chapters are tied closely to real networks. Chapters 8 through 10 are of a more fundamental and abstract nature. A course with a more current research emphasis could be built around Chapters 6 through 10. The publisher maintains a website for this book at www.mkp.com. We maintain a website for the book at ece.iisc.ernet.in/∼anurag/books. These websites contain errata, additional problems, PostScript ﬁles of the ﬁgures used in the book, and other instructional material. An instructor’s manual containing solutions to all the exercises and problems and some supplementary problems is also available from the authors. Arthur Clarke had said that the communications satellite will make inevitable the United Nations of the Earth. Wireless communication and networking are making these United Nations ﬂatter, and possibly more democratic with unbridled opportunities for all. So let’s “unwire, cut the cord, and go wireless.” And, while we do it, let us step back a bit and understand them from the ground up!

Acknowledgments We are grateful to Onkar Dabeer and P.R. Kumar who reviewed the complete manuscript and provided us invaluable comments and criticisms. Saswati Sarkar was visiting us on her sabbatical when we were writing this book; her insightful comments on several chapters of the book have helped immensely in improving the accuracy of the material. Prasanna Chaporkar, N. Hemachandra, U. Jayakrishnan, Koushik Kar, Biplab Sikdar, Chandramani Kishore Singh, and Rajesh Sundaresan read various chapters of the book at our request. They found many errors and rough edges, and we remain grateful for their time and efforts. Many students helped with reviewing the problems and verifying their solutions; these include Onkar Bharadwaj, Avhishek Chatterjee, Sudeep Kamath, Pallavi Manohar, and K. Premkumar. We would like to thank Chandrika Sridhar, our ever-helpful lab secretary in [email protected], who typed parts of an initial draft from the lecture notes of the ﬁrst author, and also prepared all the problems and solutions, both from handwritten manuscripts.

Preface

xvii

This book has been developed out of the two-part survey article “A Tutorial Survey of Topics in Wireless Networking,” by Anurag Kumar and D. Manjunath, published in Sa¯ dhana¯ , Indian Academy of Sciences Proceedings in Engineering Sciences, Vol. 32, No. 6, December 2007. We are grateful to the publishers of Sa¯ dhana¯ for permitting us to use several extracts and ﬁgures from our survey article. Parts of an early draft of this book have been used by Ed Knightly (Rice University), and Utpal Mukherji (Indian Institute of Science). We hope that the ﬁnished version will meet their expectations for the courses they teach. Book writing grants have been provided by the Centre for Continuing Education of the Indian Institute of Science to the ﬁrst author and by the Curriculum Development Program of the Indian Institute of Technology, Bombay, to the second author. Finally, we are grateful to our families for bearing patiently our absence from the regular call of duty at home, in the evenings, holidays, and weekends during the several months over which this book was developed. Anurag Kumar I.I.Sc., Bangalore

D. Manjunath I.I.T., Bombay

Joy Kuri I.I.Sc., Bangalore

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CHAPTER 1 Introduction

T

he idea of sending information over radio waves (i.e., wireless communication) is over a hundred years old. When several devices with radio transceivers share a portion of the radio spectrum to send information to each other, we say that we have a wireless communication network, or simply a wireless network. In this chapter we begin by developing a three-layered view of wireless networks. We delineate the subject matter of this book—that is, wireless networking—as dealing with the problem of resource allocation when several devices share a portion of the RF spectrum allocated to them. Next, we provide a taxonomy of current wireless networks. The material in the book is organized along this taxonomy. Then, in this chapter, we identify the common basic technical elements that underlie any wireless network as being (1) physical wireless communication; (2) neighbor discovery, association, and topology formation; and (3) transmission scheduling. Finally, we provide an overview of the contents of the remaining nine chapters of the book.

1.1

Networking as Resource Allocation

Following our viewpoint in [89] we view wireline and wireless communication networks in terms of the three-layered model shown in Figure 1.1. Networks carry the ﬂows of information between distributed applications such as telephony, teleconferencing, media-sharing, World Wide Web access, e-commerce, and so on. The points at which distributed information applications connect to the generators and absorbers of information ﬂows can be viewed as sources and sinks of trafﬁc (see Figure 1.1). Examples of trafﬁc sources are microphones in telephony devices, video cameras, and data, voice, or video ﬁles (stored on a computer disk) that are being transmitted to another location. Examples of trafﬁc sinks are telephony loudspeakers, television monitors, or computer storage devices. As shown in Figure 1.1, the sources and sinks of information and the distributed applications connect to the communication network via common information services. The information services layer comprises all the hardware and software required to facilitate the necessary transport services, and to attach the sources or sinks to the wireless network; for example, voice coding, packet buffering and playout, and voice decoding, for packet telephony; or similar

2

1

Introduction

sources and/or sinks distributed applications

Information Services

Resource Allocation

Resource Allocation

Algorithms

Algorithms

Shared Radio Spectrum

Wireline Bit Carrier Infrastructure

(a portion of the RF spectrum)

Figure 1.1 A conceptual view of distributed applications utilizing wireline and wireless networks. Wireless networking is concerned with algorithms for resource allocation between devices sharing a portion of the radio spectrum. On the other hand, in wireline networks the resource allocation algorithms are concerned with sharing the ﬁxed resources of a bit transport infrastructure.

facilities for video telephony or for streaming video playout; or mail preparation and forwarding software for electronic mail; or a browser for the World Wide Web. We turn now to the bottom layer in Figure 1.1. In wireline networks the information to be transported between the endpoints of applications is carried over a static bit-carrier infrastructure. These networks typically comprise highquality digital transmission systems over copper or optical media. Once such links are properly designed and conﬁgured, they can be viewed as “bit pipes,” each with a certain bit rate, and usually a very small bit error rate. The bit carrier infrastructure can be dynamically reconﬁgured on the basis of trafﬁc demands, and such actions are a part of the cloud labeled “resource allocation algorithms” in the ﬁgure. The left side of the bottom layer in Figure 1.1 corresponds to wireless networks. Typically, each wireless network system is constrained to operate in some portion of the RF spectrum. For example, a cellular telephony system may be assigned 5 MHz of spectrum in the 900 MHz band. Information bits are transported between devices in the wireless network by means of some physical wireless communication technique (i.e., a PHY layer technique, in terms of the ISO-OSI model) operating in the portion of the RF spectrum that is assigned to the network. It is well known, however, that unguided RF communication between mobile wireless devices poses challenging problems. Unlike wireline communication, or even point-to-point, high-power microwave links between dish antennas mounted on tall towers, digital wireless communication between mobile devices has to deal with a variety of time-varying channel impairments

1.2 A Taxonomy of Current Practice

3

such as obstructions by steel and concrete buildings, absorption in partition walls or in foliage, and interference between copies of a signal that traverse multiple paths. In order to combat these problems, it is imperative that in a mobile or ad hoc wireless network the PHY layer should be adaptable. In fact, in some systems multiple modulation schemes are available, and each of these may have variable parameters such as the error control codes, and the transmitter powers. Hence, unlike a wired communication network, where we can view networking as being concerned with the problems of resource sharing over a static bit carrier infrastructure, in wireless networking, the resource allocation mechanisms would include these adaptations of the PHY layer. Thus, in Figure 1.1 we have actually “absorbed” the physical wireless communication mechanisms into the resource allocation layer. Hence, we can deﬁne our view of wireless networking as being concerned with all the mechanisms, procedures, or algorithms for efﬁcient sharing of a portion of the radio spectrum so that all the instances of communication between the various devices obtain their desired quality of service (QoS).

1.2 A Taxonomy of Current Practice In this book, instead of pursuing an abstract, technology agnostic approach, we will develop an understanding of the various wireless networking techniques in the context of certain classes of wireless networks as they exist today. Thus we begin our treatment by taking a look at a taxonomy of the current practice of wireless networks. Figure 1.2 provides such a taxonomy. Several commonly used terms of the technology will arise as we discuss this taxonomy. These will be highlighted by the italic font, and their meanings will be clear from the context. Of course, the attendant engineering issues will be dealt with at length in the remainder of the book. Fixed wireless networks include line-of-sight microwave links, which until recently were very popular for long distance transmission. Such networks basically comprise point-to-point line-of-sight digital radio links. When such links are set up, with properly aligned high gain antennas on tall masts, the links can be viewed as point-to-point bit pipes, albeit with a higher bit error rate than wired links. Thus in such ﬁxed wireless networks no essentially new issues arise than in a network of wired links. On the other hand the second and third categories shown in the ﬁrst level of the taxonomy (i.e., access networks and ad hoc networks) involve multiple access where, in the same geographical region, several devices share a radio spectrum to communicate among themselves (see Figure 1.3). Currently, the most important role of wireless communications technology is in mobile access to wired networks. We can further classify such access networks into two categories: one in which resource allocation is more or less static (akin to circuit multiplexing), and the other in which the trafﬁc is statistically multiplexed, either in a centralized manner or by distributed mechanisms.

4

1

Introduction

wireless networks

fixed networks

ad hoc networks (wireless mesh networks)

mobile access networks

circuit multiplexing (e.g., GSM cellular networks)

centralized statistical multiplexing (e.g., CDMA cellular networks: IS 95, cdma2000, WCDMA; and IEEE 802.16 “WIMAX” networks)

distributed statistical multiplexing (e.g., IEEE 802.11 WLANs)

wireless internets

sensor networks

Figure 1.2 A taxonomy of wireless networks.

WLAN AP

Wireline Network

base stations Cellular Network

Figure 1.3 The left panel shows some access networks (a cellular telephony network, and a wireless local area network (WLAN), where the access is via an AP (access point)), and the right panel shows a mesh wireless network of portable computers.

1.2 A Taxonomy of Current Practice

5

Cellular wireless networks were introduced in the early 1980s as a technology for providing access to the wired phone network to mobile users. The network coverage area is partitioned into regions (with diameters ranging from hundreds of meters to a few kilometers) called cells, hence the term “cellular.” In each cell there is a base station (BS), which is connected to the wired network, and through which the mobile devices in the cell communicate over a one hop wireless link. The cellular systems that have the most widespread deployment are the ones that share the available spectrum using frequency division multiplexed time division multiple access (FDM-TDMA) technology. Among such systems by far the most commercially successful has been the GSM system, developed by a European consortium. The available spectrum is ﬁrst partitioned into a contiguous up-link band and another contiguous down-link band. Each of these bands is statically or dynamically partitioned into reuse subbands, with each cell being allocated such a subband (this is the FDM aspect). The partitioning of the up-link and down-link bands is done in a paired manner so that each cell is actually assigned a pair of subbands. Each subband is further partitioned into channels or carriers (also an FDM aspect), each of which is digitally modulated and then slotted in such a way that a channel can carry up to a ﬁxed number of calls (e.g., 8 calls) in a TDM fashion. Each arriving call request in a cell is then assigned a slot in one of the carriers in that cell; of course, a pair of slots is assigned in paired up-link and down-link channels in that cell. Thus, since each call is assigned dedicated resources, the system is said to be circuit multiplexed, just like the wireline phone network. These are narrowband systems (i.e., users’ bit streams occupy frequency bands just sufﬁcient to carry them), and the radio links operate at a high signal-to-interference-plus-noise-ratio (SINR), and hence careful frequency planning (i.e., partitioning of the spectrum into reuse subbands, and allocation of the subbands to the cells) is needed to avoid cochannel interference. The need for allocation of frequency bands over the network coverage area (perhaps even dynamic allocation over a slow timescale), and the grant and release of individual channels as individual calls arrive and complete, requires the control of such systems to be highly centralized. Note that call admission control, that is, call blocking, is a natural requirement in an FDM-TDMA system, since the resources are partitioned and each connection is assigned one resource unit. Another cellular technology that has developed over the past 10 to 15 years is the one based on code division multiple access (CDMA). In these networks, the entire available spectrum is reused in every cell. These are broadband systems, which means that each user’s bit stream (a few kilobits per second) occupies the entire available radio spectrum (a few megahertz). This is done by spreading each user’s signal over the entire spectrum by multiplying it by a pseudorandom sequence, which is allocated to the user. This makes each user’s signal appear like noise to other users. The knowledge of the spreading sequences permits the receivers to separate the users’ signals, by means of correlation receivers. Although no frequency planning is required for CDMA systems, the performance is interference limited as every transmitted signal is potentially an interferer for

6

1

Introduction

every other signal. Thus at any point of time there is an allocation of powers to all the transmitters sharing the spectrum, such that their desired receivers can decode their transmissions, in the presence of all the cross interferences. These desired power levels need to be set depending on the locations of the users, and the consequent channel conditions between the users and the base stations, and need to be dynamically controlled as users move about and channel conditions change. Hence tight control of transmitter power levels is necessary. Further, of course, the allocation of spreading codes, and management of movement between cells needs to done. We note that, unlike the FDM-TDMA system described earlier, there is no dedicated allocation of resources (frequency and time-slot) to each call. Indeed, during periods when a call is inactive no radio resources are utilized, and the interference to other calls is reduced. Thus, we can say that the trafﬁc is statistically multiplexed. If there are several calls in the system, each needing certain quality of service (QoS) (bit rate, maximum bit error rate), then the number of calls in the system needs to be controlled so that the probability of QoS violation of the calls is kept small. This requires call admission control, which is an essential mechanism in CDMA systems, in order that QoS objectives can be achieved. Evidently, these are all centrally coordinated activities, and hence even CDMA cellular systems depend on central intelligence that resides in the base station controllers (BSCs). Until recently, cellular networks were driven primarily by the needs of circuit multiplexed voice telephony; on demand, a mobile phone user is provided a wireless digital communication channel on which is carried compressed telephone quality (though not “toll” quality) speech. Earlier, we have described two technologies for second generation (2G) cellular wireless telephony. Recently, with the growing need for mobile Internet access, there have been efforts to provide packetized data access on these networks as well. In the FDM-TDMA systems, low bit rate data can be carried on the digital channel assigned to a user. As is always the case in circuit multiplexed networks, ﬂexibility in the allocation of bandwidth is limited to assigning multiple channels to each user. Such an approach is followed in the GSM-GPRS (General Packet Radio Service) system, where, by combining multiple TDM slots on an FDM carrier, shared packet switched access is provided to mobile users. A further evolution is the EDGE (Enhanced Data rates for GSM Evolution) system, where, in addition to combining TDM slots, higher order modulation schemes, with adaptive modulation, are utilized to obtain shared packet switched links with speeds up to 474 Kbps. These two systems often are viewed, respectively, as 2.5G and 2.75G evolutions of the GSM system. These are data evolutions of an intrinsically circuit switched system that was developed for mobile telephony. On the other hand there is considerable ﬂexibility in CDMA systems where there is no dedicated allocation of resources (spectrum or power). In fact, both voice and data can be carried in the packet mode, with the user bit rate, the amount of spreading, and the allocated power changing on a packetby-packet basis. This is the approach taken for the third generation (3G) cellular systems, which are based entirely on CDMA technology, and are meant to carry multimedia trafﬁc (i.e., store and forward data, packetized telephony, interactive

1.2 A Taxonomy of Current Practice

7

video, and streaming video). The most widely adopted standard for 3G systems is WCDMA (wideband CDMA), which was created by the 3G Partnership Project (3GPP), a consortium of standardization organizations from the United States, Europe, China, Japan, and Korea. Cellular networks were developed with the primary objective of providing wireless access for mobile users. With the growth of the Internet as the de facto network for information dissemination, access to the Internet has become an increasingly important requirement in most countries. In large congested cities, and in developing countries without a good wireline infrastructure, ﬁxed wireless access to the Internet is seen as a signiﬁcant market. It is with such an application in mind that the IEEE 802.16 standards were developed, and are known in the industry as WiMAX. The major technical advance in WiMAX is in the adoption of several high performance physical layer (PHY) technologies to provide several tens of Mbps between a base station (BS) and ﬁxed subscriber stations (SS) over distances of several kilometers. The PHY technologies that have been utilized are orthogonal frequency division multiple access (OFDMA) and multiple antennas at the transmitters and the receivers. The latter are commonly referred to as MIMO (multiple-input-multiple-output) systems. In an OFDMA system, several subchannels are statically deﬁned in the system bandwidth, and these subchannels are digitally modulated. In order to permit up-link and down-link transmissions, time is divided into frames and each frame is further partitioned into an up-link and a down-link part (this is called time division duplexing (TDD)). The BS allocates time on the various subchannels to various down-link ﬂows in the down-link part of the frame and, based on SS requests, in the up-link part of the frame. This kind of TDD MAC structure has been used in several earlier systems; for example, satellite networks involving very small aperture satellite terminals (VSATs), and even in wireline systems such as those used for the transmission of digital data over cable television networks. WiMAX speciﬁcations now have been extended to include broadband access to mobile users. We now discuss the third class of networks in the mobile access category in the ﬁrst level of the taxonomy shown in Figure 1.2—distributed packet scheduling. Cellular networks have emerged from centrally managed point-to-point radio links, but another class of wireless networks has emerged from the idea of random access, whose prototypical example is the Aloha network. Spurred by advances in digital communication over radio channels, random access networks can now support bit rates close to desktop wired Ethernet access. Hence random access wireless networks are now rapidly proliferating as the technology of choice for wireless Internet access with limited mobility. The most important standards for such applications are the ones in the IEEE 802.11 series. Networks based on this standard now support physical transmission speeds from a few Mbps (over 100s of meters) up to 100 Mbps (over a few meters). The spectrum is shared in a statistical TDMA fashion (as opposed to slotted TDMA, as discussed, earlier, in the context of ﬁrst generation FDM-TDMA systems). Nodes contend for the channel, and possibly collide. In the event of a collision, the colliding nodes back

8

1

Introduction

off for independently sampled random time durations, and then reattempt. When a node is able to acquire the channel, it can send at the highest of the standard bit rates that can be decoded, given the channel condition between it and its receiver. This technology is predominantly deployed for creating wireless local area networks (WLANs) in campuses and enterprise buildings, thus basically providing a one hop untethered access to a building’s Ethernet network. In the latest enhancements to the IEEE 802.11 standards, MIMO-OFDM physical layer technologies are being employed in order to obtain up to 100 Mbps transmission speeds in indoor environments. With the widespread deployment of IEEE 802.11 WLANs in buildings, and even public spaces (such as shopping malls and airports), an emerging possibility is that of carrying interactive voice and streaming video trafﬁc over these networks. The emerging concept of fourth-generation wireless access networks envisions mobile devices that can support multiple technologies for physical digital radio communication, along with the resource management algorithms that would permit a device to seamlessly move between 3G cellular networks, IEEE 802.16 access networks and IEEE 802.11 WLANs, while supporting a variety of packet mode services, each with its own QoS requirements. With reference to the taxonomy in Figure 1.2, we now turn to the category labeled “ad hoc networks” or “wireless mesh networks.” Wireless access networks provide mobile devices with one-hop wireless access to a wired network. Thus, in such networks, in the path between two user devices there is only one or at most two wireless links. On the other hand a wireless ad hoc network comprises several devices arbitrarily located in a space (e.g., a line segment, or a two-dimensional ﬁeld). Each device is equipped with a radio transceiver, all of which typically share the same radio frequency band. In this situation, the problem is to communicate between the various devices. Nodes need to discover neighbors in order to form a topology, good paths need to be found, and then some form of time scheduling of transmissions needs to be employed in order to send packets between the devices. Packets going from one node to another may need to be forwarded by other nodes. Thus, these are multihop wireless packet radio networks, and they have been studied as such over several years. Interest in such networks has again been revived in the context of multihop wireless internets and wireless sensor networks. We discuss these brieﬂy in the following two paragraphs. In some situations it becomes necessary for several mobile devices (such as portable computers) to organize themselves into a multihop wireless packet network. Such a situation could arise in the aftermath of a major natural disaster such as an earthquake, when emergency management teams need to coordinate their activities and all the wired infrastructure has been damaged. Notice that the kind of communication that such a network would be required to support would be similar to what is carried by regular public networks; that is, pointto-point store and forward trafﬁc such as electronic mails and ﬁle transfers, and low bit rate voice and video communication. Thus, we can call such a network a

1.3 Technical Elements

9

multihop wireless internet. In general, such a network could attach at some point to the wired Internet. Whereas multihop wireless internets have the service objective of supporting instances of point-to-point communication, an ad hoc wireless sensor network has a global objective. The nodes in such a network are miniature devices, each of which carries a microprocessor (with an energy efﬁcient operating system); one or more sensors (e.g., light, acoustic, or chemical sensors); a low power, low bit rate digital radio transceiver; and a small battery. Each sensor monitors its environment and the objective of the network is to deliver some global information or an inference about the environment to an operator who could be located at the periphery of the network, or could be remotely connected to the sensor network. An example is the deployment of such a network in the border areas of a country to monitor intrusions. Another example is to equip a large building with a sensor network comprising devices with strain sensors in order to monitor the building’s structural integrity after an earthquake. Yet another example is the use of such sensor networks in monitoring and control systems such as those for the environment of an ofﬁce building or hotel, or a large chemical factory.

1.3 Technical Elements In the previous section we provided an overview of the current practice of wireless networks. We organized our presentation around a taxonomy of wireless networks shown in Figure 1.2. Although the technologies that we discussed may appear to be disparate, there are certain common technical elements that constitute these wireless networks. The efﬁcient realization of these elements constitutes the area of wireless networking. The following is an enumeration and preliminary discussion of the technical elements. 1. Transport of the users’ bits over the shared radio spectrum. There is, of course, no communication network unless bits can be transported between users. Digital communication over mobile wireless links has evolved rapidly over the past two decades. Several approaches are now available, with various tradeoffs and areas of applicability. Even in a given system, the digital communication mechanisms can be adaptive. First, for a given digital modulation scheme the parameters can be adapted (e.g., the transmit power, or the amount of error protection), and, second, sophisticated physical layers actually permit the modulation itself to be changed even at the packet or burst timescale (e.g., if the channel quality improves during a call then a higher order modulation can be used, thus helping in store and forward applications that can utilize such time varying capacity). This adaptivity is very useful in the mobile access situation where the channels and interference levels are rapidly changing.

10

1

Introduction

2. Neighbor discovery, association and topology formation, routing. Except in the case of ﬁxed wireless networks, we typically do not “force” the formation of speciﬁc links in a wireless network. For example, in an access network each mobile device could be in the vicinity of more than one BS or access point (AP). To simplify our writing, we will refer to a BS or an AP as an access device. It is a nontrivial issue as to which access device a mobile device connects through. First, each mobile needs to determine which access devices are in its vicinity, and through which it can potentially communicate. Then each mobile should associate with an access device such that certain overall communication objectives are satisﬁed. For example, if a mobile is in the vicinity of two BSs and needs certain quality of service, then its assignment to only a particular one of the two BSs may result in satisfaction of the new requirement, and all the existing ones. In the case of an access network the problem of routing is trivial; a mobile associates with a BS and all its packets need to be routed through that BS. On the other hand, in an ad hoc network, after the associations are made and a topology is determined, good routes need to be determined. A mobile would have several neighbors in the discovered topology. In order to send a packet to a destination, an appropriate neighbor would need to be chosen, and this neighbor would further need to forward the packet toward the destination. The choice of the route would depend on factors such as the bit rate achievable on the hops of the route, the number of hops on the route, the congestion along the route, and the residual battery energies in devices along the route. We note that association and topology formation is a procedure whose timescale will depend on how rapidly the relative locations of the network nodes is changing. However, one would typically not expect to associate and reassociate a mobile device, form a new topology, or recalculate routing at the packet timescale. If mobility is low, for example in wireless LANs and static sensor networks, one could consider each ﬁxed association, topology, and routing, and compute the performance measures at the user level. Note that this step requires a scheduling mechanism, discussed as the next element. Then that association, topology, and routing would be chosen that optimizes, in some sense, the performance measures. In the formulation of such a problem, ﬁrst we need to identify one or more performance objectives (e.g., the sum of the user utilities for the transfer rates they get). Then we need to specify whether we seek a cooperative optimum (e.g., the network operator might seek the global objective of maximizing revenue) or a noncooperative equilibrium. The latter might model the more practical situation, since users would tend to act selﬁshly, attempting to maximize their performance while reducing their costs. Finally, whatever the solution of the problem, we need an algorithm (centralized or distributed) to compute it online.

1.3 Technical Elements

11

If the mobility is high, however, the association problem would need to be dynamically solved as the devices move around. Such a problem may be relatively simple in a wireless access network, and, indeed, necessary since cellular networks are supposed to handle high mobility users. On the other hand such a problem would be hard for a general mesh network; highly mobile wireless mesh networks, however, are not expected to be “high performance” networks. 3. Transmission scheduling. Given an association, a topology, and the routes, and the various possibilities of adaptation at the physical layer, the problem is to schedule transmissions between the various devices so that the users’ QoS objectives are met. In its most general form, the schedule dynamically needs to determine which transceivers should transmit, how much they should transmit, and which physical layer (including its parameters, e.g., transmit power) should be used between each transceiver pair. Such a scheduler would be said to be cross-layer if it took into account state information at multiple layers; for example, channel state information, as well as higher layer state information, such as link buffer queue lengths. Note that a scheduling mechanism will determine the schedulable region for the network; that is, the set of user ﬂow rates of each type that can be carried so that each ﬂow’s QoS is met. In general, these three technical elements are interdependent and the most general approach would be to jointly optimize them. For example, in a mobile Internet access network the mobile devices are associated with base stations. The channel qualities between the base stations and the mobile devices determine the bit rates that can be sustained, the transmission powers required, and transmission schedule required to achieve the desired QoS for the various connections. Thus, the overall problem involves a joint optimization of the association, the physical layer parameters, and the transmission schedule. In addition to the preceding elements that provide the basic communication functionality, some wireless networks require other functional elements that could be key to the networks’ overall utility. The following are two important ones, which are of special relevance to ad hoc wireless sensor networks. • Location determination. In an ad hoc wireless sensor network the nodes

make measurements on their environment, and then these measurements are used to carry out some global computation. Often, in this process it becomes necessary to determine from which location a measurement came. Sensor network nodes may be too small (in terms of size and available energy) to carry a GPS (global positioning system) receiver. Some applications may require the nodes to be placed indoors, where GPS signals may not penetrate. Hence GPS-free techniques for location determination become important. Even in cellular networks, there is a requirement in some countries that, if needed, a mobile device should be

12

1

Introduction

geographically locatable. Such a feature can be used to locate someone who is stranded in an emergency situation and is unaware of the exact location. • Distributed computation. This issue is speciﬁc to wireless sensor net-

works. It may be necessary to compute some function of the values measured by sensors (e.g., the maximum or the average). Such a computation may involve some statistical signal processing functions such as data compression, detection, or estimation. Since these networks operate with very simple digital radios and processors, and have only small amounts of battery energy, the design of efﬁcient self-organizing wireless ad hoc networks and distributed computation schemes on them is an important emerging area. In such networks there is communication delay and also data loss; hence existing algorithms may need to be redesigned to be robust to information delay and loss.

1.4

Summary and Our Way Forward

We began with a discussion of our view of networking as resource allocation. Figure 1.1 summarizes our view. This was followed by a taxonomy of current wireless practice in Section 1.2. Next, the common technical elements that underlie the apparently disparate technologies were abstracted and discussed in Section 1.3. Before we can proceed to the core topic of this book—resource allocation to meet speciﬁed QoS objectives—we will need to understand basic models of, and notions associated with, the wireless channel. Along with this, the important techniques employed in digital communication will be covered in Chapter 2. These concepts will be like the building blocks in terms of which our resource allocation problems will be posed, and answers sought. Essentially, in Chapter 2, our discussion will be conﬁned to the so-called PHY layer. However, before commencing our study of resource allocation problems, we will pause and take a look at the applications that usually are carried on communication networks. Our objectives will be to understand the characteristics of the bit streams or the packet streams generated by various applications (the top layer of Figure 1.1), as well as the performance requirements the streams demand. This will be the topic of Chapter 3. Beginning with Chapter 4, we will consider, one by one, the different wireless networks shown at the second level of our taxonomy (Figure 1.2). In each case, the emphasis will be on posing and solving resource allocation problems speciﬁc to that type of network. In Chapter 4, narrowband cellular systems will be studied. Power, bandwidth, and time are the resources here, and the principal objective is to maintain the signal-to-interference ratio (SIR) at an adequately high level. Our discussion will give rise to several important concepts, including

1.4

Summary and Our Way Forward

13

frequency reuse, sectorization, spectrum efﬁciency, handover blocking, and channel reservation. Continuing with cellular access networks, we will focus on CDMA systems in Chapter 5. The distinguishing feature here is that of universal frequency reuse. As before, the main theme is to assign power so as to ensure that the signalto-interference-plus-noise ratio (SINR) is adequately high. We will see how the notions of other-cell interference, power control, and hard as well as soft handover arise in this context. In Chapter 6, we will turn to OFDMA-TDMA systems, where power, frequency, and time constitute the basic resources to be allocated. Unlike FDMTDMA and CDMA systems, where to each ﬂow a ﬁxed bit rate is assigned, in OFDMA-TDMA systems, the resources are assigned dynamically over time, depending on time varying user requirements and channel conditions. Generally speaking, the objective is to maximize the aggregate bit capacity of a time-varying channel, subject to a constraint on the average power. The important notion of the water-ﬁlling power allocation will emerge from our discussions. In Chapter 7, the focus shifts to random-access systems and, in particular, IEEE 802.11 WLANs. The principal resource here is channel time, and distributed control of access to the channel is of interest. In a system of n colocated WLAN nodes, what is the saturation throughput that each can achieve? We will analyze this important question. Various issues pertaining to the transport of voice and data trafﬁc over WLANs will also be discussed. Continuing with our discussion of the various networks according to our taxonomy, we will study multihop wireless mesh networks in Chapters 8 and 9. In Chapter 8, we assume that a wireless mesh network is given. On this network, we will address the fundamental question of optimal routing and link scheduling of packet ﬂows for a given set of source-destination pairs. Again, the basic resources here are bandwidth, time, and power, and it is of interest to know which nodes should get access to the bandwidth at what times so as to achieve the objective of maximizing throughput. Our analysis will lead to the notions of optimal scheduling and routing. We ﬁrst consider open loop ﬂows. The ﬂow rates may be given or they may be unknown. For the latter case, the important maximum weight scheduling is described in detail. We also consider routing and scheduling for elastic ﬂows so as to maximize a network utility function. In Chapter 9, we will address some fundamental questions that arise in the context of wireless mesh networks. First, we ask, what is the minimum power level that nodes can use while ensuring that the network of nodes remains connected? After a suitable deﬁnition of the network capacity we also obtain the capacity of arbitrary and random networks. Although asymptotic analyses provide interesting insights, wherever possible, we also consider ﬁnite networks. Finally, in Chapter 10, we will turn to wireless sensor networks. Apart from power and bandwidth, each sensor itself can be considered as a resource now.

14

1

Introduction

A variety of new problems arise; for example, if sensors are deployed in a random manner over a given area, how many of them are required so that every point in the area is sensed by not less than k sensors? As mentioned before in Section 1.3, wireless sensor networks often have special needs; for example, localization and distributed computation. Resource allocation problems for meeting such objectives will also be discussed.

CHAPTER 2 Wireless Communication: Concepts, Techniques, Models

W

e recall from Figure 1.1 in Chapter 1 that, when studying wireless networks, we will not take the links as given bit carriers but will be concerned with the sharing of the wireless spectrum resource as well. The strictly layered approach would view the wireless physical layer as providing a bit carrier service to the link layer. The link layer just offers packets to the physical layer, which does the best it can. If on the other hand, there is interaction between the layers and the link layer can be aware of the time varying quality of the wireless communication, then it could prioritize, schedule, defer, or discard packets in order to attempt to meet the QoS requirements of the various ﬂows. It is therefore important to obtain an understanding of how digital radio communication is performed, and the issues, constraints, and trade-offs that are involved. The material in this chapter is well established and is available in great detail and in much more generality in many books on digital communications. An excellent up-to-date coverage of this topic is provided in [131] and [43]. Readers familiar with digital wireless communication can skip this chapter with no loss of continuity.

Overview Our approach to modeling, analyzing, and designing resource allocation in wireless networks will be based on simple models of the techniques that are used for carrying bit streams over wireless channels. Because of their place in the seven-layer OSI model, these are also called physical layer techniques or, as an abbreviation, PHY techniques. In this chapter we will provide these models, and show how they arise. In Section 2.1 we will study, in some detail, the simplest binary modulation over a very simple radio channel in which the only phenomenon that corrupts the user’s data is additive noise. We will see that the receiver can make errors when attempting to extract the transmitted bits from the noisy received signal, and we will relate the bit error rate (BER) to the received signal-to-noise ratio (SNR). We will see how higher bit rates can be obtained by using higher order constellations into which blocks of user bits can be mapped. We will brieﬂy discuss how adding redundant bits at the transmitter, or channel coding, can be used to reduce the BER

16

2 Wireless Communication: Concepts, Techniques, Models

at the expense of a reduction in the user level bit rate. Then, in Section 2.1.4, we will understand other ways in which propagation over a radio channel can corrupt the user’s data: these are path loss, shadowing, and multipath fading. The latter two are stochastic phenomena, and we will see how they are modeled. Section 2.1 will close with an understanding of how random fading causes a deterioration in the BER achievable for a given SNR. In Section 2.2 we will explain the idea of channel capacity, and we will provide Shannon’s formula for the capacity of an additive white Gaussian channel. The idea of the ergodic capacity of a fading channel will also be introduced. In Section 2.3 we will study how diversity can mitigate the effect of a fading channel. Diversity can be obtained in various ways, one of them being by the use of multiple receive antennas. We will then see that multiple transmit and receive antennas (i.e., MIMO antenna systems) can also provide a capacity gain by making the channel look like several independent parallel channels. Recent mobile wireless access networks have relied heavily on the techniques of code division multiple access (CDMA), and also, more recently, orthogonal frequency division multiple access (OFDMA). In these systems, the resources (e.g., bandwidth and time) are not statically partitioned over the users. Instead, the available spectrum is shared dynamically between the users, with the resource allocation being dynamically adjusted as the user demands and channel conditions vary over time. We study CDMA and OFDMA in Section 2.4.1 and in Section 2.4.2, respectively.

2.1

Digital Communication over Radio Channels

The primary resource that is shared in a wireless network is the radio spectrum. We will limit ourselves to the situation in which the communicating nodes share a radio spectrum of bandwidth1 W , centered at the carrier frequency fc (see Figure 2.1). W 2fc

W

0

fc

f

Figure 2.1 The nodes in a wireless network share a portion of the radio spectrum.

1 The term bandwidth has varied and confusing usage in the wireless networking literature. The RF spectrum

in which a system operates has a bandwidth. When a digital modulation scheme is used over this spectrum then a certain bit rate is provided; often this aggregate bit rate may also be referred to as bandwidth, and we may speak of users sharing the bandwidth. This latter usage is unambiguous in the wire-line context. In multiaccess wireless networks, however, users would be sharing the same RF spectrum bandwidth, but would be using different modulation schemes and thus obtaining different (and time varying) bit rates, rendering the use of a phrase such as “bandwidth assigned to a user” very inappropriate.

2.1

Digital Communication over Radio Channels

17

C1p(t )

C2p(t 2T ) 101101

100101 Modulator

Channel

Demodulator

noise

Figure 2.2 A sequence of pulses is modulated with the bits to be transmitted. The √ basic pulse is p(t). Notice that the bit sequence 101101 is transmitted as + E p(t), s √ √ √ − Es p (t − T ), + Es p (t − 2T ),. . ., + Es p (t − 5T ). There is an error in the third bit, so that, after detection, the received sequence is 100101.

It is assumed that fc >> W ; for example, fc = 2.4 GHz and W = 5 MHz. All communication between any pair of nodes in the network can utilize this entire spectrum.

2.1.1 Simple Binary Modulation and Detection As shown in Figure 2.2, digital communication is achieved over the given radio spectrum by modulating a sequence of pulses by the given bit pattern. The pulse, p(t) (also called the baseband pulse), is chosen so that when translated to the carrier fc its spectrum ﬁts into the given radio spectrum; that is, in this case, the spectrum W W 1 of the baseband pulse will occupy the frequencies − 2 , + 2 . Taking T = W , it , +W , and is such is possible to deﬁne a pulse p(t), that is bandlimited2 to − W 2 2 that p(t − kT), k ∈ {. . . , −3, −2, −1, 0, 1, 2, 3, . . .}, constitute an orthonormal set, +∞ +∞ that is, −∞ p(t)p(t − kT)dt = 0 for k = 0. Further, −∞ p2 (t)dt = 1, that is, the energy of the pulse is 1. The pulses are repeated every T seconds. In the situation depicted in Figure 2.2, √ the modulation is very simple: each pulse√in the pulse train is multiplied by + Es if the bit to be transmitted is 1, and by − Es if the bit to be sent is 0. Notice that the energy of the modulated pulse becomes Es . It is said that the modulator maps √ √ bits into channel symbols. Thus, in this example, the symbol set is − Es , + Es . In general, there could be more than just two possible symbols; for example, four symbols would permit two incoming bits to be mapped into each channel symbol. Continuing our simple 2 Mathematically, a pulse, p(t), that is bandlimited (e.g., to (− W , + W )) occupies inﬁnite time. Practically, 2 2

a pulse that is chosen for a digital modulation scheme has negligible energy beyond a small multiple of T on either side of its main lobe.

18

2 Wireless Communication: Concepts, Techniques, Models

example, let Ck denote the symbol into which the k-th bit is mapped. When the pulses are repeated every T seconds, the modulated pulse stream can be written as X(t) =

∞

Ck p(t − kT)

(2.1)

k=−∞

Given this continuous time signal, and recalling the orthonormality of the various shifts of p(t) by kT , it is easy to see that the following operation recovers the information carrying sequence Ck . +∞ Ck = X(t)p(t − kT)dt −∞

The baseband signal X(t) is then translated to the radio spectrum shown in Figure 2.1 by multiplying it with a sinusoid at the carrier frequency. The resulting signal is S(t) =

∞ √ 2 Ck p(t − kT) cos(2πfc t)

(2.2)

k=−∞

√ The multiplication by 2 is to make the energy in the modulated symbols equal3 to Es . Thus, the symbol energy in the transmitted signal is Es Joules/symbol, and since the symbol rate is T1 symbols/second, the transmitted signal power is therefore Es T Watts. In Figure 2.2 we do not show the translation of the signal by the carrier. It is as if the channel has been shifted to the baseband. As shown in Figure 2.2, as the modulated signal passes through the channel, and is processed in the front-end of the receiver, it is corrupted by noise. This is taken to be zero mean additive white Gaussian noise (AWGN), which means that noise just adds to the signal and is a Gaussian random process with a power spectrum that is constant over the passband of the channel (hence the term “white,” since all frequencies (“colours”) have the same power). The signal occupies a band of W Hz around the carrier frequency fc ( W Hz below and W Hz above ±fc ; 2 2 see Figure 2.1). Hence, we need only be concerned with noise that occupies this band. Such bandpass white Gaussian noise, with a power spectral density of N20 , is mathematically represented as (see [113]) N(t) = U(t) cos(2πfc t)

(2.3)

√

3 To see why we have chosen the symbols C to be ±√E , and the reason for the factor 2, notice s k +∞ (Ck )2 p2 (t) cos2 (2πfc t) dt which can be shown to be that the energy in each transmitted pulse is 2 −∞ equal to Es .

2.1

Digital Communication over Radio Channels

19

where the process U(t) is a zero Gaussian process with power spectral mean white W W density N0 , bandlimited to − 2 , + 2 . We can view the noise process U(t) as a baseband noise process that is translated to the carrier frequency and placed in the passband of the channel. It can now be shown (see this chapter’s Appendix) that the previously described modulation scheme, and the additive white Gaussian noise model, along with receiver processing, results in the following symbol-by-symbol channel model that relates the source symbol sequence Ck and the predetection statistic Yk , from which the source symbol sequence has to be inferred. Yk = Ck + Zk

(2.4)

where Zk is a sequence of i.i.d. zero mean Gaussian random variables with variance N20 . Figure 2.3 depicts the probability density of Yk under the two possible values of Ck . These are both Gaussian densities with variance N20 . The detector concludes that the bit sent was 0 if the value of Yk is smaller than the threshold and 1 if the value of Yk is more than the threshold. An error occurs if 1 is sent and Yk falls below the threshold, and vice versa. When the source produces 0s and 1s with equal probabilities then the threshold is midway between the means of the two densities, that is, the threshold is 0. The probability of error if a 0 was sent is then given by:

Pr(Yk > 0 | 0 was sent) = Q

2Es N0

depends on signal energy

probability density of value at detector if ‘0’ was sent

depends on noise energy

2 Es

threshold

Es

Figure 2.3 The probability densities of the statisticYk under the two possible symbols.

20

2 Wireless Communication: Concepts, Techniques, Models

x2 where Q(τ) : = τ∞ √1 e− 2 dx. This can be seen to be the same as the probability 2π of error if a 1 was sent. Hence the probability of error of the binary modulation scheme that we have described, under AWGN, is given by

2Es Perror − AWGN = Q (2.5) N0

Note that in this example, since each symbol is used to send one bit, the error rate obtained is also the bit error rate (BER). In Problem 2.1 we ﬁnd that Perror − AWGN decreases exponentially with NEs . In particular, for BERs of 10−3 and 10−6 the Es N0

0

values required are approximately 7 dB and 10.5 dB, respectively. We note that if 1500 byte packets have to be transmitted over a wireless link, then in order to obtain a packet error probability of 0.01, we need BER ≤ 10−6 . We see that the probability of correct detection depends on NEs , which is 0 the ratio of the symbol energy to the noise power spectral density. Increasing the symbol energy increases the separation between the two Gaussian probability densities in Figure 2.3, and hence, for given noise variance, reduces the probability of Yk falsely crossing the threshold. Similarly, decreasing the noise reduces the width of the two Gaussian probability densities, thus also reducing the error probability for a given signal energy. 2.1.2 Getting Higher Bit Rates In the simple example in Section 2.1.1, since each pulse is modulated by one of two possible symbols, and the symbol rate is T1 , the bit rate is therefore 1 bps. One of the goals in designing a digital communication system over a radio T spectrum is to use this spectrum to carry as high a bit rate as possible. With the binary modulation example in mind there are two possibilities for increasing the bit rate. 1. Increase the symbol rate; that is, decrease T . 2. Increase the number of possible symbols, from 2 to M > 2. log M

Then, in general, the bit rate will be given by T2 . There are, however, limits on both these possibilities. Note that if the pulse bandwidth is limited to W , the channel bandwidth, 2 then the pulse duration will not be time limited, and in fact the received signal in a symbol interval will be the sum of the pulse in that interval and parts of pulses in neighboring intervals. The pulses therefore have to be appropriately designed to take care of this effect. This leads to the so-called Nyquist criterion, which limits the pulse rate to no more than W (i.e., T1 ≤ W ).

2.1

Digital Communication over Radio Channels

21

Before we proceed, it is useful to make an observation. We saw in Section 2.1.1 that the probability of error for that binary signaling system depended on the ratio NEs . If the signaling rate is T1 , then the average power in the transmitted 0

signal is Es × T1 . The noise power in the channel bandwidth is W N0 . Hence the Es signal power to noise power ratio (SNR) is given by TWN . If, in addition, the 0 Es symbol rate is such that T × W = 1, then the SNR is just N0 . Thus we see that for this example the probability of error depends on the SNR. This is sometimes called the predetection SNR, as it is the SNR before the receiver attempts to decide which symbol was sent. Let us now consider the other alternative for increasing the bit rate; that is, increasing the number of possible symbols that can modulate the pulses. Figure 2.4(a) shows the binary symbol set that we have already discussed. This is called binary pulse amplitude modulation (PAM), or 2-PAM. An example of the simplest possibility is shown in Figure 2.4(b); this is called 4-PAM. Since each of the 2-bit patterns 00, 01, 10, 11 can be mapped to one of the symbols, this scheme can transmit 2 bits per symbol. However, in order to achieve a particular probability of error with a given noise power, the distance between the symbols has to be retained as in the binary case; to see this consider Figure 2.3, add a Gaussian density for each new symbol added, and then consider the probability of error between neighboring symbols. This means that the symbol energy when transmitting the left-most and right-most symbols in Figure 2.4(b) will be 32 times larger than that for the other two symbols. This in turn implies a larger average signal power, and hence a larger SNR (assuming the same noise power) for achieving the same probability of error. Yet another alternative is shown in Figure 2.5(a) where we have two-dimensional symbols. Each symbol can be written in the form ce jθ , with c = 1

π 3π and θ ∈ 0, 2 , π, 2 . This symbol set is called QPSK (quadrature phase shift

⎯ 2√ Es

⎯ √ Es

(a)

Figure 2.4

⎯ 23√Es

⎯ 2√ Es

⎯ √ Es

⎯ 3√Es

(b)

Some symbol sets: (a) binary antipodal, (b) 4-level amplitude modulation.

22

2 Wireless Communication: Concepts, Techniques, Models

(a)

Figure 2.5 added.

(b)

(a) A complex symbol set with 4 symbols; (b) the symbol set with noise

keying) since all the symbols have the same amplitude but they have different phases. Now, instead of the form in (2.2), the transmitted signal takes the general form S(t) =

∞ √ 2 Ck cos(Θk )p(t − kT) cos(2πfc t) k=−∞ ∞ √ Ck sin(Θk )p(t − kT) sin(2πfc t) − 2

(2.6)

k=−∞

Here, the sequence (Ck , Θk ) depends on the modulating bits. Thus, basically, the x-coordinate (i.e., Ck cos(Θk )) of the symbol modulates the carrier cos(2πfc t) and the y-coordinate (i.e., Ck sin(Θk )) of the symbol modulates −sin(2πfc t), which is

also called the quadrature carrier (since it is π2 out of phase with the in-phase carrier). The bandpass additive noise N(t) has the general form N(t) = U(t) cos(2πfc t) − V(t) sin(2πfc t)

where U(t) and V(t) are independent Gaussian processes with power zero mean W , . We can interpret U(t) and V(t) as spectral density N0 , bandlimited to − W 2 2 the in-phase and quadrature noise processes, respectively. In fact, we notice that the QPSK signal shown in (2.6) is the superposition of two orthogonal 2-PAM signals; the in-phase and quadrature signals √ are both 2-PAM signals.√After down conversion (multiplying the signal by 2 cos(2πfc t) and also by − 2 sin(2πfc t) and ﬁltering out the high frequency terms), and multiplication and integration with the pulse p(t), we will obtain the following pair of statistics: (i)

(i)

(q)

(q)

Yk = Ck cos(Θk ) + Zk Yk = Ck sin(Θk ) + Zk

2.1

Digital Communication over Radio Channels

23

where (i) and (q) denote the in-phase and quadrature components. The sequences (q) (i) Zk and Zk are independent, and each is a sequence of i.i.d. zero mean Gaussian random variables with variance N20 . We can write this more compactly by using (q) (q) complex numbers. Deﬁne Yk = Yk(i) + jYk and Zk(i) + jZk . Then, we can write Yk = Xk + Zk

(2.7)

where Xk = Ck ejΘk is the k-th channel symbol. We say that the sequence of complex random variables Zk are circularly symmetric complex Gaussian. In Figure 2.5(b) we show the received symbols after corruption by noise; the noise now has the two-dimensional Gaussian density that is circularly symmetric about each symbol. Notice from the geometry in Figure 2.5(a) that, by utilizing both dimensions, for a given probability of error, a smaller symbol spacing can be used than for the symbol set in Figure 2.4(b), and hence a given BER can be achieved with less average power. Thus, we have noisy observations of the two coordinates of the transmitted complex symbol, from which the transmitted symbol has to be detected. Since in each symbol only one of the phases is used (and the other is 0, owing to the simple QPSK symbol set), the average signal power is that of a 2-PAM signal. As is evident from Figure 2.5 many more symbol sets are possible. If the amplitude as well as phase of the symbols can vary then it is called QAM (quadrature amplitude modulation), whereas if only the phase can vary then it is called a PSK symbol set. Symbol sets are also called constellations. The probability of error of all the digital modulation and demodulation schemes based on the basic ideas discussed earlier can be expressed as a function of the SNR at the receiver. 2.1.3 Channel Coding In a given situation, owing to physical limitations it may not be possible to increase the SNR so as to achieve the desired BER. The application being transported on the wireless link may require a lower BER in order to achieve reasonable performance. For example, if the link is used to transport packets and the packet length is L bits, then a BER of yields a packet error rate of 1 − (1 − )L . We will see in Section 3.4.3 that a high packet error rate can seriously affect the performance of TCP transfers. Hence, this may place a minimum BER requirement on the link. For a given digital modulation scheme, the BER as seen by the data source can be reduced by channel coding. The simplest viewpoint is shown in Figures 2.6 and 2.7. The channel with the given modulation scheme is viewed as an error prone binary channel. Blocks of the incoming bits of length K are coded into codewords of length N(> K), thus introducing redundancy. If the code length and the codes are judiciously chosen, even after the channel introduces errors, an errored codeword can be expected to stay close to the original codeword. In Figure 2.7 we show source bit strings of length K being mapped into code blocks of length N . Since the number of possible code strings (2N ) is larger than

24

2 Wireless Communication: Concepts, Techniques, Models error control coder adds redundant bits

Figure 2.6

binary channel (introduces bit errors)

error control decoder extracts transmitted bits from received code words

Channel coding: adding redundant bits to protect against channel errors.

code words

set of possible blocks of length K (2K blocks)

set of possible blocks “sphere” of highly probable of length N errored code words N (2 blocks)

Figure 2.7 A channel code maps source bit strings into longer code bit strings (or codewords); decoding involves identifying the codeword nearest to the received bit string.

the number of possible source strings (2K ), the code words can be chosen so that there is sufﬁcient spacing between them. Now even if the channel causes errors, the errored codewords will occupy spheres of high probability around the transmitted codewords. Hence, by using nearest codeword decoding, the transmitted codeword, and hence the original source string, can be inferred with a small residual error probability. The trade-off is that the information bit rate of K the communication link becomes N , which is less than 1 information bit per code bit. This is called the rate of the code, denoted by R. One trivial way of improving error performance is to increase N , because this results in the codewords being spaced farther apart; but this reduces the information rate. It is possible, however, to increase K with N , keeping the information rate R constant, while reducing the bit error rate to arbitrarily small values. Shannon’s noisy channel coding theorem states that there is a number C , called the channel capacity, such that if R < C , then, as the block length increases, an arbitrarily small bit error rate can be achieved (of course, at the cost of a large block coding delay). If we attempt to use R > C , then the bit error rate cannot be reduced to 0. Recall our analysis of the two-level modulation carried out earlier in this chapter. We recall that for bit error rates of 10−3 and 10−6 the NEs0 values required

2.1

Digital Communication over Radio Channels

25

were approximately 7 dB and 10.5 dB, respectively. As an example, with a high quality rate 12 code, the required NEs can be reduced by 2 dB for 10−3 and by 5 dB 0

for 10−6 . Of course, the user bit rate drops to 12 bit per symbol. This reduction in Es is called coding gain. N0 A coder is followed by a digital modulation scheme that maps code bits into channel symbols. As discussed in Section 2.1.1, the modulator maps a certain number of code bits (e.g., 2 in 4-QPSK) into each channel symbol. Thus the capacity of the overall system (coder—modulator—channel—demodulator— decoder) can be expressed in terms of bits per symbol. At this point, it is obvious that in order to achieve this capacity the receiver must know the channel coding and modulation scheme that the transmitter is using. Shannon also provided the fundamental relationship between the channel capacity (C ) and the signalto-noise ratio for an additive white Gaussian noise channel. We will introduce this relationship later in this chapter. First we need to study models for signal power attenuation between the transmitter and the receiver. 2.1.4 Delay, Path Loss, Shadowing, and Fading In the previous discussion we assumed that the transmitted signal was contaminated by only additive white Gaussian noise. This yielded the simple model shown in (2.4). However, in practical channels, signals undergo attenuation and delay. In wireless channels, because of propagation over multiple paths, and mobility of the scatters or of the communicating devices, the attenuation can vary with time and the relative location between the transmitter and the receiver. We have seen that the BER performance of a digital communication system depends on the received SNR. Hence, we are interested in the received signal power after the signal has passed through the channel. Radio waves are scattered by the objects on which they impinge. Hence, unless a very narrow antenna beam is used, the receiver’s antenna receives the transmitted signal along several paths. There is often a direct or line-of-sight path, and there are several paths along which the signal reaches the receiver after one or more reﬂections from various objects. Energy is lost in reﬂections, and is absorbed by media through which the signal passes (partitions and walls). Hence the received signal is a sum of attenuated and delayed versions of the original signal.

Delay Spread and Intersymbol Interference Superposition of the delayed signals from the various paths can cause a symbol from one path to overlap with a neighboring symbol from another path. Let us examine this issue ﬁrst. These are electromagnetic signals and hence they travel at the speed of light; let us take the propagation time to be roughly 0.33 μsec per 100 meters. Hence, this is the kind of delay that can be expected if the various path lengths differ by no more than 100 m. If the symbol time is several μseconds (e.g., 100,000 symbols per second) then there will not be signiﬁcant overlap between the neighboring symbols, and

26

2 Wireless Communication: Concepts, Techniques, Models

we can assume that the symbols are still separately discernible, except that each is multiplied by a complex “attenuation.” If this happens then the channel is said to have ﬂat fading. We will understand the term “ﬂat” when we interpret this phenomenon in the frequency domain. Then, motivated by (2.7), we can write the k-th received symbol after down conversion as Yk = Gk Xk + Ik + Zk

(2.8)

where the various new terms are understood as follows. 1. Gk is the random attenuation of the k-th symbol. Gk , k ≥ 1, is a complex valued random process. Thus, a transmitted symbol is not only attenuated, but can also be rotated. Note that the symbol energy is multiplied by |Gk |2 . Let us write Hk = |Gk |2 ; Hk , k ≥ 1, is a random process, and we need to characterize it in order to understand the effect of the channel on the received signal power, and hence the SNR. We note that the Hk are also called channel gains. 2. Ik is a complex random variable that models the interference (from other transmissions in the same or nearby spectrum4 ). We recall that Zk is a sequence of complex random variables that models the additive noise (for example, the thermal noise in the electronic circuitry of the receiver) and is taken to be a white Gaussian random process. A commonly used simpliﬁcation is to use the same model even for the interference process, with the noise and interference processes being modeled as being independent. The BER then becomes a function of the signal to interference plus noise ratio (SINR). For a transmitter receiver pair, the difference between the smallest signal delay and the largest signal delay is called the delay spread, Td . For example, if the path lengths differ by no more than 100s of meters then the delay spread would be in 100s of nanoseconds. When the delay spread is not very small compared to the symbol time then the superposition of the signals received over the variously delayed paths at the receiver results in intersymbol interference (ISI). We then obtain the following linear model: Jd −1

Yk =

Gk ( j)Xk−j + Ik + Zk

(2.9)

j=0

For every k, Gk ( j), 0 ≤ j ≤ Jd − 1, are complex random variables that model the way the channel attenuates and phase shifts the transmitted symbols. Gk ( j) models 4 Note that we are taking the simpliﬁed approach of treating other users’ signals as interference. More generally, it is technically feasible to extract multiple users’ symbols even though they are superimposed. This is called multiuser detection.

2.1

Digital Communication over Radio Channels

27

the inﬂuence that the input j symbols in the past has on the channel output at k. Thus, in general, a channel has memory; in the model, the memory extends over Jd symbols. The memory arises as a consequence of there existing several paths from the transmitter to the receiver, with the different paths having different delays. The notation shows that the channel gain at the k-th symbol could be a function of the symbol index k; this models the fact that fading is a time-varying phenomenon. As the devices involved in the communication move around, the radio channel between them also keeps changing. The delay spread, Td , has been explained previously as a time domain concept. It can also be viewed in the frequency domain as follows. The symbols Xk are carried over the RF spectrum by ﬁrst multiplying them with a (baseband) pulse of bandwidth approximately W (e.g., 200 KHz), and then upconverting the resulting signal to the carrier frequency (e.g., 900 MHz) (recall (2.2)). The delay spread in the channel (i.e., Td ) can be such that superposition of variously delayed versions of some frequency components in the baseband pulse can cancel out. In such a case, some of the frequency components in the pulses can get selectively attenuated, resulting in the corruption of the symbols they carry; this is called 1 frequency selective fading. On the other hand, if Td > W then all the frequencies fade together and we have ﬂat fading. The assumption of ﬂat fading is reasonable for a narrowband system, where the available radio spectrum is channelized and each bit stream occupies one channel. Then the symbol duration becomes larger than the delay spread, and the model of (2.8) is applicable. This will be the channel model that we will use when analyzing FDM-TDMA cellular systems in Chapter 4. On the other hand, consider the situation in which Wc is small compared 1 to the system bandwidth (Td is large compared to W ); that is, the channel is frequency selective. Then, in relation to the model in (2.9), and recalling that 1 the intersymbol interval is W , we observe that frequency selectivity corresponds to the channel memory extending over more than 1 symbol, and hence to the existence of ISI. Thus, when high bit rates are carried over wideband channels (i.e., large W ) then techniques have to be used to combat ISI, or to avoid it altogether. We will encounter CDMA and OFDMA later in this chapter, as two wideband systems that actually exploit delay spread or frequency selectivity to achieve diversity (a concept explained in Section 2.3). In some systems, we can combat ISI by passing the received signal through a channel equalizer, which can compensate for the various channel delays, making the overall system (i.e., the channel followed by the equalizer) appear like a ﬁxed delay channel. In a mobile wireless situation, owing to mobility, the paths that a signal takes between a transmitter and a receiver may keep changing; hence a channel equalizer needs to be adaptive. In some systems the problem of signals arriving over multiple paths is turned into an advantage. If the paths can be

28

2 Wireless Communication: Concepts, Techniques, Models

resolved, and if they fade independently, then their signals can be combined to reduce the probability of error, for a given received signal-to-noise ratio. Such a receiver is said to exploit multipath diversity.

A Characterization of the Power Attenuation Process It follows from the linear model with ﬂat fading, shown in Equation 2.8, that the received sequence, Yk , k ≥ 1, is also a complex valued random process. The problem for the receiver, on receiving the sequence of complex numbers Yk , k ≥ 1, is to carry out a detection of which symbols Xk , k ≥ 1, were sent and hence which user bits were sent. This problem is particularly challenging in mobile wireless systems since the channel is randomly changing with time. The analysis and design of modulation schemes often is based on the analysis of received signal power to noise power ratios. Hence, it is important to have an effective but simple model of the channel power attenuation process, Hk . The process {Hk } is characterized by writing it in terms of three multiplicative components, that is, Hk =

dk d0

−η

· Sk · R2k

(2.10)

Let us write the marginal terms of the stationary random processes in this expression by dropping the symbol index k. We will now discuss each of these terms. −η The term dd is the path loss factor. Here, d is the distance between the 0 transmitter and the receiver when the k-th symbol is being received, d0 is the “far ﬁeld” reference distance beyond which this model is applicable, and η is the path loss exponent, which is typically in the range 2 to 5. The value of d0 relates to the antenna dimensions and the propagation environment. For distances less than d0 , a different path loss exponent may be used, or, when d0 is very small, we may assume no path loss. If the attenuation is measured at various points at a distance d from the transmitter, then the attenuation will be found to be random, owing to variations in the terrain, and in the media through which the signal may have passed. Empirical studies have shown that this randomness is captured well if the second ξ factor S, in (2.10), has the form 10− 10 , with ξ being a Gaussian random variable with mean 0 and variance σ 2 . This is called the shadowing component of the attenuation, and, since log10 of this term has a Gaussian (or normal) distribution, it is called log-normal shadowing. It is often convenient to express values of power and power ratios in the decibel (dB) unit which is obtained by taking 10 log10 of the value. Hence the shadowing attenuation in signal power is 10 log10 S = −ξ dB, which is zero mean Gaussian with variance σ 2 . A typical value of σ is 8 dB. Considering two standard deviations above and below the mean, this value means

2.1

Digital Communication over Radio Channels

29

that, a high with2×8 probability, shadowing can result in a variation of channel gain of −2×8 10 10 times to 0.025 ≈ 10 times the mean path loss. 40 ≈ 10 Shadow fading is spatially varying, and hence if there is relative movement between the transmitter and the receiver then shadow fading will vary. The correlation in the shadow fading in dB between two points separated by a −

D

distance D is given by σ 2 e D0 , where D0 is a parameter that depends on the terrain. Some measurements have given D0 = 500 m for suburban terrains, and D0 = 50 m for urban terrains. Hence if the distance is varying by a few meters per second (note that 36 Kmph = 10 meters/second) then the shadowing will vary over seconds, which means that the variations will occur over hundreds of thousands of symbols. We now turn to the third factor, R2 , in the expression for attenuation in (2.10). Typical carrier frequencies used in mobile wireless networks are 900 MHz, 1.8 GHz (e.g., these two frequency bands are used in cellular wireless telephony systems), or 2.4 GHz (e.g., used in IEEE 802.11 wireless LAN systems). Hence, the carrier wave periods are a few picoseconds. Thus, when the transmitted signal arrives over several paths then very small differences in the path lengths (a few centimeters) can cause large differences in the phases of the carriers that are being superimposed. Thus, although these time delays may not result in ISI, the superposition of the delayed carriers results in constructive and destructive carrier interference, leading to variations in signal strength. This phenomenon is called multipath fading. This is a random attenuation that has strong autocorrelation over a time duration called the coherence time, Tc ; that is, the attenuations at two time instants separated by more than the coherence time are weakly correlated. The coherence time is related to the Doppler frequency, fd , which is related to the carrier frequency, fc , the speed of movement, v, and the speed of light, c, by fd = fc vc . Roughly, the coherence time is the inverse of the Doppler frequency. For example, if the carrier frequency is 900 MHz, and v = 20 meters/sec, then fd = 60 Hz, leading to a coherence time of 10s of milliseconds. In the indoor ofﬁce or home environment, the Doppler frequency could be just a few Hz (e.g., 3 Hz), with coherence times of 100 s of milliseconds. The marginal distribution of R2 depends on whether all the signals arriving at the receiver are scattered signals, or if there is a line-of-sight signal as well. In the former case, assuming uniformly distributed arrival of the signal from all directions, the distribution of R2 is exponential with mean E R2 , that is, fR2 (x) =

2 1 e −x/E R E R2

The distribution of the amplitude attenuation (i.e., R) is Rayleigh; hence this is also called Rayleigh fading. On the other hand if there is a line-of-sight component so

30

2 Wireless Communication: Concepts, Techniques, Models

that a fraction K K+ 1 of the signal arrives directly, and the remaining signal arrives uniformly over all directions, then K+1 fR2 (x) = 2 e E R

−K− (K+1)x 2

( )

E R

where I0 (x) =

1 2π

2π

⎛ ⎞ K(K + 1)x ⎠ I0 ⎝2 E R2

e−x cos(θ) dθ

0

This is called the Ricean distribution. With this characterization of the attenuation in the received signal power we can now write the received SNR (denoted by Ψrcv ) in terms of the ratio of the transmitted signal power to the received noise power (denoted by Ψxmt ). We have Ψrcv = Ψxmt · H = Ψxmt ·

d d0

−η

−ξ

· 10 10 · R2

(2.11)

Then, in dB, we can write the received SNR as (Ψrcv )dB = (Ψxmt )dB + 10 log10 H = (Ψxmt )dB − 10η log10

d d0

− ξ + 10 log10 R2

(2.12)

BER with Fading We now turn to the calculation of the performance of the wireless link in the presence of fading. We have seen that, although the transmitter may send at a ﬁxed power, in the presence of fading, the received power, and hence the received SNR, is time varying. The rate of variation of the SNR depends on the mobility of the receiver. A receiver that moves short distances over the duration of a “conversation” (e.g., a voice call, or a ﬁle transfer) would sample the distribution of the Rayleigh fading but would see roughly constant values of path loss and shadowing. On the other hand a receiver that makes large movements during a call duration would see variations in all the three attenuation factors during the call. Let us consider the former situation. In this case the in-call performance depends on the value of path loss and shadow fading sampled by the call, and on the distribution of Rayleigh fading, but the performance across calls depends on the variation in path loss and shadowing as well. We would like the performance not to fall below some value. For example, there could be a desired upper bound

2.1

Digital Communication over Radio Channels

31

on BER; exceedance of this bound would be termed an outage. Let us examine this point in the context of the binary modulation scheme discussed in Section 2.1.1. The BER for this modulation scheme was given by (2.5): Perror − AWGN (Ψrcv ) = Q

2Ψrcv

If the path loss and shadowing factors during a call are ﬁxed, then we can calculate the in-call, BER averaged over the fading, as follows:

∞

0

Perror − AWGN

d d0

−η

· 10

−ξ 10

· γ · Ψxmt fR2 (γ)dγ

where, as mentioned earlier, 2 for Rayleigh fading, fR2 (·) is the exponential probability density with mean E R . Let us write the SNR during the call, with the fading averaged out, as Ψrcv :=

d d0

−η

−ξ · 10 10 · E R2 Ψxmt

In many cases it can be shown that the preceding integral expression for in-call BER can be simpliﬁed to the form Perror−fading Ψrcv

for some function Perror − fading . For example, for the binary modulation scheme Ψrcv , discussed earlier, it can be shown that Perror − fading Ψrcv = 12 1 − 1+Ψ rcv

which for large Ψrcv can be observed to decrease reciprocally with SNR (i.e., as 1 ), rather than exponentially, as for unfaded AWGN (see Problem 2.1; see also Ψ rcv

Problem 2.4). During a call, we can write the average SNR (with the averaging being over the fading), Ψrcv , in dB as

Ψrcv

dB

= Ψxmt E R2

dB

− 10η log10

d d0

−ξ

The term Ψxmt E R2 dB is the Rayleigh faded SNR “referred to” d0 . We see that the received SNR, in dB, ata distance d from the transmitter is Gaussian with mean 2 d Ψxmt E R dB − 10η log10 d and variance σ 2 . In order to achieve a certain BER, 0 say, , the received SNR will be required to be above a threshold, say, β; that is, Ψrcv > β ⇒ Perror − fading Ψrcv <

32

2 Wireless Communication: Concepts, Techniques, Models

Violation of this requirement would be called an outage, the probability of which we would like to limit to Poutage . We note that, since we assumed that during a call the path loss and shadowing are ﬁxed, Poutage is the outage probability across calls; that is, the fraction of calls that experience a BER larger than . The BER and outage requirement can then be expressed in the following form: Pr Ψrcv dB < (β)dB < Poutage

Equivalently, Pr Ψxmt E R2

dB

− 10η log10

d d0

− ξ < (β)dB

< Poutage

Let us look at an example. Given that dd0 = 10, η = 3, the shadowing standard deviation σ = 8 dB, the received SNR threshold is β = 10 dB, and Poutage = 0.01, the requirement just displayed is satisﬁed if

Ψxmt E R2

dB

− 30 − 2.3 × 8 = 10

where the factor 2.3 is obtained from a table of the Gaussian distribution. This yields Ψxmt E R2 = 58.4 dB dB

2.2

Channel Capacity

2.2.1 Channel Capacity without Fading Consider the following simple version of the general linear model that was shown in (2.9) Yk = Xk + Zk

(2.13)

where we notice that we have removed the model of ISI, the multiplicative fading term, and also the additive interference term, leaving just a model in which the output random variable at symbol k is the input symbol Xk with an additive noise term Zk . Thus, there is no attenuation of the transmitted symbol, but there is perturbation by additive noise. When the Xk are taken from a one-dimensional constellation (as in the beginning of Section 2.1), then the model for the random process Zk , k ≥ 1, is that these are i.i.d. Gaussian random variables with mean 0 and variance σ 2 (see Equation 2.4). This is called an additive white Gaussian noise (AWGN) channel. The information bits are mapped to the channel symbols

2.2

Channel Capacity

33

Xk , which are corrupted by additive noise. The observations Yk have to be used to infer which symbols were transmitted. Suppose that the input symbols have the following power constraint: 1 lim |xk |2 ≤ P n→∞ n n

(2.14)

k=1

that is, the average energy per symbol is bounded by P Joules/symbol. This is a practical constraint as power ampliﬁers operate well only in certain limited power ranges. Also, microwave radiations can be harmful to the body; hence there are safety regulations on how much power can be radiated by radio transmitters. Further, when several systems coexist then intersystem interference needs to be managed. Hence, some form of power constraint usually is required in wireless communication systems. If the input symbols are allowed to be only real numbers, then Shannon’s celebrated Noisy Channel Capacity Theorem states that the maximum rate at which information can be transmitted over this AWGN channel, in bits/symbol, is given by Prcv 1 bits/symbol C = log2 1 + 2 2 σ

(2.15)

where, Prcv is the received signal power per symbol, and Pσrcv is the received 2 signal-to-noise power ratio. Evidently, here, in the no fading case, we have Prcv = P. What this result means is that this rate can be achieved with the bit error rate going to zero as channel coding is done over longer and longer blocks, with the block length going to ∞. In Section 2.1.1, we derived the symbol-by-symbol channel model by starting with a continuous time model for a modulation scheme that used only real valued symbols. Let us now apply this formula to derive the capacity of that system. We saw that the additive noise sequence has variance N20 . If the power constraint on the transmitted signal (i.e., S(t) in (2.2)) is P Watts, then the power constraint per P symbol is P = PT = W Joules/symbol. Since we are assuming no channel loss, using (2.15), we obtain the capacity C=

1 2P log2 1 + bits/symbol 2 N0 W

(2.16)

If, in (2.13), the input symbols are complex numbers, then the additive noise is modeled as a sequence of complex valued random variables, which is taken to be a sequence of i.i.d. zero mean, circularly symmetric Gaussian random variables with variance σ 2 (recall (2.7)). This means that the real and the imaginary parts

34

2 Wireless Communication: Concepts, Techniques, Models

are independent sequences of zero mean i.i.d. Gaussian random variables with the 2 same variance, σ2 . The capacity formula then takes the simple form Prcv bits/symbol C = log2 1 + 2 (2.17) σ where Prcv is the average received power per symbol. Without channel loss Prcv = P. Now let us apply this to the modulation with complex symbols that led to the channel model in (2.7). There Zk are i.i.d. zero mean circularly symmetric Gaussian with variance with the real and imaginary parts have variance N20 . Then, without channel loss, and a power constraint P on the transmitted continuous time P signal, the constraint on the average received energy per symbol is W , yielding the channel capacity P bits/symbol C = log2 1 + (2.18) N0 W It is instructive to compare the expressions (2.16) and (2.18); see Problem 2.5. We note that these capacity expressions gave the answer in bits per symbol. Often, in analysis it is better to work with natural logarithms. With this in mind we can rewrite (2.17) as Prcv C = ln 1 + 2 nats/symbol σ Since ln x = log2 x × ln 2, the capacity in nats per symbol is obtained by multiplying the capacity in bits per symbol by ln 2 ≈ 0.693. If the symbol rate is T1 then, for the AWGN channel with complex symbols, Shannon’s formula yields the bit rate Prcv 1 log2 1 + bits/second T N0 W where Prcv is average power in the received signal. For the system bandwidth W , the bit rate, therefore, is limited to Prcv W log2 1 + (2.19) bits/second N0 W An important measure of performance of a digital modulation scheme is bits/Hz; that is, the number of information bits that can be carried per Hertz of system bandwidth. Let us write Prcv = Eb × C , where we can call Eb the received energy per bit. (2.19) can then be written as C Eb C = log2 1 + W N0 W C W

2.2

Channel Capacity

35

from which we obtain C

Eb 2W − 1 = C N0 W

The quantity on the left is the ratio of the received energy per bit to the power spectral density of the additive noise, and is called the signal-to-noise ratio per bit. We C 2W − 1 Eb C conclude that, in order to achieve W bits/Hz, we require an N of at least . C 0

For example, for

C W

W

= 1 bit/Hz (a typical number for a FDM-TDMA system such

as GSM), the minimum value of schemes need larger values of

Eb N0

Eb , N0

= 1 or 0 dB. Practical modulation and coding

as seen in the examples earlier in this chapter.

2.2.2 Channel Capacity with Fading How does a time varying channel attenuation affect the Shannon capacity formula? If the channel attenuation is h, and the noise is AWGN, then, for transmitted power Pxmt , the channel capacity is given by (2.19): hPxmt W log2 1 + N0 W Suppose that the transmitter is unaware of the extent of the channel fading, and uses a ﬁxed power and a ﬁxed modulation and coding scheme. Suppose also that the fading level varies slowly. Then, for a given level of fading, the receiver must know h in order for the communication to achieve the Shannon capacity. To see this, let us look at Figure 2.4(b). √ If the channel’s power attenuation is h, the received symbols are multiplied by h. This results in the symbols being “squeezed” together or spread apart. Obviously, the detection thresholds will need to depend on the level of fading. Suppose that Hk is a stationary and ergodic process. It can then be shown that if the transmitter cannot adapt its coding and modulation, but the receiver can exactly track the fading, then the channel capacity with fading is given by hPxmt gH (h)dh Cfading − CSIR = W log2 1 + (2.20) WN0 where gH (·) is the marginal density of the channel attenuation process Hk . For example, gH (h) is exponential for Rayleigh fading (see Section 2.1.4). The acronym CSIR stands for channel state (or side) information at the receiver. Thus the transmitter can encode at any ﬁxed rate R < Cfading−CSIR , and for large enough code blocks the error rate can be made arbitrarily small, provided the receiver can track the channel. It is important to bear in mind that this is an ideal result; to

36

2 Wireless Communication: Concepts, Techniques, Models

achieve it, the channel fades will have to be averaged over and this will result in large coding delays. xmt In Problem 2.6 we see that Cfading − CSIR ≤ W log2 1 + E(H)P , that is, the WN0 capacity with fading is less than that with no fading with the same average SNR. With fading, there will be times when the SNR is higher than the average and times when the SNR will be lower than the average. Yet this result shows that the resulting channel capacity is less than that without fading, as long as the same average SNR is maintained.

2.3

Diversity and Parallel Channels: MIMO

We emphasise that we are discussing direct point-to-point communication between a transmitter and a receiver. We have already seen that the signal from the transmitter can reach the receiver over multiple paths. Since it can be expected that random fading along these paths will be independent, combining the signals from these paths in some manner might lead to better performance than working with the aggregate signal. Such diversity can be obtained in various ways. If the receiver has multiple antennas (see Figure 2.8), and if the antennas are spaced sufﬁciently far apart (at least half the carrier wavelength) then, for the same transmitted signal,

G1 G2

receiver

X

^ X

GK

Figure 2.8 A single-input-multiple-output (SIMO) system comprising one transmit antenna and K receive antennas.

2.3

Diversity and Parallel Channels: MIMO

37

the signals received at the different antennas fade approximately independently.5 To see how such independently faded copies can be exploited, let us consider the following model for the signal received along each path. Yk = Gk X + Zk

where k, 1 ≤ k ≤ K, indexes the diversity “paths,” and X is the transmitted (complex) symbol. The Zk , 1 ≤ k ≤ K, are zero mean, i.i.d. circularly symmetric normal random variables, with variance σ 2 . Recalling the notation Hk = each 2 jθ k |Gk | , let us write Gk = Hk e , that is, on the k-th path, the transmitted symbol X is scaled by Hk and rotated by θk . Assuming that the receiver knows the values of θk , 1 ≤ k ≤ K, it can be shown that the optimum strategy is to form a linear combination of the K received signals by using complex weights μk e−jθk , to obtain K

Y=

μk e−jθk Yk

k=1

⎛ =⎝

K

⎞

μ k Hk ⎠ X +

k=1

K

μk e−jθk Zk

k=1

Note that rotation by θk does not destroy the circular symmetry of the noise, Zk . Let the transmitted power be P, that is, E(|X|2 ) = P. If the symbol detection is based on the statistic Y , then the performance of this receiver algorithm will be based on the received SNR Ψrcv =

2 Hk P K 2 2 k=1 μk σ K k=1 μk

Now, by the Cauchy-Schwartz inequality, we have ⎛ ⎝

K

k=1

μk

⎞2 Hk ⎠ ≤

K k=1

μ2k

K

Hk

k=1

5 To understand the relationship between antenna spacing and low correlation between received signals, let

us recall the concept of coherence time. Multipath fading observed by a mobile has low correlation between f

time instants separated by Tc , which is roughly the reciprocal of fd = cc v, where fc is the carrier frequency, c is the speed of light, and v is the speed of the mobile. Equivalently, fd = λvc , where λc is the wavelength of the carrier. It follows that fade correlations are weak over a distance equal to the carrier wavelength. A precise analysis of the phenomenon actually shows that the correlations are weak over distances as little as half the wavelength. Note that λc = 30 cm for fc = 1 GHz, and λc = 6 cm for fc = 5 GHz.

38

2 Wireless Communication: Concepts, Techniques, Models

with equality when μk = a H some a (i.e., the vector (μ1 , μ2 , . . . , μK ) k for √ √ √ when is a multiple of the vector H1 , H2 , . . . , HK ). Choosing the weights μk , 1 ≤ k ≤ K, in this way maximizes the predetection SNR, yielding ⎛ Ψrcv = ⎝

K

⎞ Hk ⎠ Ψxmt

k=1

where, as before, Ψxmt = σP2 is the transmit SNR. We now wish to study the bit error probability for this approach. Suppose that the bit error probability with AWGN decreases exponentially with the received SNR (see Problem 2.1). Then, the average bit error rate is proportional to

E e

−Ψrcv

=E e

−

K k=1

Hk Ψxmt

Recall our discussion in Section 2.1.4, and hence, write Hk = πΦk where π is the path loss and shadowing factor from the transmitter to the receiver (taken to be a constant over the time scale to which this analysis applies), and Φk , 1 ≤ k ≤ K, represent Rayleigh fading over the various paths. This yields K − k=1 πΦk Ψxmt E e−Ψrcv = E e

Assuming that the fading at the different antennas are independent and identically distributed, we take the Φk , 1 ≤ k ≤ K, to be i.i.d. exponentially distributed with mean, say, φ. We then have K E e−Ψrcv = E e−Φ1 πΨxmt =

1 1 + φπΨxmt

K

−K ≈ Ψrcv

where the approximation holds for large average received SNR Ψrcv = φπΨxmt . Recall that for Rayleigh fading the probability of error decreased only as the reciprocal of Ψrcv . Thus, by combining the received signals over multiple paths, the bit error probability performance has been substantially improved. From the form for the decay of the bit error probability with Ψrcv , we say that we have a diversity gain of K. The transmitter could also just repeat the signal over time, and if the repetitions are spaced apart by more than the coherence time (see Section 2.1.4)

2.3

Diversity and Parallel Channels: MIMO

39

then the received signals fade independently. It turns out that commonly used channel codes provide a better chance of successful decoding if the channel error process is uncorrelated over the code symbols. We saw earlier that the channel fade process, Gk , is correlated over periods called the channel coherence time, which depends on the speed of movement of the mobile device. Interleaving is a way to obtain an uncorrelated fade process from a correlated one. Basically the transmitter does not send successive symbols of a codeword over contiguous channel symbols, but successive symbols are separated out so that they see uncorrelated fading. In between, other codewords are interleaved. We say that interleaving exploits time diversity, that is, the fact that channel times separated by more than the coherence time fade independently. Observe that interleaving introduces interleaving delay, which adds to the link delay, and hence to the end-to-end delay over the wireless network. Also, interleaving fails if the fading is very slow, for example if the relative motion stops, and the transmitter-receiver pair are caught in a bad fade. In the discussion earlier in this section, we considered the case in which multiple independently faded copies of a transmitted symbol arrive at the receiver. By appropriate combining of these received symbols, the probability of error is reduced. Suppose that the channel is such that the transmitter can, in parallel, transmit several symbols, each of which is then independently faded and received. Then the available power P can be distributed over the parallel channels to obtain a higher bit rate than if all the power was used on a single channel; see Problem 2.7, and, for more details, Chapter 6. Physically, parallel channels between a transmitter-receiver pair can arise if the system bandwidth is partitioned into several orthogonal channels (e.g., by partitioning in frequency and time), and then several of these channels are simultaneously available for communication between the transmitter-receiver pair. Even for narrow-band systems, multiple parallel channels can arise if the transmitter and receiver use multiple antennas (see Figure 2.9). As before, let the system bandwidth be W Hz. Suppose antennas that there are N transmit and M receive antennas. Let Gk, i, j 1 ≤ i ≤ M, 1 ≤ j ≤ N denote the channel gain between the transmit antenna j and the receive antenna i, at the k-th symbol. As we know, these channel gains will capture the path loss, the shadowing, and the multipath fading, and will be modeled as complex valued random variables. Let Gk denote the M × N channel gain matrix at symbol k. T Let Xk = Xk,1 , Xk,2 , . . ., Xk,N denote the input symbols into the N transmit antennas at the k-th symbol time. These too are complex valued, as would be the T case for general two-dimensional constellations. Let Yk = Yk,1 , Yk,2 , . . ., Yk,M denote the corresponding complex valued channel outputs. Hence, we can write Yk = Gk Xk + Zk

40

2 Wireless Communication: Concepts, Techniques, Models G1,1 G2,1

1

1

GM,1 2

G2,N

N GM,N

M

Figure 2.9 A multiple-input-multiple-output (MIMO) system comprising N transmit antennas and M receive antennas.

where Zk is the M × 1 additive noise process. The components of Zk are zero mean i.i.d. circularly symmetric Gaussian random variables, each with variance σ 2 ; also the Zk sequence is i.i.d. over k. This also means that Zk,i , 1 ≤ i ≤ M, are complex with their real and imaginary parts being zero mean independent Gaussian random 2 variables, each with variance σ2 . There is a total transmit power constraint of P: 1 |Xk,j |2 < P n→∞ n n

N

lim

k=1 j=1

Deﬁne, as before, Ψxmt = σP2 . Let us also assume i.i.d. Rayleigh fading between each transmit-receive antenna pair. Then Hk,i,j = |Gk,i,j |2 are i.i.d. exponentially distributed with a common mean over the antennas, say, φ. If the distance between the transmit antennas and the receive antennas is large, then the path losses between the antenna pairs would be the same. Let us denote this common path loss by π, as in the diversity analysis shown earlier. We “pull” the average path

2.3

Diversity and Parallel Channels: MIMO

41

loss, and the mean of the Rayleigh fading out of the channel gain matrix, leaving the mean power gain of the elements in the channel gain matrix to be 1. Then the received SNR, averaged over Rayleigh fading, is (as before) written as Ψrcv = φπΨxmt

The transmitter does not know the channel gains, and it can be shown that the best strategy is for the transmitter to split its power equally over the N transmit antennas. Then, given a sample of the gain matrix, say, G, it can be shown that the capacity of this channel is given by Ψrcv † C = W log2 det IM + (2.21) bits/second G·G N where det(·) denotes the “matrix determinant,” IM denotes the M × M identity matrix, and G† denotes “conjugate-transpose.” Now G·G† is an M×M Hermitian matrix (i.e., its conjugate-transpose is the same as itself). The theory of matrices provides the following facts: 1. The eigenvalues of G · G† are real and nonnegative. 2. The number of positive eigenvalues is no more than min{M, N}. Let us index the eigenvalues in decreasing order of magnitude and denote them by λ1 ≥ λ2 ≥ . . . ≥ λmin{M,N} . Then using the fact that the determinant of a square matrix is equal to the product of its eigenvalues, and that the eigenvalues of IM + ΨNrcv G · G† are of the form 1 + λj ΨNrcv , we obtain the following simpliﬁcation: min{M,N} Ψrcv C=W log2 1 + λj (2.22) bits/second N j=1

We see that, under the assumptions we have made, the multiple transmit antenna and multiple receive antenna system (also called a multiple-input-multiple-output (MIMO) system) is equivalent to several parallel channels. Note that, for different realizations of the channel gain matrix, the gains of the parallel channels (the eigenvalues λj , 1 ≤ j ≤ min{M, N}) will be different. In effect, we have parallel channels with random gains. Let us consider the situation in which M = N , and all the eigenvalues are equal, say, λ. Then λΨrcv C = WM log2 1 + bits/second M

42

2 Wireless Communication: Concepts, Techniques, Models

We see that for a single transmit and receive antenna system the capacity (i.e., W log2 (1 + Ψrcv )) scales as log Ψrcv for large Ψrcv , whereas for an M × M MIMO system (with equal eigenvalues) the capacity scales as M log Ψrcv . This is called multiplexing gain. Thus, we ﬁnd that a multiple antenna system can be used to obtain diversity gain (as explained above for one transmit antenna and M receive antennas), or can be used to increase the channel capacity by the creation of parallel spatial channels between the transmit and receive antenna groups. For an N transmit antenna, and M receive antenna system, the diversity gain is bounded by M × N , whereas the multiplexing gain is limited to min{M, N}. We note that the above discussion assumed that the channel gains are unknown at the transmitter. If channel gain estimates could be provided to the transmitter, then it could judiciously choose the transmitted symbols and their powers so that the better of the parallel spatial channels are assigned the larger transmit powers. We will study such optimal power allocation problems in the OFDMA context in Chapter 6.

2.4 Wideband Systems Unlike the narrow-band digital modulation used in FDM-TDMA systems, in CDMA and OFDMA the available spectrum is not partitioned, but all of it is dynamically shared among all the users. The simplest viewpoint is to think of CDMA in the time domain and OFDMA in the frequency domain. In a wideband system, a user’s symbol rate is much smaller than the symbol rate that the channel 1 can carry (i.e., W ). 2.4.1 CDMA In CDMA a user’s symbol, which is of duration L channel symbols (also called chips), is multiplied by a spreading code of length L chips. This is called direct sequence spread spectrum (DSSS), since this multiplication by the high rate spreading code results in the signal spectrum being spread out to cover the system bandwidth. If the user’s bit rate is R and the chip rate is Rc (> R), then L = RRc (> 1) and is called the spreading factor. The spreading codes take values in the set {−1, +1}L and are chosen so that each code is approximately orthogonal to all the time shifts of the other codes, and also to its own time shifts. Then the spread symbols are transmitted. All the signals interfere because they occupy the same radio bandwidth. We provide a simple analysis of such a system, with reference to the depiction in Figure 2.10. There are M users. The symbol duration is R1 , during which there are L chips. Denote the chip time by τc = R1c . We can write the transmitted signal from User 1 (see Figure 2.10) as x1

L−1 j=0

S1,j p(t − jτc )

2.4 Wideband Systems

43

3 (2)

(2)

S2,0

S2,(L 2 1)

2 x1

1 S1,0

S1,(L 2 1)

Figure 2.10 Depiction of the superposition of CDMA symbols. The transmissions of three users are shown. The tall ticks denote symbol boundaries and theshort ticks denote chip boundaries. A symbol of User 1 that has the value x1 ∈ {+ E1 , − E1 } has been shown. It has been spread by the code S1,j , 0 ≤ j ≤ L − 1. Interfering symbols of the other users are also shown. The interfering users are assumed to be chip synchronous but their symbols are randomly offset from that of the symbols of User 1.

where x1 is the user’s information carrying symbol, S1,j , 0 ≤ j ≤ L − 1,is User 1’s W spreading code, and p(t) is the baseband pulse that is bandlimited to − W 2 , 2 , and has the property ∞ p2 (u) du = 1 −∞

√ √ Let xi ∈ {+ Ei , − Ei }, where Ei corresponds to the transmit power used by User i. Let hi,1 denote the magnitude of the channel attenuation from the transmitter of User i to the receiver of User 1. For simplicity, let us work at the baseband, and then we can write the received signal at the receiver of User 1, over the duration of one symbol, 0 ≤ t ≤ R1 , as

y(t) =

L−1 j=0

h1,1 x1 S1,j p(t − jτc ) +

M L−1

(i)

hi,1 xi,j Si,j p(t − jτc ) + N(t)

i=2 j=0

where xi,j denotes the value of the symbol of User i that interferes with User 1 at (i) the j-th chip in User 1’s symbol (see Figure 2.10), Si,j denotes that a shifted version (denoted by the superscript (i)) of the spreading code of User i interferes with the chips of User 1, and N(t) is additive white Gaussian noise with power spectral

44

2 Wireless Communication: Concepts, Techniques, Models

density N0 , bandlimited to − W , W . The receiver of User 1 now performs the 2 2 following operation:

+∞

−∞

y(u)

L−1

S1,j p(u − jτc ) du

j=0

yielding the following statistic,6 based on which the transmitted symbol from User 1 has to be detected:

h1,1 x1 L +

M L−1

(i)

hi,1 xi,j Si,j S1,j + Z

i=2 j=0

where Z is zero mean Gaussian with variance N0 L (to see how this is obtained, see the derivation in the appendix of this chapter). The spreading codes are pseudo(i) random sequences taking values in {−1, +1}, and hence we model xi, j Si,j S1,j , 0 ≤ √ √ j ≤ L − 1, as i.i.d. random variables taking values in {+ Ei , − Ei }, each with equal probability. Thus we obtain the following symbol-by-symbol model for the CDMA channel: Yk = L h1,1 Xk + Ik + Zk (2.23) where Ik is the interference, Zk is additive noise (which is an i.i.d. Gaussian sequence with zero mean and variance N0 L), and we have assumed that the channel gains are not varying with time. Since the interference is the sum of contributions from many independent random variables, we model it also as having a Gaussian distribution. Note, from this calculation, that Ik has zero mean, and variance M

L hi,1 Ei

i=2

Hence, the detection performance will depend on (see Section 2.1.1)

M

L2 h1,1 E1

i=2 L

hi,1 Ei + N0 L

=

M

L h1,1 P1

i=2 hi,1

Pi + N0 W

6 The integration over (−∞, +∞) will actually cover neighboring symbols as well. But, because the pulses p(t − jτc ) are orthogonal, the terms that we display are all that we will get.

2.4 Wideband Systems

45

where we have taken Rc = W , and Pi = Ei ×Rc as the transmit power of User i (the power is the energy per chip times the chip rate). Thus the detection performance depends on the ratio M

L h1,1 P1

i=2 hi,1

Pi + N0 W

(2.24)

Now we can see why L is also called the processing gain. The effective predetection signal-to-interference-plus-noise ratio (SINR) for a user is the received SINR h P (i.e., M 1,1 1 ) multiplied by the processing gain L. i=2

hi,1 Pi +N0 W

Recalling the notation Td for the delay spread of the channel, let us write Ld = Tτcd : Ld is the number of chip-times that correspond to the delay spread. Now consider the signal arriving over paths that have delays that are multiple of the chiptimes. If the receiver can lock into any of these paths, then the transmitted symbol can be decoded as described earlier. If the paths fade independently, however, then we can exploit multipath diversity in much the same way as explained in Section 2.3. Because of the orthogonality property mentioned earlier, at the receiver, multiplication of the received signal by various shifts of the spreading code, and appropriate linear combination of the results, yields a detection statistic that is the sum of several faded copies of the user symbol. Since these shifts correspond to as many paths from the transmitter to the receiver, this is called multipath resolution. In the context of CDMA systems this is achieved by the Rake receiver. We note that this is exactly the same procedure as explained for receive antenna diversity in Section 2.3. Thus the Rake receiver permits a desired bit error rate to be achieved with a smaller SINR. Advanced receiver techniques such as interference cancellation now also are employed in CDMA systems. Scheduling transmissions in a CDMA system involves a decision as to the spreading codes and the power levels to be allocated to the users. These determine the rate at which a particular bit ﬂow can be transmitted. Of course, this decision will have to be made jointly for all users, since the decision for one user impacts every other user. We turn to such resource allocation problems in Chapter 5. 2.4.2 OFDMA We begin by recalling some notation. The system bandwidth is denoted by W , and the delay spread by Td . In a wideband system we are dealing with a situation 1 in which Td >> W , so that intersymbol interference has to be dealt with if we 1 . For example, we may have directly do digital modulation at a symbol rate of W W = 5 MHz, and Td = 5 μsec. OFDMA is based on OFDM (orthogonal frequency division multiplexing) (see [43]), which can be viewed as statically partitioning the available spectrum into several (e.g., 128 or 512) subchannels, each of bandwidth B, such that B > Td ); see Figure 2.11. The term orthogonal in OFDM refers to the fact that the center frequencies of the subchannels are separated by the reciprocal of the OFDM block time, T (see Figure 2.12). This makes the carriers approximately orthogonal over the block time. The subchannels can then be overlapping (i.e., B > T1 ), while the orthogonality between the subcarriers facilitates demodulation at the receiver. Let Xj,k , 1 ≤ j ≤ n, denote the j-th symbol in the k-th OFDM block (see Figure 2.11). The batch of n symbols, which are transmitted in parallel, is also called an OFDM symbol. Then the earlier discussion suggests that the predetection channel output can be written as Yj,k = Gj,k Xj,k + Zj,k

(2.25)

X1,k

X1,k11

X2,k

X2,k11

X3,k

X3,k11

X4,k

X4,k11

X5,k

X5,k11

{ {

User bit stream

{

11010011100111 0110 011011

OFDM Carriers

} } }

} } } }

where j, 1 ≤ j ≤ n, indexes the subcarrier and k ≥ 1 indexes the successive OFDM symbols. Gj,k denotes the fading on the j-th subcarrier during the k-th OFDM symbol. Zj,k denotes an additive noise sequence, which is taken to be i.i.d. zero mean, Gaussian.

T T T Successive OFDM blocks

Figure 2.11 Depiction of the mapping of user bits into OFDM symbols. Here there are ﬁve OFDM carriers. Serially arriving user bits are split into pairs that are mapped successively into ﬁve parallel channel symbols (X1, k , X2, k ,. . ., X5,k ), k ≥ 1 (for example, the 4-QPSK constellation could be used).These ﬁve channel symbols comprise an OFDM block, which is transmitted over the block time T.

2.4 Wideband Systems

47 1 T

B W

Figure 2.12 In OFDMA, the system bandwidth, W, is partitioned into overlapping subchannels, each of bandwidth B, with their center frequencies spaced apart by 1 , where T is the OFDMA symbol duration.

T

Let us see how this model can be justiﬁed. By the orthogonality requirement, the carrier spacing is the reciprocal of the OFDM block time, T1 . Then the number of carriers, n, is related to the system bandwidth, W , by 1 ×n=W T

As an example, consider T = 100 μsec, so that the carrier spacing is 10 KHz and, for W = 5 MHz, n = 500. Suppose the channel delay spread, Td , is such that 1 × n >> Td W

even though

1 W

< Td . Then, combining the previous two equations we ﬁnd that T >> Td

that is, the delay spread is much smaller than the OFDM symbol duration. We see that this is true in our numerical example, where T = 100 μsec and Td = 5 μsec. 1 Thus, the model in (2.25) is justiﬁed if the condition W × n >> Td holds. We see 1 that a frequency selective channel (for which W < Td ) gets converted to n parallel channels, each of which is frequency nonselective. If it is further true that T × N > W ), and we are left with ∞

1 Ck p(t − kT) + √ U(t) 2 k=−∞

(2.27)

U(t) is white Gaussian with power spectral density N20 Watts/Hz. W , + Since the signal is now bandlimited to the interval − W 2 2 , the average noise

The noise

√1 2

power is W ×

N0 2

=

WN0 2

Watts; this means that 1 t→∞ t

t

lim

0

U(x) √ 2

2 dx =

WN0 2

where the integrand on the left is the power dissipation if the noise was put across a 1 ohm resistor; the integration yields energy over (0, t), and the division by time yields the average power. The receiver also needs to synchronize to the pulse boundaries. Once this is done the demodulator then needs to look at each received pulse and determine which symbol it is carrying. This step is called detection. Let us now see √ how the k-th√symbol is detected, that is, how it is determined whether Ck = + Es , or Ck = − Es . The received signal is multiplied by the pulse p(t − kT) and integrated (−∞, +∞), the pulse p(t) being assumed to be known at the receiver.7 Since over +∞ 2 −∞ p (t) dt = 1, and the shifted pulses are orthogonal, this yields Ck +

+∞

−∞

U(t) √ p(t − kT) dt 2

Now U(t) is a zero mean Gaussian process; hence, using the fact that a linear combination of Gaussian random variables is again Gaussian, we conclude that 7 Since the pulses are practically time limited to some small multiple of T, such an integration can be

performed by storing the received signal for some multiple of T, before starting the integration.

50

+∞ −∞

2 Wireless Communication: Concepts, Techniques, Models U(t) √ p(t 2

− kT) dt is a zero mean Gaussian random variable, which we denote

by Zk . Thus, E(Zk ) = 0, and the variance of Zk is obtained as E

+∞

−∞

U(t) √ p(t − kT) dt 2

1 = E 2

2

+∞ +∞ −∞

−∞

U(t)p(t − kT)U(x)p(x − kT) dt dx

Since U(t) is a white Gaussian noise process, with power spectral density N0 , W W bandlimited to − 2 , + 2 , it can be shown that the covariance function of U(t) is given by E(U(t)U(x)) = N0

sin πW(x − t) π(x − t)

It then follows that +∞ +∞ 1 E U(t)p(t − kT)U(x)p(x − kT) dt dx 2 −∞ −∞ +∞ N0 +∞ sin πW(x − t) = p(t − kT) dt p(x − kT) dx 2 −∞ π(x − t) −∞ However,

sin πWx is just the πx W − 2 ,+ W 2 . Since

pass band frequencies, we have

+∞

−∞

transfer function of an ideal low pass ﬁlter with the pulse p(t) is bandlimited to this same range of

sin πW(x − t) p(t − kT) dt = p(x − kT) π(x − t)

We therefore conclude that E

Zk2

=E

+∞ −∞

U(t) √ p(t − kT) dt 2

2 =

N0 2

In a similar manner it can be shown that E(Zk Zl ) = 0, for k = l . Hence, since they are jointly Gaussian, Zk and Zl are independent for k = l . Thus, we ﬁnd that we have the symbol-by-symbol channel model Yk = Ck + Zk

where Zk is a sequence of i.i.d. zero mean Gaussian random variables with variance N20 .

Problems

51

Problems 2.1

Show that Perror−AWGN decreases exponentially with x, Q(x) ≈

2.2

2

x √1 e− 2 x 2π

Es N0 .

(Hint: for large

.)

Consider a mobile radio environment in which we model only path loss and Rayleigh fading. The path loss exponent is η. The transmit power, averaged over Rayleigh fading, at the reference distance d0 from a transmitter is P. a. Write down an expression for the random received power Prcv (d) at a receiver at a distance d = ad0 , and obtain the distribution of Prcv (d). b. Two cochannel transmitters (indexed 1 and 2) are simultaneously transmitting at distances d1 = a1 d0 and d2 = a2 d0 from the receiver. A transmission can be decoded if its signal to interference ratio exceeds γ. Ignoring the receiver noise, obtain the probability that the transmission from Transmitter 1 is decoded, treating the signal from Transmitter 2 as interference. This is called the capture probability (of Transmitter 1 over Transmitter 2). c. Determine β such that if a2 > (1 + β)a1 then the probability of transmission 1 being decoded is greater than 1− ( > 0 is very small).

2.3

Consider the binary modulation scheme analyzed in Section 2.1.1. Obtain the bit error rates for various SNR values γ = 12 dB, 11 dB, 10 dB, and 9 dB. In each case, calculate the probability of packet error for 1500 byte packets. Hence compare the plots in Figure 2.9 with the AWGN plot in Figure 2.12. Hint: Use the approximation Q(x) ≈

2

x √1 e− 2 x 2π

.

2.4

For the same situation as Problem 2.3 consider Rayleigh fading. For average (Rayleigh-faded) SNRs γ = 12 dB, 24 dB, and 36 dB, obtain the fraction of time that the SNR is less than 9 dB. Hence explain why a very large SNR is required in Figure 2.12 to obtain a high throughput.

2.5

By using the concavity of log(1 + x), show that the capacity in (2.16) is less than that in (2.18). What practical insight do we get from this? xmt Use Jensen’s inequality to show that Cfading − CSIR ≤ W log2 1 + E(H)P . WN

2.6 2.7

0

Consider two AWGN channels with the same (power) fading h, and noise power σ 2 . We have an amount of power P to assign. If the power Pi is hPi assigned to Channel i, the capacity achieved is ln 1 + σ 2 . Is it better to put all the power into one channel or to split the power over the two channels? What is the optimal power assignment, assuming that the transmitter knows that the two channels have the same power gain?

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CHAPTER 3 Application Models and Performance Issues

I

n Chapter 2 we provided an understanding of the issues and techniques involved in carrying bit streams over wireless channels. The resources required to carry a bit stream depend on the characteristics of the stream (e.g., the average rate, peak rate, and rate variability), and the performance required by the application generating the bit stream (e.g., an interactive voice call requires end-to-end delay bounds, but can tolerate some data loss). In this chapter we will discuss the major types of applications that telecommunication networks are used for, and the performance issues related to these applications, particularly when wireless networks are involved.

Overview We begin by providing a “big-picture” view of telecommunication networks as they exist today, showing all the elements, including the phone network, the Internet, and various wireless access networks. Then we outline various application scenarios that can arise in these interconnected networks. We classify network applications into three types according to the trafﬁc they generate for the network to carry, namely, elastic trafﬁc, real-time stream trafﬁc, and store-andforward stream trafﬁc. Taking interactive telephony as the principal example of real-time stream trafﬁc, we point out that the trafﬁc offered to the network can be either constant bit rate (CBR) or variable bit rate (VBR). We provide the quality of service (QoS) objectives for each case. The predominant use of the Internet is for applications such as e-mail and web browsing, which generate elastic trafﬁc. Such applications are also increasingly important for wireless access networks, as users begin to use their handheld devices for Internet access. The remainder of the chapter provides an understanding of this important type of trafﬁc. We show the need for feedback control of elastic trafﬁc sources. In the Internet such feedback control is exercised by the Transmission Control Protocol (TCP), which uses an adaptive window mechanism for managing the rates of elastic trafﬁc sources. The latter part of this chapter will be devoted to a discussion of the performance of TCP over wireless links.

54

3.1

3 Application Models and Performance Issues

Network Architectures and Application Scenarios

In Figure 3.1 we provide a simpliﬁed view of telecommunication networks as they exist today. The public switched telephone network (PSTN) has carried telephone calls for nearly a century. In this network, the calls are multiplexed onto the links by using circuit switching. Packet switched public data networks have evolved from the early X.25 networks to the present ubiquitous Internet. Cellular networks have provided mobile access since the early 1980s, and already have evolved through three generations of commercial deployment. As discussed in Chapter 1, a variety of resource allocation techniques are employed in cellular networks; these techniques will be the subject of Chapters 4, 5, and 6. In campuses and enterprises, mobile devices such as laptops and personal digital assistants (PDAs) obtain access to Internet services via wireless local area networks (WLANs). Figure 3.1 also shows some emerging technologies, namely, wireless metropolitan area networks (WMANs), and ad hoc multihop wireless mesh networks. Figure 3.1 also shows certain devices, generically called gateways (GW), interconnecting the PSTN, the Internet, and cellular networks. Note that we show only bearer gateways that are in the path of the actual application trafﬁc. Signaling gateways are also needed because signaling protocols are different in networks

Ad Hoc Multihop Wireless Network

D WLAN

R TX

GW

Internet

TX

A

WMAN

R

R

PSTN

CO

AP/BS

R R GW

LAN Switch

GW Cellular Network

VoIP Client

B

media server

E

C

Figure 3.1 A simpliﬁed view of the public switched telephone network (PSTN) and the Internet, and how they connect with each other and various wireless networking technologies. GW denotes a gateway; there are gateways for signaling and call control, and also for transferring trafﬁc across network boundaries. Other abbreviations are: CO – Central ofﬁce, TX – Trunk exchange; R – Router.

3.1

Network Architectures and Application Scenarios

55

that have evolved independently. For example, a signaling protocol called Session Initiation Protocol (SIP) is used to set up voice calls in the Internet, whereas the Signaling System No. 7 (SS #7) is used in the PSTN. In this setting we can identify several different instances of point-to-point communication, each of which gives rise to certain resource allocation issues. A common instance is that of a voice call between a ﬁxed line telephone instrument on the PSTN (e.g., A in Figure 3.1) and a cellular phone (e.g., C). We will often use the more generic term mobile station (MS) for a mobile device such as a cell phone or a PDA; in some technologies the terms station (STA) or subscriber station (SS) are used instead of MS. One of the functions of the gateway, GW, between the PSTN and the cellular network is to convert between the constant bit rate (CBR) ﬂow of voice bytes in the PSTN to a lower bit rate voice coding scheme over the resource limited cellular wireless network. In cellular networks, typically, voice is handled as a CBR stream of a lower bit rate than that in the PSTN. In an FDM-TDMA system, such as GSM, there are channels, each of which can carry one call at a ﬁxed rate. An accepted call is assigned to a channel for the entire duration of the call. Thus, this is essentially a circuit multiplexing system. A call for which no free channel is available is blocked and the main performance measure for the system is the blocking probability, deﬁned as follows. Let A(t) be the number of call arrivals until time t , and B(t) be the number of calls blocked in the same time; then the blocking probability Pb is given by B(t) t→∞ A(t)

Pb = lim

whenever the limit exists. In a CDMA cellular system, the voice connection is handled by assigning to it the required coded rate. However, unlike typical FDMTDMA systems, a variety of rates can be assigned. Each accepted call needs to be assigned a transmit power level, and whether or not a call is accepted depends on the rate requirements of the other calls that have already been accepted, and the resulting interference levels once the new call is accepted. Again, the performance measure of interest is the probability of call blocking. Another possible type of connection is a voice call between the PSTN instrument A and a voice over IP (VoIP) endpoint, say, B. The GW between the PSTN and the Internet then would convert between the CBR ﬂow of voice bytes in the PSTN to an asynchronous ﬂow of voice packets in the Internet. Since the packets ﬂow asynchronously in the packet network, and can be queued in buffers in the network routers, certain new issues arise, which will be discussed in Section 3.3. Similarly, there could be a voice call between either A or B and the endpoint D, which accesses the Internet via a WLAN or a WMAN. In such cases, the packet multiple access mechanism over the wireless access network affects the performance of the voice call. Yet another scenario is that of the MS C being used to browse the contents of the web server E which is attached to the Internet. We will see later, in Section 3.4,

56

3 Application Models and Performance Issues

that this kind of application is quite different from voice, as there is no intrinsic rate at which the data should be transferred from the web server to the mobile phone. Feedback-based rate control algorithms are employed to ensure some sort of rate fairness between such connections, and efﬁcient utilization of network resources. In the Internet, such control is exercised by TCP in conjunction with implicit or explicit congestion feedback from the network. In addition, a cellular access network such as a CDMA cellular network, or an OFDMA network, would have its own rate control algorithms. Unlike these centrally controlled cellular systems, a random access WLAN (see Chapter 7) does not have an explicit rate allocation mechanism. Hence, when a device such as D is engaged in web transfers from the Internet, it is of interest to determine what the throughput is, and what kind of fairness is achieved. Finally, the MS C or the device D could be displaying a video that is stored in the server E. Now, once the video starts playing, the network should provide this connection the average rate required to transport the video stream. Variability in the rate at which the network transports the video can be compensated by buffering a sufﬁcient amount of the video in the playout device. Such buffering must be done in such a way that the playout does not starve (i.e., the buffer empties out), nor does the buffer overﬂow. Figure 3.1 also shows an ad hoc multihop wireless mesh network attached to the Internet. Although multihop mobile wireless networks have been studied for more than three decades (in the early years under the name packet radio networks), even today the deployments of such networks are still experimental. It is one of the more active research areas in wireless networking, and we will provide a research oriented discussion in Chapters 8, 9, and 10. The IEEE 802.16 suite of protocols (popularly known as WiMax) now contains the deﬁnition of a mesh networking standard. Under this standard, nodes that cannot directly access a WiMax base station (BS) can form a static mesh network that is connected at some point to the WiMax BS. We can think of these as managed mesh networks. Such networks will be expected to carry all Internet services, respecting the QoS objectives that we will describe. On the other hand, ad hoc wireless mesh networks, such as community networks formed from WiFi access points in homes, cannot be expected to provide any consistent QoS to the applications they carry. We might expect that these would be used primarily for nonreal-time store-and-forward applications, such as e-mail and web browsing.

3.2 Types of Trafﬁc and QoS Requirements Based on the discussion of the various scenarios in Section 3.1, we can infer that applications generate one of the types of trafﬁc in the following list. Some example applications that generate each type of trafﬁc are also listed. • Elastic trafﬁc; e.g., WWW browsing, FTP ﬁle transfers, and electronic

3.2 Types of Trafﬁc and QoS Requirements

57

• Real-time stream trafﬁc; e.g., packet voice telephony • Store-and-forward stream trafﬁc; e.g., streaming movies or music over

the Internet. In the remainder of this section we discuss the characteristics of these trafﬁc types, and also their quality of service (QoS) requirements.

Elastic Trafﬁc Consider a data ﬁle, residing on the disk of the server E (shown in Figure 3.1) that needs to be transferred to the disk of a portable computer attached to the Internet (e.g., the laptop D, which connects via a WLAN), or to the memory of the cell phone C. Although the human (or some machine application) that wishes to achieve this ﬁle transfer would like to have the transfer completed in, say, a second or two, the source of data itself does not demand any speciﬁc transfer rate. If the data transfer does not lose data, no matter how fast or slow it is (but as long as the rate is positive), the ﬁle will sooner or later get transferred to the destination device. We say that, from the point of view of the network, this source of trafﬁc is elastic. Many store-and-forward services (with the exception of media streaming services) are elastic; e.g., ﬁle transfer, WWW download, electronic mail (e-mail). In this list, the ﬁrst two are distinguished by the fact that they are nondeferable (i.e., the network should initiate the transfer immediately), whereas e-mail is deferable. We observe that elastic trafﬁc does not have an intrinsic temporal behavior, and can be transported at arbitrary transfer rates. Thus the following are the QoS requirements of elastic trafﬁc. • Transfer delay and delay variability can be tolerated. An elastic transfer

can be performed over a wide range of transfer rates, and the rate can even vary over the duration of the transfer. • The application cannot tolerate data loss. This does not mean, however,

that the network cannot lose any data. Packets can be lost in the network (owing to uncorrectable transmission errors or buffer overﬂows) provided that the lost packets are recovered by an automatic retransmission procedure. Thus effectively the application would see a lossless transport service. Since elastic sources do not require delay guarantees, the delay involved in recovering lost packets can be tolerated. In practice, of course, users will not tolerate arbitrarily poor throughput, high throughput variability, and large delays. Hence a network carrying elastic trafﬁc will need to manage its resource-sharing mechanisms in a way such that some minimum level of throughput is provided. Further, some sort of fairness must also be ensured between the ongoing elastic transfers. Elastic trafﬁc can also be carried over circuit multiplexed networks (e.g., the PSTN or GSM cellular networks), or over networks that allocate a ﬁxed rate to

58

3 Application Models and Performance Issues

the elastic connection (e.g., a second generation CDMA cellular network). In this case, shaping of the trafﬁc so as to match the allocated rate should be carried out by the source. Obvious examples would be Internet access over a dial-up line in the PSTN, or a ﬁxed rate connection over a cellular access network being used for Internet access.

Real-Time Stream Trafﬁc Consider digitized speech emanating from an end-device involved in interactive telephony. This could be a periodic stream of bytes or packets, or, if silence suppression is employed then it could be an on-off stream of bytes or packets. Obviously, this source of trafﬁc has an intrinsic temporal behavior, and this pattern needs to be preserved for faithful reproduction of the speech at the receiver. The network will introduce delay: ﬁxed propagation delay, and, in packet networks, queuing delay that can vary from packet to packet (see Figure 3.2). Playout delay introduced at the receiver (to mitigate the effect of random packet delay variation) will be larger the more variable the packet delay. Hence, the network cannot serve such a source at arbitrary rates, as it could in the case of elastic trafﬁc. In fact, depending on the adaptability of such a real-time stream source, the network may need to reserve bandwidth and buffers in order to provide an adequate transport service to the source. Applications such as real time interactive speech or video telephony are examples of real-time stream sources.

first packet in a burst

peak rate 5 R bits/sec h

source output

X3 X2

network output

X1 t1

t2

t3 h

b input to playout device playout delay

Figure 3.2 A sequence of packets from a voice talk-spurt being transported across a packet network, and then being played out at the receiver after a playout delay. Each source packet contains h seconds of voice, the packet delays are X1 , X2 , . . ., the packets arrive at the receiver at times t1 , t2 ,. . ., and the playout delay is b. Notice that immediate playout at time t1 would have resulted in the talk-spurt being broken.

3.2 Types of Trafﬁc and QoS Requirements

59

The following are the typical QoS requirements of real-time stream sources. • Delay (average and variation) needs to be controlled. Real-time interactive

trafﬁc such as that from packet telephony would require tight control of source-to-sink delay; for example, for wide area packet telephony the delay may need to be controlled to less than 200 ms with a probability more than 0.99. Packets that do not conform to the delay bound are considered to be lost. • There is tolerance to data loss. Note that, from the point of view of the

receiver, packets can be lost for two reasons: (1) buffer overﬂows, or unrecovered link losses in the network, or (2) late arrivals at the receiver. Owing to the high levels of redundancy in speech and images, a certain amount of data loss is imperceptible. As an example, for packet voice in which each packet carries 20 ms of speech, and the receiver does lost-packet interpolation, 5 to 10% of the packets can be lost without signiﬁcant degradation of the speech quality [81], [54]. Because of the delay constraints, the acceptable data loss target cannot be achieved by ﬁrst losing and then recovering the lost packets; in other words, stream trafﬁc expects the intrinsic loss rate from the packet transport service to be bounded.

Store-and-Forward Stream Trafﬁc We can distinguish what we have just described as real-time stream trafﬁc from the kind of trafﬁc that is generated by applications such as streaming audio and video. Such applications basically involve a one-way transfer of an audio or video ﬁle stored on the disk of a media server. Consider a video stored in a server being played over a network. For example, the computer D or the handheld device C in Figure 3.1 may be used to watch a movie stored in the Server E. In order for the received video to be useful, the playout device should be continuously “fed” with video frames so that it is able to reproduce a smooth video output. This can be achieved by providing a guaranteed rate to the transfer, as would be done, for example, in a CDMA cellular system in the context of the device C. Alternatively, owing to the fact that the transfer is one way, a more economical way is to treat the transfer as elastic, and buffer the video frames as they are received. This would be the approach taken when the video is transferred over the random access WLAN to the computer D. Playout is initiated only after a sufﬁcient number of video frames has been buffered so that a smooth video playout can be achieved in spite of a variable transfer rate across the contention-based WLAN. Note that the same description holds for streaming audio. Thus, the problem of transporting streaming audio or video becomes just another case of transferring elastic trafﬁc, with appropriate receiver adaptation. Note, however, that the elasticity here is constrained since the average rate at

60

3 Application Models and Performance Issues

which the network transports the video bit stream must match the rate at which the video has been coded. Simple interactivity, such as the ability to rewind, can also be supported by the receiver storing frames that have already been played out. This, of course, puts a burden on the amount of storage that the playout device needs to have. An alternative is to trade off sophistication at the receiver with the possibility of interactivity across the network; that is, the press of the rewind button stops the video playout, frames stored in the playout device are used to create a rewind effect, and meanwhile additional past frames are fetched from the server. But this would need some delay and throughput guarantees from the network, requiring a service model somewhere in between the elastic and the real-time stream model that we have described earlier. We conclude that the QoS requirements of a store-and-forward stream transfer would be the following: • The average transfer rate provided in the network should match (in fact,

should be greater than) the average rate at which the stored media has been encoded. • The transfer rate variability should not be too large.

Thus store-and-forward stream trafﬁc is like stream trafﬁc since it has an intrinsic average rate at which it must be transported, but it does not have strict delay bounds, and hence the network can provide it a time varying transfer rate. In fact, TCP can be used to transport store-and-forward streaming media, provided the average TCP throughput does not drop below the average coded rate of the media. The added beneﬁt of TCP is that it recovers lost packets.

Closed and Open Loop Trafﬁc: It is appropriate to refer to real-time stream trafﬁc as open loop as it has an intrinsic temporal behavior. Typically, the rate of ﬂow on a connection is determined by the application, and these sources are not controlled by the network. In some systems, a limited amount of controllability is possible, by the sink alerting the source of poor playout quality, to which the source can respond by using a lower bit rate coder. On the other hand, closed loop controls invariably are used when transporting elastic trafﬁc, and, hence, such trafﬁc can be called closed loop. By means of implicit feedback (packet loss) or explicit feedback (control bits in packet headers) the source of the trafﬁc is made to continually adjust its rate of emitting data.

3.3

Real-Time Stream Sessions: Delay Guarantees

In this section we will discuss trafﬁc modeling and QoS issues for real-time stream sessions in the context of voice telephony. 3.3.1 CBR Speech Consider a voice call between a pair of endpoints in Figure 3.1. For example, the PSTN phone A and the cellular phone C, or between B and D, or between C

3.3

Real-Time Stream Sessions: Delay Guarantees

61

and B. In each end device, electrical signals from a microphone are digitized and coded by a speech codec. A typical approach is to sample the analog signal from the microphone at 8000 samples per second, to quantize the resulting continuous amplitude samples into 256 predetermined levels, and then encode each of these levels into 8 bits (one byte). The output of such a speech coder is called PCM (Pulse Code Modulation) coded speech (ITU’s G.711 standard). The PCM encoder, thus, yields a CBR source that produces 1 byte every 125 μseconds. A PCM source can be compressed to yield CBR sources at various rates. For example, ITU’s G.729 vocoder takes PCM as the input and produces 10 bytes of coded speech every 10 ms, thus yielding a coded bit rate of 1 KBps (kilobytes per second). However, this speech coder has a coding delay of 15 ms and a decoding delay of 7.5 ms. An important measure of the performance of network telephony is the Mouth-to-Ear (MtoE) delay—the delay between a sound being produced at the source device and this being heard at the other end. Thus, if the G.729 speech coder is employed, then there is a minimum MtoE delay of 22.5 ms. In order to carry a CBR voice source, it is necessary for the network to use a service rate greater than or equal to the voice bit rate. Further, if the source is allocated exactly the constant bit rate then there will not be any queuing. Hence, for CBR sources it is sufﬁcient to allocate the CBR rate. Consider a voice call between the PSTN phone A and the cell phone C. If the gateway GW converts PCM speech arriving over the PSTN to CBR speech at rate R, then the cellular network can just allocate resources so that the voice call is provided a service rate of R. This is typically what is done in an FDM-TDMA cellular system (such as GSM), or in a CDMA cellular system. We will discuss resource allocation issues in these two types of systems in Chapters 4 and 5, respectively. 3.3.2 VBR Speech In speech generated by interactive telephony, there are low energy periods that correspond to silences while the speaker listens, or to gaps between words, sentences, and utterances. The coder output corresponding to these inactive periods can be discarded or encoded at a lower rate. This yields a variable bit rate (VBR) coded speech. The VBR speech can be handled as a variable rate byte stream, or can be packetized for transport over a packet network. One approach is to take a certain number of bytes from the source (e.g., 160 bytes or 20 ms of speech from a PCM source) and generate a packet from these. It may happen that a talkspurt ﬁnishes before 160 bytes have been collected; in such a case a short packet is generated. The packetizer must wait to accumulate a packet; thus bytes that arrive early in the packet have to wait for those that arrive later, until the packet is formed. This results in a packetization delay, which can, obviously, be reduced by using shorter packets. Packets cannot be very short either, as there could be a signiﬁcant amount of header overhead in each packet (e.g., in the Internet there would be at least 12 bytes for RTP (Real-time Transport Protocol), 8 bytes for UDP (User Datagram Protocol), and 20 bytes for IP). If the coder output

62

3 Application Models and Performance Issues

during speech inactivity periods is discarded, then the output of a packetizer will comprise bursts of packets (during which packets are generated at a constant rate) and periods during which no packets are generated. Note that although the inactive periods do not have speech information in them, the duration of the gaps is indeed information that needs to be conveyed to the receiver. One of the difﬁculties in the transport of packetized VBR speech is in the retention of such timing information. Since packets are transmitted only during active periods, the inactive periods can only be approximately replicated at the receiver. It has been found that the resulting errors are not noticeable if the inactive periods are long. Thus the voice activity detection function (after the speech encoder) does not discard bytes from short inactive periods. Consider again the voice conversation between the PSTN phone A and the cell phone C in Figure 3.1, with VBR speech being used in the cellular network; that is, the voice arrives over the PSTN as a CBR ﬂow, but is encoded into a VBR ﬂow at the gateway, GW. Suppose the VBR speech source is allocated the service rate C in the cellular network (see Figure 3.3). Let us denote by R the peak rate of the VBR source, and by r¯ the average rate. Thus, for example, if the on-off VBR source has an average on duration of 400 ms and average off duration of 600 ms, then with R = 64 Kbps, we will have r¯ = 400400 + 600 × R = 25.6 Kbps. It is clear that it is a waste of bandwidth to make C > R, and that it is necessary that C ≥ r¯. Now suppose we take C < R. Notice that when the voice source is emitting data at the rate R, then the link buffer builds up at the rate (R − C ) Kbps. Any byte that arrives when the buffer level is, say, B bits will be delayed by CB ms. A priori, we do not know for how long this rate mismatch will last (the average rate r¯ = 25.6 Kbps could have been obtained with a 4 sec on time and a 6 sec off time too!). Hence, if we want to bound the delay of the voice bytes in the link buffer, in the absence of any other information about the source, our only recourse is to use C = R. This approach of peak rate allocation could be one way in which the cellular network manages its resources, and is typically the approach adopted in FDM-TDMA and CDMA cellular systems. In CDMA systems, even though the peak rate is allocated to a call, the on-off nature of VBR speech is exploited because during the voice silence periods a call does not cause interference (see

R r r T then the packet is discarded.

3.3

Real-Time Stream Sessions: Delay Guarantees

65

the receiver can stretch out the delay of each arriving packet to T ; thus packets that are delayed more than T are lost and may be interpolated. We are still left with the problem of determining a value for T . There are two alternatives. The packet network may have the ability to provide a delay guarantee at call setup (e.g., Pr(X > T) < where X is the delay of packets in the network; see Figure 3.5). In such a case, the endpoints specify the trafﬁc characteristics of the source they want to be carried, and the values of T and . The network evaluates whether the call can be accepted, and, if the call is accepted, the network sets up the appropriate mechanisms along the path of the call so that the delay objective is met. Now T is known to the receiver at call setup time, and the procedure shown in Figure 3.5 can be performed. If the network cannot provide delay guarantees, then the receiver would need to estimate T as the call progresses. Time stamps carried by the voice packets in their headers would be used to obtain a statistical estimate of T . This estimate can then be used to set the playout delays of arriving packets. Since there is no guarantee, the value of T could be larger than desired and could vary over time as congestion in the network varies. 3.3.4 QoS Objectives We gather that the MtoE delay for a packet voice call is the sum of several terms as shown in the following equation: MtoE Delay = coding delay + packetization delay + network propagation delay + network transmission and queuing delay + receiver playout delay + decoding delay

In this expression, the network propagation delay is just the signal propagation delay over the various media interconnecting the routers and switches in the path of the call. A rule of thumb is to compute this ﬁxed delay as 5 ms per 1000 Km of cabled transmission. Thus, for example, between points in the continental United States and India separated by a distance of about 20,000 Km, the one-way WAN propagation delay would be about 100 ms. For a geostationary satellite link, the one-way propagation delay is computed as the time taken for radio waves to travel from the transmitter up to the satellite and then down to the receiving ground station, or about 250 ms. In addition to the MtoE delay, some voice packets can be lost, either owing to buffer overﬂows in routers, or because they arrive after their scheduled playout time at the receiver. Thus, an example of the QoS expected by a voice call could be: Pr MtoE Delay > 200 ms < 0.02 and

Pr Packet Loss < 0.05

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3 Application Models and Performance Issues

We recall that packet loss includes loss due to late arriving packets, as well as loss due to buffer overﬂows and errors in the network. All such missing packets will need to be interpolated by the voice decoder, which leads to degradation in voice quality. Notice that the MtoE delay has some ﬁxed parts (coding and decoding delay, packetization delay, and propagation delay), and some variable parts (transmission and queueing delay, and the playout delay). It is these delays that are governed by the characteristics of the trafﬁc emitted by the source, and the way the trafﬁc is handled in the network (i.e., the other trafﬁc with which it is multiplexed, and the resource allocation decisions made by the network). Letting X denote the variable (random) delay, and then subtracting the ﬁxed delays from the MtoE delay target, the network performance requirement can be reduced to Pr(X > T) <

where, for example, = 0.02. Now consider a call between the devices B and D. The computer B is attached to the Internet by a high-speed enterprise or campus LAN, whereas D is attached to the Internet by a contention-based WLAN. One approach to the analysis of such a situation is to break down the end-to-end QoS objective into subnetwork-wise objectives. Thus, one could break up the end-toend delay bound T as T = T1 + T2 , and the probability of violating the delay bound can be split up as = 1 + 2 . We can call T1 and T2 the delay budgets in the respective subnetworks. Let the stationary random delay over the Internet segment of the call be denoted by X1 and that over the WLAN be denoted by X2 . Suppose we ensure that, for each i = 1, 2, Pr(Xi > Ti ) < i

It will then follow that Pr(X > T) = Pr(X1 + X2 > T1 + T2 ) ≤ Pr({X1 > T1 } ∪ {X2 > T2 }) < 1 + 2 =

where the ﬁrst inequality follows from the simple observation that if X1 + X2 > T1 + T2 , then it cannot be that X1 ≤ T1 and X2 ≤ T2 ; the last inequality is just the union bound. The resource allocation in the WLAN can then be performed so as to ensure that the voice packet delay exceeds T2 with a probability less than 2 . The same approach can be used if end-device D is attached to the Internet via a WMAN (see Figure 3.1).

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Elastic Transfers: Feedback Control

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3.3.5 Network Service Models In the previous section we showed how to derive, from the end-to-end delay QoS problem for a connection, a QoS objective for the access network. If the access network accepts calls based on peak rate allocation, then the only issue is to design the network for a desired call blocking probability. For this purpose, the Erlang blocking model (see Appendix D) can be used. If the access network assigns a ﬁxed rate less than the peak rate of a VBR call, then the model depicted in Figure 3.3 can be used. The analysis of such models has been discussed at length in [89, Chapter 5]. In some access networks, however, the service rate applied to a connection may not be constant. For example, in OFDMA systems, the number of bytes to be served from a queue can vary from frame to frame depending on the fading in the various carriers, the competing trafﬁc, and the power constraint. Thus, in this case we have a dynamically controlled server; see Chapter 6. Detailed analysis of such systems to obtain buffer occupancy distributions or delay distributions is difﬁcult. Wireless LANs, based on the IEEE 802.11 standard, are contentionbased systems. Hence the service applied to a queue is time varying because the number of contending nodes varies over time, as some queues empty out while others receive new trafﬁc. Some progress has been made on developing analytical models for the performance analysis of wireless LANs; some of these approaches will be discussed in Chapter 7.

3.4

Elastic Transfers: Feedback Control

Elastic trafﬁc is generated by applications whose basic objective is to move chunks of data between the disks of two computers connected to the network. Elastic ﬂows can be speeded up or slowed down depending on the number of ﬂows contending for the capacity of the network. Figure 3.6 shows that, at the most basic level, an elastic session simply involves the transfer of some ﬁles from one host attached to the packet network to another host. For example, the two hosts could be e-mail relays; each ﬁle transfer would then correspond to an e-mail being forwarded toward its destination mail server. Alternatively, the source host could be an FTP archive, and at the destination host, a user is downloading several ﬁles during an FTP session. A similar example would be that of the source being a web server, and the destination being a client with a web browser, using the HTTP (Hyper-Text Transfer Protocol) protocol to browse the ﬁles at the server. In an internet, for example, when a user requests a web page (using an HTTP GET request), ﬁrst, a base ﬁle is downloaded, which in turn may trigger the transfer of several embedded objects, such as images. When there are embedded objects, the exact mechanism for downloading the objects depends on the version of HTTP in use. In HTTP 1.0, for the transfer of the base ﬁle, and for each embedded object ﬁle, a separate TCP connection would be set up between the client and the server.

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source of files Packet Network

destination of files

Figure 3.6 An elastic session simply involves the transfer of one or more ﬁles from one host to another.

In HTTP 1.1, in order to reduce connection set up overheads, a TCP connection once set up would be reused for several ﬁle transfers between the same client and server. In all these cases the basic problem is to transfer each ﬁle in its entirety from the source machine to the destination machine. This is the primary objective. There is no intrinsic rate at which the ﬁles must be transferred. In fact the transfer rate can vary as a ﬁle is transferred. Although the user downloading the ﬁles would want to receive the ﬁles quickly, this requirement is not really a part of the service deﬁnition of an elastic session. Further, we note that there is no intrinsic packet size that the ﬁles need to be segmented into during their transfer. The transfer protocol can just view each ﬁle as a byte stream, and transfer varying amounts of it in each packet. We will deal primarily with point-to-point elastic sessions, and this is what we will mean when we use the term “elastic session.” Thus an elastic session involves a connection between two endpoints. The network determines a route between the endpoints in each direction; if the session lasts long enough, there is a possibility that the route, in either direction, changes during the session. During a session, data transfers may take place in either direction, with possible gaps between the successive transfers. For example, if a user at a computer logs on to an FTP server, then FTP’s get or put commands can be used to download or upload ﬁles. The user may need to do some other activities in between the ﬁle transfers (e.g., read what is downloaded and make notes); in user models, these gaps often are referred to as think times. Similarly, a user browsing a web server would download a web page, and spend some time looking at it, before downloading another web page from the same site. If the user shifts to browsing another web server, then we view this as another session starting, typically over a different pair of network routes.

3.4

Elastic Transfers: Feedback Control

HOST 1

69

bandwidth of this pipe is shared between a varying number of elastic flows

WEB

HOST 2

SERVERS

feedback control necessary to slow down or speed up the traffic sources HOST n

users downloading data from the web servers

Figure 3.7

Several users dynamically share a link to download ﬁles from web servers.

3.4.1 Dynamic Control of Bandwidth Sharing Figure 3.7 shows a very simple “network” comprising a single link over which several users, on their respective hosts, are downloading ﬁles from some web servers. Let us take the link capacity to be C bps, and assume that the local networks attaching the users and the web servers to this link are inﬁnitely fast. We use this simple scenario to illustrate and discuss some basic issues that arise when several elastic sessions share the network bandwidth. In fact, the situation depicted in Figure 3.7 is similar to what happens when several mobile users download ﬁles (text, music, video, etc.) from a server attached to a cellular operator’s own high speed local area network. The important difference is that in cellular systems the system bandwidth is not managed as one “fat pipe.” Suppose, to begin with, a single user initiates a download from a web server over the link. It is reasonable to expect that an ideal data transfer protocol will (and should) provide this ﬁle transfer with a throughput of C bps. This much bandwidth is available, and if all of it is provided to the transfer, the session will be out of the system as early as possible. Now suppose another user starts a session, while the ﬁrst ﬁle transfer is still progressing. When the corresponding web server starts transferring data toward the user, the total input rate into the link (from the web server’s direction) will exceed C bps. If the ﬁrst ﬁle transfer is proceeding at C bps, then the link’s service rate will be exceeded no matter how slowly the second server sends its data. This will lead to link congestion. The network device that interconnects the server’s LAN to the backbone link will have buffers “behind” this link. These buffers can absorb excess data that accumulate during this overload, provided the situation does not sustain for long. In addition,

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this situation is clearly unfair, with one user getting the full link rate, and the other user getting practically no throughput; this situation should not be allowed to persist. Both of these issues (congestion and unfairness) require that there be some kind of feedback (explicit or implicit) to the data sources so that the rate of the ﬁrst transfer is reduced, and that of the second transfer is increased, so that ultimately each transfer obtains a rate of C2 bps. Now suppose the ﬁrst ﬁle transfer completes; then, if the second transfer continues to proceed at C2 bps, the link’s bandwidth is wasted, and the second session is unnecessarily prolonged. Hence, when the ﬁrst session departs, the source of the second session should increase its transfer rate so that a throughput of C bps is obtained. In summary, from this discussion we conclude that an explicit or implicit feedback control mechanism needs to be in place so that as the number of sessions varies, the transfer rate provided to each session varies accordingly. By an explicit feedback we mean that control packets ﬂow between the trafﬁc sources, sinks, and the network, and these packets carry information (e.g., an explicit rate or a rate reduction signal) that is used by the sources to adapt their sending rates. On the other hand, implicit feedback can be provided by packet loss or increase in network delay; that is, a source can reduce its rate on sensing that one of the packets it sent may not have reached the destination. For example, in much of the current Internet, TCP (Transmission Control Protocol) uses an implicit feedbackbased congestion control mechanism. Explicit rate control was proposed for the ABR (available bit rate) service in ATM (asynchronous transfer mode) networks. The idea was to associate with each ABR session a ﬂow of control cells (called Resource Management (RM) cells) generated by the source. As the RM cells of a session ﬂow through the network, the ATM switches in their path could set an explicit rate value in these cells. The sink would return the RM cells back to the source, and the source could use the explicit rate in the returned RM cells to adjust its cell emission rate. On the other hand, TCP uses a windowbased transmission protocol. For a ﬁxed round trip delay, the TCP throughput is proportional to the average window size. Thus, the window can be adapted to vary the TCP transfer rate. Window adaptation works by a TCP source detecting lost packets, taking these as indications of rate mismatch and network congestion, and voluntarily reducing its transmission rate by reducing the transmission window. 3.4.2 Control Mechanisms: MAC and TCP As mentioned in the previous section, and as is clear from our discussions in Chapter 2, in wideband cellular wireless networks, the entire system bandwidth is not used as one fat pipe. Instead, radio resource allocation is done on an MS by MS basis, depending on the channel conditions to the mobiles. Thus, even at the medium access layer it is possible to implement control strategies that achieve some sort of a rate allocation objective over the MSs. For example, the objective could be

3.4

Elastic Transfers: Feedback Control

71

equal rate allocation (this is an example of a more general fairness objective called max-min fairness); such an approach might be very inefﬁcient as the MS with the weakest link will determine the rate that all MSs obtain. Another objective could be to allocate rates so as to maximize the total rate over the MSs; such an approach might be very unfair as MSs with poor connectivity might obtain no throughput. We will examine MAC level rate allocation for elastic trafﬁc in CDMA cellular systems in Chapter 5. In CSMA/CA based wireless LANs, the medium access control protocol results in some default bandwidth sharing. If only downlink ﬁle transfers are considered then it is found that the IEEE 802.11b standard results in equal rate sharing, irrespective of the physical rate at which MSs are connected. We will provide an analytical model for understanding this in Chapter 7. Bandwidth sharing in the wide area Internet is controlled by the Transmission Control Protocol (TCP) that resides in all end-systems attached to the Internet, including Internet-enabled cellular phones. In OSI terminology, TCP is a Transport Layer protocol. Thus TCP sits between the applications and IP, the Internet’s packet routing and forwarding protocol. TCP is connection oriented, which means that a connection has to be established between the endpoints before data transfer can start, and this connection is taken down when the data transfer completes. TCP enhances the unreliable, nonsequential packet transport service provided by IP to a reliable and sequential packet transport service. It uses a window-based packet loss recovery mechanism to achieve this function. In addition, the windowbased mechanism is employed for two other major functions that TCP provides: (1) sender-receiver ﬂow control, which prevents a fast source of packets (at the application level) from overwhelming a slow sink, and (2) adaptive bandwidth sharing in the network. The TCP transmitter maintains a congestion window that increases if packets are acknowledged in sequence. On the other hand if the desired acknowledgment (ACK) fails to show up then the transmitter takes this as an indication of congestion, and reduces the transmission window. The transmission window can also be controlled by the receiver, by a window advertisement in the ACK packets. By the latter means, the receiver can exercise ﬂow control over the transmitter. For a connection, the number of packets in the network is roughly related to the TCP window, and the average window divided by the mean round trip packet delay is an estimate of the TCP throughput. The adaptive window-based congestion control mechanism of TCP has evolved over several versions. In the earliest version, any packet loss resulted in a transmitter time-out, and the reduction of the congestion window to one. A TCP receiver continues to accept packets even if previous packets are missing. For all such out-of-order packets, the transmitter returns an ACK packet “asking” for the ﬁrst missing packet. These ACKs are called duplicate ACKs. Thus, duplicate ACKs are an indication of out-of-order packets at the receiver, and multiple duplicate ACKs are indicative of packet loss. In a later version, called TCP Tahoe, loss recovery was initiated at the transmitter by the receipt of three duplicate ACKs; this was called fast retransmit. However, the transmitter dropped the congestion

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3 Application Models and Performance Issues

window to one. Thus, although loss recovery started earlier than the older version (say, OldTahoe), the window had again to be built up from one. TCP Tahoe was followed by TCP Reno in which, on the receipt of three duplicate ACKs, the congestion window was cut by half, and loss recovery was initiated; this was called fast retransmit and fast recovery. A more aggressive loss recovery than TCP Reno was implemented in TCP NewReno. This version was followed by another improvement in TCP SACK, in which the TCP transmitter uses the ACK packets to send the pattern of missing TCP packets back to the TCP transmitter. From the foregoing, we see that the rate allocation achieved by elastic transfers to or from devices that attach to the Internet via wireless access networks will be governed by the interaction between the rate allocation strategies in the wireless MAC, the behavior of the wireless channel, and TCP’s window-based end-to-end control mechanisms. 3.4.3 TCP Performance over Wireless Links In [89, Chapter 7] we have discussed the TCP protocol at length and have studied models for evaluating the performance of TCP-controlled ﬁle transfers in several situations. We studied a model that can be used to obtain the performance of TCP controlled ﬁle transfers with random packet loss. We saw that the performance of TCP can be signiﬁcantly affected by packet loss. In these discussions, the concern was with congestion-related loss; that is, either a packet was lost owing to buffer overﬂow, or a packet was deliberately dropped at a router queue owing to imminent congestion. We were not concerned with the possibility of packet loss in the physical bit carriers. In a sense, we were assuming a wired physical infrastructure. Wired links can be properly established so that they have small BERs. On the other hand, mobile wireless links can have high packet loss rates, and are subject to random variations in their quality. Also, in CSMA/CA based wireless LANs, it is unrealistic to model the service provided to a ﬂow as being at a constant bit rate. It is therefore of interest to study the performance of TCP transfers over wireless access networks, particularly in light of the growing importance of mobile wireless access to the Internet. It is well known that the bandwidth delay product (BDP) (normalized to the packet length) along a path in a network is deﬁned as 2Cδ L

where C is the bottleneck link rate along the path of the TCP connection, 2δ is the round-trip propagation delay (RTPD), and L is the packet length. If the TCP window grows to the BDP and stays at that value, then the bottleneck link can be kept fully occupied. This yields the highest possible TCP throughput on that path. With this in mind, let us consider elastic transfers from the server E to an MS associated with the cellular network in Figure 3.1. Let us suppose that the cellular system assigns a ﬁxed service rate to each transfer, where the rate

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Elastic Transfers: Feedback Control

73

depends on the condition of the channel to the MS and on the other MSs that are being served by the system. Typical service rates would be in 100s of Kbps, and hence the cellular network would be the bottleneck in the path of the TCP connection. Even if the server E is 20,000 Km away (halfway around the earth), thus yielding an RTPD of about 200 ms, we have a BDP of about four packets for a bottleneck rate of 250 Kbps and L = 1500 bytes. The TCP maximum window size implemented in various operating systems is 20 packets or more. Hence, assuming that the wide area Internet has negligible packet loss, the TCP window is well above the minimum to keep the bottleneck link busy.1 This observation permits us to make the simpliﬁcation that we may ignore the wide area packet network, and study only the interaction between TCP and the behavior of the wireless link. It is as if the server E was attached to the local area network of the cellular operator.

Independent Packet Losses With this discussion in mind, Figure 3.8 shows a simple scenario in which a mobile host is doing a TCP controlled ﬁle transfer from a ﬁle server on a wired LAN. The LAN wireless router network would be located at the base station. The propagation delay between the base station and the mobile host is negligible. The BER on the wireless link is such that packets are lost with probability p. The packets are lost independently; correlated losses owing to channel fading are not modeled here. Only ACKs are sent from the mobile host to the LAN, and since these are small (40 bytes), their loss probability is ignored; recall that TCP uses cumulative ACKs, which further limits the effect of ACK loss. The link layer random packet losses

Server

TCP IP

Mobile Station

Base Station LAN – Wireless Router

TCP IP

IP link

link modem

modem

local area network

Figure 3.8 A mobile station transferring data over a wireless link from a server on the LAN attached to the base station.

1 A simple way to quantify the effect of random losses in the wide area Internet is to use the square root formula 1.5 p where p is the packet loss probability. This formula gives an approximation to the mean

window size of a TCP connection over a wide area network, if the loss probability is p and the connection stays in congestion avoidance. Typical values of p in a well-engineered ISP network would be 0.001 or 0.005. The resulting average window is well above the 4 required to keep the bottleneck link busy.

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3 Application Models and Performance Issues

protocol is unable to recover all the wireless packet losses; hence any residual packet losses have to be recovered by TCP. As the TCP transmitter on the ﬁle server grows its window, the wireless link buffer builds up. The buffer can hold as many packets as needed; that is, there is no buffer loss. Eventually, a loss occurs in the wireless link, and one of the loss recovery mechanisms is invoked. The throughput of a large ﬁle transfer can be analyzed via a stochastic model of the TCP protocol with random packet losses. A sample of results obtained from this analysis is shown in Figure 3.9. The parameters and the results are normalized. We plot the ﬁle transfer throughput versus the packet loss probability. The throughput is normalized to the bit rate of the wireless link. One set of parameters that would correspond to the results is LAN speed; 10 Mbps wireless link bit rate: 2 Mbps; TCP packet length: 1500 bytes (hence the packet transmission time is 6 ms); time-out granularity: 420 ms; minimum time-out: 600 ms; and Wmax = 24 packets, where Wmax is the maximum TCP window. The performance of four versions of TCP is compared: OldTahoe (which is the name we give to the version of TCP that predates Tahoe and always requires time-outs to recover losses), Tahoe, Reno, and NewReno. We observe that even with a packet loss probability of 0.001, the throughput with OldTahoe is less than the full link rate,

Packet throughput, normalized to speed of lossy link

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1023

NewReno: K = 3 Tahoe: K = 3 Reno: K = 3 OldTahoe 1022 1021 Packet Error Probability

100

Figure 3.9 File transfer throughput (normalized to the link’s bit rate) vs. packet loss probability for various versions of TCP; OldTahoe refers to a version that recovers losses only by timeout. K is the duplicate ACK threshold for fast-retransmit in the TCP loss recovery protocol. Adapted from Kumar [85].

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Elastic Transfers: Feedback Control

75

and drops to just over 50 percent of the link rate for a packet loss probability of 0.01. The other three TCP versions implement fast-retransmit and they yield 100 percent throughput at p = 0.001, and better than 95 percent throughput up to p = 0.01. Beyond 1 percent packet loss, the performance of these versions too begins to drop, and is not much better than OldTahoe for a 10 percent packet loss rate. Reno is slightly better than Tahoe up to p = 0.02, but becomes worse for large loss rates since multiple losses cause it to waste more time than Tahoe. The more aggressive fast-recovery of NewReno results in this version yielding almost 90 percent throughput up to p = 0.03. We can make a broad observation that random packet loss probabilities larger than 1 percent can signiﬁcantly affect the performance of TCP, with these parameters. Note that the coarse time-out and minimum time-out values are large in this example. Smaller values for these parameters will yield better performance, as losses will then result in less wastage of link capacity. Thus, we see that there is a maximum packet loss probability below which the TCP throughput is just the bit rate of the wireless link. If the packet loss probability is ensured to be less than this maximum then the effect of TCP can be ignored, and we can just take the bit rate provided by the MAC mechanisms as the transfer rate obtained by the elastic application. If a packet loss probability of p is desired, and the packet length is L bits, then the BER on the wireless link should satisfy the requirement p = 1 − (1 − )L . An upper bound on p thus yields an upper bound on the BER. Hence we see that the performance of the application we wish to carry on the wireless link puts a requirement on the performance of the link. We saw in Chapter 2 that the BER on a wireless link is a function of the SNR. Hence, for a given modulation and coding scheme, the desired BER places a requirement on the minimum SNR at which the link can operate. We also note that a desired packet loss probability can be obtained by using an ARQ protocol over a physical link with a higher BER than calculated from the formula above. Since the propagation delay on cellular links is negligible (i.e., the number of bits “in ﬂight” is much smaller than the packet length), a stop-and-wait ARQ sufﬁces. The overall effect of using an ARQ protocol is that we have a lower bit rate link (due to ARQ overheads and retransmissions) with the desired packet loss rate.

Correlated Packet Losses We now turn to the performance of TCP controlled ﬁle transfers over a fading channel. In Section 2.1.4 we discussed models for channel fading. We pointed out that the fading is correlated in time. Thus for a given average BER there would be periods when the BER is greater than the average, and periods during which the BER is less than the average. A similar statement can be made for the packet error rate if ﬁxed length packets are being used, as is typically the case with large ﬁle transfers over TCP. A simple approach is to model the channel as being in one of two states: a Good state (during which a packet transmission is

76

3 Application Models and Performance Issues (12g)

g

Good

Bad

b

(12b)

Figure 3.10 Transition structure of the two-state Markov model for a fading channel.

successful), and a Bad state (during which a packet transmission is unsuccessful). A further simpliﬁcation is to model the state process as a two-state Markov process, on the state space {Good , Bad } (see Figure 3.10). The durations in each state are taken to be multiples of the packet transmission time. The transition probabilities of the Markov chain are obtained by specifying the amount of fading that leads to a bad transmission (at other times a good transmission is assumed). The marginal distribution of the fading process and results about correlations in the fading process can be used to obtain the transition probabilities. For a packet length L, channel bit rate C , and Doppler frequency fd , the parameter fd CL is a measure of the fade durations relative to the packet transmission time; thus L fd C = 0.01 means that channel coherence time is roughly 100 packet transmission times. Using this Markov model for the channel state, the analysis of the throughput of a long ﬁle transfer under TCP can be performed by developing a certain stochastic model. In performing this analysis, in addition to the state of the TCP window adaptation process, the state of the channel will also need to be maintained. Figure 3.11 shows some typical numerical results with Rayleigh fading. The normalized throughputs with TCP Tahoe and Reno are plotted versus the average packet error probability, with and without fading. For the results with fading, the parameter fd CL = 0.01. The other parameters are the same as in Figure 3.9, except that the local area network is taken to be inﬁnitely fast. Notice that the performance without fading is similar to that depicted in Figure 3.9. With fading, the performance is signiﬁcantly different. For the same probability of error, we ﬁnd that the performance of TCP Tahoe increases substantially, whereas that of TCP Reno drops for p < 3 × 10−2 , and improves for large packet loss probabilities. This can be understood as follows. With independent losses, the repeated reductions in the window lead to a small effective window; hence when a loss occurs there are not enough packets in circulation to generate the number of duplicate ACKs required for a fast retransmit. Thus with uncorrelated losses, time-outs are more frequent. When packet errors are clustered (as in the case of fading), the durations between packet loss events are larger. Hence with correlated packet losses, the TCP transmitter is able to grow its window to larger values than

3.4

Elastic Transfers: Feedback Control

77

packet throughput, normalized to link bit rate

1.0

0.8

0.6

0.4

0.2

0.0 1023

Tahoe analysis (i.i.d.) Tahoe analysis (fading) Reno analysis (i.i.d.) Reno analysis (fading) Tahoe simulation (i.i.d.) Tahoe simulation (fading) Reno simulation (i.i.d.) Reno simulation (fading)

1021 1022 packet error probability

100

Figure 3.11 File transfer throughput (normalized to the link’s bit rate) vs. packet loss probability for TCP Tahoe and Reno; with independent losses (denoted as i.i.d.), and with Rayleigh fading with fd L = 0.01. Adapted from Zorzi et al. [144]. C

in the independent packet loss case (for the same average packet error probability). When a loss does occur, it is more likely that there are enough successful packets sent subsequently in the window to trigger a fast retransmit. Even if a time-out does occur, it is long enough to last out the fade, so that when transmission resumes, the channel is likely to be in the Good state. For small values of p, the performance of Reno is worse since Reno requires additional duplicate ACKs for recovering each lost packet. With correlated losses, multiple losses are more likely and this results in Reno wasting more time than Tahoe. Reno attempts to perform a fast-retransmit for each lost packet, spends time in this process waiting for duplicate ACKs, and then times out anyway. For large values of p, the two protocols have similar behavior since with the high loss rate the window grows to small values, the number of duplicate ACKs are insufﬁcient to trigger a fast-retransmit, and hence it is very likely that both protocols recover with a time-out. Although this discussion illustrates the effect of correlated errors on TCP controlled ﬁle transfer performance, it is important to make a comparison by ﬁxing the average SNR. The same two-state Markov model can be used. The SNR that corresponds to a Bad state is ﬁrst ﬁxed. Then for each SNR and Doppler frequency, the two-state Markov model can be parametrized. Sample results

3 Application Models and Performance Issues Packet throughput, normalized to speed of wireless link

78 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3

TCP Tahoe; no fading TCP Tahoe; fade = 1 pkts TCP Tahoe; fade = 2 pkts limit, speed –>0

0.2 0.1 0 10

15

20 25 30 Mean Signal to Noise Ratio (dB)

35

40

Figure 3.12 File transfer throughput (normalized to the link’s bit rate), withTCP-Tahoe, vs. SNR in dB, with no fading (AWGN only) and with Rayleigh fading. The legend fade = n pkts means that the mean Bad state duration is n packets, where a Bad state occurs if the SNR < 10 dB. Adapted from Kumar and Holtzman [87].

for TCP Tahoe are shown in Figure 3.12. In order to compare with the results presented earlier, no channel coding or link level retransmissions are taken into account. The normalized throughput is plotted against the average SNR in dB. We observe that without fading, an SNR of about 12 dB sufﬁces to obtain a TCP throughput of over 90 percent of the link rate. This is because the packet error probability itself is very small without fading. With fading, however, much larger Rayleigh faded SNRs are required; between 25 to 30 dB for a throughput of 90 percent of link rate. We notice that slower fading and hence more correlated errors improves the TCP throughput, but the throughput even with this improvement is much worse than that without fading. We also show the case of speed → 0. This corresponds to the fade level being constant during the entire TCP transfer; either the channel is good throughout the transfer, or is bad throughout. This is a bound on the achievable throughput with fading. As the faded SNR decreases, the probability of the Good state reduces and hence the bound rapidly decreases for decreasing average faded SNR.

3.5

Notes on the Literature

Extensive analytical treatments of QoS issues and models is provided in [89], [133], and [119]. References [81] and [54] provide a discussion of issues in transporting

3.5

Notes on the Literature

79

voice over packet networks. The material on analysis of TCP controlled ﬁle transfer throughput over lossy wireless links has been taken from the papers by Kumar ([85], which assumes i.i.d. packet loss) and by Zorzi et al. ([144], which accounts for correlated packet losses). An approach for two-state Markov modeling of a fading channel is provided by Zorzi et al. in [146]. Additional references on TCP throughput analysis with correlated packet losses in the wireless setting are Kumar and Holtzman [87], Zorzi and Rao [145], and Anjum and Tassiulas [3].

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CHAPTER 4 Cellular FDM-TDMA

T

he FDM-TDM technique for allocating spectrum and time resources to calls is the most classical one, and systems based on this technique carry a substantial majority of the cellular telephony trafﬁc around the world. Many of the basic techniques of cellular telephony emerged from the design of such systems; for example, techniques such as frequency reuse management, cell sectorization, power control, and handover management.

Overview One of the main ideas developed in this chapter is that of spatial reuse by partitioning the available FDM carriers into reuse groups, and then allocating these reuse groups to cells in such a way that cochannel interference is within acceptable limits. It is shown that the cochannel interference constraint places a constraint on the D R ratio, the ratio between the shortest distance between cochannel cells and the cell coverage radius. This analysis is based on signal-to-interference ratio (SIR) modeling, where we use a power attenuation model that includes path loss and shadowing. Assuming that the cells form a hexagonal tessellation of the plane, the D R ratio is related to the number of cochannel reuse groups into which the cells must be partitioned. It is shown how the partitioning of the channels and other system parameters affect the spectrum efﬁciency. Once channel allocation constraints are understood, various channel allocation strategies are considered and a call blocking analysis is developed. Finally, we consider intercell handovers. We show how signal strength measurements from neighboring BSs are used to determine that a call needs to be handed over between cells. An approximate handover blocking analysis is also shown. The chapter ends with an overview of call handling in GSM, the most widely deployed FDM-TDMA cellular system.

4.1

Principles of FDM-TDMA Cellular Systems

Suppose that a system bandwidth of Wsystem is to be used for providing FDMTDMA based telephony services in a certain coverage area, say a town in some country. The operator would have to pay a substantial fee to the authority managing the spectrum in that country, and hence it is in the operator’s interest to maximize the revenue from operating the service while keeping costs down. We begin by providing an understanding of the issues involved in the efﬁcient design

82

4

Cellular FDM-TDMA

of an FDM-TDMA cellular telephony system. We will refer to a mobile handset as an MS (mobile station) and to the ﬁxed stations that are connected to the wire-line network as base stations (BSs). There are several commercial implementations of FDM-TDMA technology for mobile telecommunication, the one with the most widespread deployment being the GSM system (see Section 4.7). In any of these implementations, the system bandwidth, Wsystem , is partitioned into several nonoverlapping FDM channels, each of which is then digitally modulated, and then time slotted to yield FDM-TDM channels. A guard band is left vacant at either end of an operator’s spectrum allocation to prevent power from one operator’s system from interfering with another system. For example, in the GSM system the FDM channel spacing is 200 KHz. After digital modulation, each such channel is time slotted to provide eight TDM channels, each of which can carry one direction of a digitized voice call. Each voice call has two directions, and hence for each call we need two links to be established, one from the MS to the BS, and one from the BS to the MS. The way these two links are established is called the duplexing technique. In FDMTDMA systems the common duplexing mechanism employed is frequency division duplexing (FDD); that is, two separate FDM carriers are used to carry the two directions of a call. Thus, the operator actually gets two nonoverlapping segments of the radio spectrum, each of bandwidth Wsystem (see Figure 4.1). One of these is the uplink band and the other is the downlink band. Each band is partitioned into an equal number of nonoverlapping FDM channels. The FDM channels in the uplink and downlink bands are then paired, as shown in Figure 4.1, for two FDM channels j and k. Thus, when we say that FDM channel j is assigned to an MS, for the purpose of making a telephone call, then actually two TDM slots, one in each of the two FDM channels with center frequencies fju and fjd , are assigned to the call, for the entire duration of the call. For example, if the operator leases a Wsystem of 5 MHz, then (allowing for a total guard band equal to the bandwidth of one FDM channel), the system can be used to carry 24 × 8 = 192 simultaneous calls. Let us denote the FDM channel

fj u

fku

fj d

fkd

frequency uplink band

downlink band

Frequency division duplexing in FDM systems: The FDM channels fju and fjd are paired, as are the channels fku and fkd . Figure 4.1

4.1

Principles of FDM-TDMA Cellular Systems

83

bandwidth by W , the number of FDM channels in the system bandwidth by C (24 in the preceding example), and the number of trafﬁc carrying TDM slots per FDM channel by s (8 in the preceding example). Let N = C × s be the number of calls that can be carried simultaneously. We will further denote the set of FDM carriers by {f1 , f2 , . . . , fC }. In view of our earlier discussion on duplexing, each element of the set of carriers {f1 , f2 , . . . , fC } actually denotes an uplink-downlink pair. The system can now be set up as shown in the left panel of Figure 4.2. We observe that a large power will need to be used in order to serve the MSs at the periphery of the coverage area, in order to ensure that the power received at either end of an MS-BS link is such that the SNR exceeds the minimum required for the desired bit error rate for the voice coder being used. Such MSs will quickly drain their batteries, and will also cause interference to systems in neighboring coverage areas. Also, the maximum number of users that can be simultaneously served in this simple system is N . Let B(ρ, n) denote the blocking in an Erlang blocking system with a load of ρ Erlangs and n servers (see Appendix D, Section D.5.1). If the arrival rate of new calls into the system is λ per second, and the mean holding time of a call is h seconds, then for this system ρ = λh, and the blocking probability becomes B(ρ, N). Table 4.1 shows a sample Erlang table from which the number of servers that would be required to obtain a speciﬁed blocking probability for a given trafﬁc intensity can be obtained.

BS

BS

BS

BS

BS

BS

Figure 4.2 Spatial reuse: In the left panel, all the MSs associate with one BS, and the entire band is used to serve all the calls in the desired coverage area. In the right panel the band is reused at multiple BSs.

84 n↓

15 16 17 18 19 20 21 22 23 24 25

n↓

Loss Probability 0.0001

0.001

0.002

0.005

0.01

0.02

0.05

0.1

0.2

4.7812 5.3390 5.9110 6.4959 7.0927 7.7005 8.3186 8.9462 9.5826 10.227 10.880

6.0772 6.7215 7.3781 8.0459 8.7239 9.4115 10.108 10.812 11.524 12.243 12.969

6.5822 7.2582 7.9457 8.6437 9.3515 10.068 10.793 11.525 12.265 13.011 13.763

7.3755 8.0995 8.8340 9.5780 10.331 11.092 11.860 12.635 13.416 14.204 14.997

8.1080 8.8750 9.6516 10.437 11.230 12.031 12.838 13.651 14.470 15.295 16.125

9.0096 9.8284 10.656 11.491 12.333 13.182 14.036 14.896 15.761 16.631 17.505

10.633 11.544 12.461 13.385 14.315 15.249 16.189 17.132 18.080 19.031 19.985

12.484 13.500 14.522 15.548 16.579 17.613 18.651 19.692 20.737 21.784 22.833

15.608 16.807 18.010 19.216 20.424 21.635 22.848 24.064 25.281 26.499 27.720

Table 4.1 Part of the Erlang table showing the trafﬁc intensity that can be offered to a link of capacity n (rows) circuits for speciﬁed blocking probabilities (columns).

15 16 17 18 19 20 21 22 23 24 25

4.1

Principles of FDM-TDMA Cellular Systems

85

Telephony systems typically are designed for blocking probabilities such as 0.01 or 0.02. If a single phone is expected to provide a load of 0.1 Erlangs (e.g., two calls per hour, with an average holding time of 3 min), then for a coverage area with 2500 MSs the Erlang load is 250, and we see that this proposed system (with N = 192) will yield an unacceptable probability of call blocking B(250, 192).1 The right panel of Figure 4.2 shows the idea of spatial reuse. The MS-BS communication is done using smaller powers. The same channel can be used in several places in the system, provided that the cochannel interference is such that the signal-to-interference plus noise ratio (SINR) of any MS-BS link is maintained above a threshold. Suppose that, by exploiting spatial reuse, each channel could be reused, say, ﬁve times in the same coverage area; then we would have effectively multiplied the number of calls that can be simultaneously handled by a factor of 5. On further thought, however, a problem becomes evident with the simple arrangement in Figure 4.2. If an MS is not close to any BS, then in order to serve it from some BS, on any channel, a large transmission power will need to be used. This will cause cochannel interference if the same channel is reused elsewhere in the coverage area, thus rendering spatial reuse less effective. In order to address this problem, the cellular FDM-TDMA approach is to tessellate the coverage area into cells, each of which has a BS. The set of FDM carriers is partitioned into subsets called reuse groups. These channel groups are then assigned to the cells in such a way that cells with the same group of channels (called cochannel cells) are not close together. How close cochannel cells can be depends on the SINR required for reliable communication. Each cell then acts as an Erlang blocking system for the calls that require a channel in it. We observe that, if the SINR required is large, then the cochannel cells will need to be kept far apart. This will require more channel reuse groups, and hence fewer channels per reuse group. This brings us to another issue. If the reuse groups have only a small number of carriers in them, then trunking efﬁciency is lost. By this we mean the following. For a ﬁxed probability of blocking, , let ρ (n) denote the Erlangs that can be carried when the number of servers is n, B(ρ (n), n) =

(4.1)

To understand this, notice that each column of Table 4.1 corresponds to a value of , and each element in that column gives the value of ρ (n) for the corresponding n in the ﬁrst column. Now deﬁne g (n) = ρn(n) , that is, g (n) is the Erlangs per server that can be offered, when the number of servers is n and the target blocking 1 Note that the maximum load that 192 servers can carry is just 192, so a load of 250 Erlangs will give a

blocking probability close to 1. As a rule of thumb, for B(ρ, n) to be as small as 0.01 or 0.02, it is necessary that ρ < n. This follows because, with a blocking probability of , the rate of calls that are carried is (1−)λ, and time average number of busy servers is (1 − )λh = (1 − )ρ (by Little’s Theorem; see Appendix D). For low blocking, however, the average number of busy servers will be substantially less than n (see Table 4.1).

86

4

Cellular FDM-TDMA

ρ probability is . For n = 1, note that B(ρ, n) = 1+ρ , which yields g (1) = 1− . Thus, a very small load (per server) can be handled if there is just one server. However, g (n) increases monotonically to 1 as n increases. Thus, we see that it is beneﬁcial to not partition the set of carriers into small groups, as this reduces trunking efﬁciency. We conclude that a larger target SINR results in smaller reuse groups, which results in lower trunking efﬁciency. It follows that there is a trade-off between keeping the SINR above a required threshold and keeping the trunking efﬁciency high. The number of reuse groups we use in a system will be denoted by Nreuse . We observe from this discussion that the SINR requirements, the spatial reuse, and the system efﬁciency are intimately linked, and some analysis is required to evaluate the trade-offs.

4.2

SIR Analysis: Keeping Cochannel Cells Apart

In Figure 4.3, we depict uplink and downlink cochannel interference in a conﬁguration in which a channel is reused at the ﬁve BSs shown. The circular boundaries indicate the coverage of each BS; these are assumed to be of radius R. The distance between the centers of each of the outer BSs and the one in the middle is D. It is intuitively clear that a large D R ratio will be required if the cochannel interference has to be kept very small. In this section we will study how to carry out the cochannel interference analysis with a target SINR, in order to determine the required D R ratio.

BS1 MS1 D 2R R BS4

MS0 BS2

D 2R MS4

R

D 2R MS2 MS0

D 2R

D MS3 BS3

Figure 4.3 Depiction of downlink (left panel) and uplink (right panel) cochannel interference. In each case the MS0 is taken to be in the most unfavorable position, such that the desired signal will suffer the maximum attenuation and the interference will suffer the least attenuation. D is the shortest distance between BSs of cochannel cells, and R is the coverage radius of each BS.

4.2

SIR Analysis: Keeping Cochannel Cells Apart

87

For the purpose of studying the cochannel interference, the MS whose signal performance is being analyzed is considered to be in the most unfavorable position (at the periphery of its BS’s coverage area), and the interferers are also assumed to be in the most unfavorable position, as close as possible to the receiver of the desired transmission. For example, in the right panel of the ﬁgure, the uplink is being considered, and therefore the interferers are cochannel MSs in the other cells. Notice that these are being assumed to be at the peripheries of their own cells, and placed so that they are as close as possible to BS0. Such worst case conﬁgurations are used to determine how far away cochannel cells need to be kept. At this point we recall the material in Section 2.1.4. Let H denote the channel power gain (actually, an attenuation) between the transmitter of the desired signal and its receiver, and let Hi denote the power gain from the i-th cochannel interferer to the receiver of the desired signal. Let there be NI interferers. We will view all transmitter powers as being the Rayleigh faded mean values at the reference distance d0 (see Section 2.1.4). With this convention, let P be the power used by the transmitter of the desired signal, and Pi , 1 ≤ i ≤ NI be the powers of the interfering transmitters. It then follows that the SINR at the receiver is given by Ψ=

PH I N0 W + N i=1 Pi Hi

We note that these FDM-TDMA systems use narrowband modulation, and hence the SINR requirements are in the range of 10 dB to 20 dB. Also the noise power, N0 W , is very small; approximately −120 dBmW (i.e., 10−12 mW). It is, therefore, assumed that the noise power is much less than the received signal power, and we neglect this term in the denominator. Let d be the distance between the transmitter and its receiver, and di the distance between the i-th interfering transmitter and the receiver. We can then write (see Section 2.1.4) H= Hi =

d d0 di d0

−η −η

10−

(ξ+ξ0 ) 10

10−

(ξi +ξ0 ) 10

where ξ, ξ0 , and ξi , 1 ≤ i ≤ NI , normally are distributed and correspond, respectively, to the shadowing at the transmitter of the desired signal, at the receiver, and at the NI interferers. Here the ξ, ξ0 , ξi , 1 ≤ i ≤ NI , are i.i.d. normally 2 distributed, 0 mean, and with variances σ2 ; thus, the lognormal shadowing standard deviation on any path is σ dB. This form of the lognormal shadowing is used since shadow fading comprises a part due to the shadowing near the receiver (which is common to all paths to the receiver), and a part near the transmitters

88

4

Cellular FDM-TDMA

(which is assumed to be independent for widely separated transmitters). Hence, we can write the SINR (or, simply, the SIR) Ψ as

−η

(ξ+ξ0 )

10− 10 Ψ= −η (ξi +ξ0 ) NI di P 10− 10 i i=1 d P

d d0

0

Notice that the terms ξ0 all cancel. We can then rewrite the SIR expression in the following form: Ψ=

1 − 10 −10 log10 P+10η log10

10 NI

10

i=1

d +ξ d0

1 − 10 (−10 log10 Pi +10η log10

di +ξi ) d0

(4.2)

Notice that in the numerator we have a log-normally distributed random variable 1 of the form 10− 10 Q , where Q has units of dB, and is normally distributed with d E Q = m := −10 log10 P + 10η log10 dB d0 VAR(Q) = υ :=

σ2 2

and in the denominator we have a sum of NI log-normally distributed random 1 variables of the form 10− 10 Qi , where Qi also has units of dB, and is normally distributed with di E Qi = −10 log10 Pi + 10η log10 dB d0 VAR(Qi ) =

σ2 (= υ) 2

Thus, we have 1

10− 10 Q

Ψ= NI

i=1

1

10− 10 Qi

where Q, Qi , 1 ≤ i ≤ NI , are independent normally distributed random variables that essentially model the shadowing. Since shadow fading is correlated over distances of several 10s of meters, we assume that the shadowing is “sampled” once during a call, and independent samples of the shadow fading random variables are taken from call to call. We also assume that a call, during its holding time, samples the entire distribution of the Rayleigh fading; it does not get “stuck” in

4.2

SIR Analysis: Keeping Cochannel Cells Apart

89

a deep fade. This corresponds to our use of mean transmit powers averaged over Rayleigh fading. We are now interested in ensuring that the SIR exceeds a threshold γ with a high probability, say, 1 − . Note that, given a target BER, γ will be obtained from an analysis of the underlying modulation scheme under Rayleigh distributed ﬂat fading and additive white Gaussian noise; see Section 2.1.4. Then would be the outage probability. What does this probability mean? Consider instances of calls from or to MSs at the boundaries of the coverage areas of the cells in which they are handled. Then the fraction of such calls that will experience a BER higher than the target will be less than . This is because for such calls, we have assumed that the cochannel interferers are placed at the most unfavorable locations; refer back to Figure 4.3. One approach to carrying the analysis forward is to approximate the I 1 − 10 Qi distribution of N by a log-normal distribution. Such an approximation i=1 10 is known to work well. Also, the resulting SIR threshold analysis becomes simple, since the ratio of two independent log-normal random variables is obviously log-normal. So, let us write NI

1

1

10− 10 Qi ≈ 10− 10 QI

i=1

The approximation is performed by matching the mean and second moment of the random variables on the two sides. This is called the Fenton-Wilkinson method, the details of which can be found in standard texts on wireless digital communication (see, for example, [123]). Let us suppose that this procedure yields QI as normally distributed with mean mI and variance vI . Then, we have 1

Ψ = 10− 10 (Q−QI )

or, equivalently, (Ψ)dB = QI − Q

where QI − Q is in dB, and is normally distributed with mean mI − m and variance v + vI . Thus, mI − m is the mean SIR in dB, and the SIR variance is v + vI . We need that QI − Q > (γ)dB with a large probability, where, as usual, (γ)dB = 10 log10 γ. From Figure 4.3, we notice that in the downlink worst case situation (left panel of the ﬁgure), the BSs are all taken as transmitting to MSs at the peripheries of their coverage areas. Similarly, in the uplink worst case situation (right panel) the BSs are all receiving from MSs at the peripheries. We thus assume that the transmission powers P, Pi , 1 ≤ i ≤ NI , are all equal. It follows from (4.2) that the transmitter powers cancel in the SIR expression. The mean value, m, corresponds to the path loss from the transmitter of the desired signal to its receiver. Further, mI depends on the path losses from the interferers to the receiver, and is the effective path loss of the interfering transmitters, in dB. We can thus require that mI − m > 0,

90

4

Cellular FDM-TDMA

that is, the interferers have a larger effective power attenuation to the receiver than does the desired transmitter. Figure 4.4 depicts a typical situation, when the target probability of exceeding γ is being met. The normal density of (Ψ)dB has been plotted in this ﬁgure. It follows that for an outage probability (i.e., to ensure that Pr (QI − Q) < γ < ,) there is a τ , such that we need to ensure that √ mI − m > γ + τ v + v I Such a τ will be obtained from a table of the tail of the normal distribution. For example, τ0.01072 = 2.3, as can be seen from Table 4.2. This inequality provides the insight that shadowing variance of the signal and of the interference add up, and a larger value of this total variance requires the cochannel reuse to be designed so that there is a larger difference between the mean interference attenuation and the signal attenuation, mI − m. As an application of this analysis, consider the uplink conﬁguration shown in the right panel of Figure 4.3. Since, in this case, the interferers are as close as possible to the receiver of the desired signal, this situation is worse than the downlink situation shown in the same ﬁgure. It can be shown that the FentonWilkinson analysis yields m = 10η log10 R

2 ea v NI3 1 mI = 10η log10 (D − R) − ln 2 2a ea v + (NI − 1) 2 1 ea v − 1 vI = 2 ln +1 NI a

␥

mI 2 m

c in dB

Figure 4.4 A sketch of the normal probability density of the SIR, Ψ, in dB. target SIR, and mI − m is the mean SIR.

γ is the

4.2

SIR Analysis: Keeping Cochannel Cells Apart

91

z

Q(z)

z

Q(z)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

0.50000 0.46017 0.42074 0.38209 0.34458 0.30854 0.27425 0.24196 0.21186 0.18406 0.15866 0.13567 0.11507 0.09680 0.08076 0.06681 0.05480 0.04457 0.03593 0.02872

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

0.02275 0.01786 0.01390 0.01072 0.00820 0.00621 0.00466 0.00347 0.00256 0.00187 0.00135 0.00097 0.00069 0.00048 0.00034 0.00023 0.00016 0.00011 0.00007 0.00005

Table 4.2 The probability under the right “tail” of the ∞ −x2 normal (Gaussian) distribution: Q(z) = √1 e 2 dx. 2π z

where a :=

ln 10 10

≈ 0.23026. We thus get the requirement

10η log10

2 a v−1 D 1 e − 1 > γ + τ v + 2 ln +1 R NI a 2 ea v NI3 1 + ln 2 2a ea v + (NI − 1)

(4.3)

We notice that this inequality places a constraint on the ratio between D, the distance between cochannel cells, and R, the cell radius. Let us look at two numerical examples, both with η = 4. Consider ﬁrst v = 0, that is, there is no log-normal shadowing, just path loss. The D R constraint reduces to 40 log10

D − 1 > γ + 10 log10 NI R

(4.4)

92

4

Cellular FDM-TDMA

Exercise 4.1 Obtain the expression in (4.4) directly from the SIR expression in (4.2). For NI = 6, we ﬁnd 40 log10

D − 1 > γ + 7.78 R

On the other hand, if the shadow fading standard deviation is 8 dB, then 2 v = σ2 = 32. For η = 4, NI = 6, and outage probability = 0.01, (4.3) yields 40 log10

D − 1 > γ + 25.25 R

We conclude that shadow fading has a signiﬁcant effect on the

D R

ratio.

Discussion a. We observe, from the preceding analysis, that as long as the transmitter powers are all assumed to be equal, the required D R ratio does not depend on the actual values of the transmit powers. b. We notice also that only the ratio D R is determined, but not the absolute values of D and R. This provides the important insight that the cell sizes can be shrunk while retaining the D R ratio. This increases the system call handling capacity, since the channel groups in each cell are used to serve a smaller cell area. This approach to increasing the system capacity has its limitation, however. As the cell size decreases the MSs tend to more frequently require intercell handovers. Since the blocking of handover requests leads to the dropping of ongoing calls, an increase in handover rates needs more channels to be reserved for handover handling (see Section 4.6), thus leading to a possible reduction in call handling capacity. In addition, the higher frequency of handovers results in more signaling load, thus possibly overloading the call handling processors in the system.

4.3

Channel Reuse Analysis: Hexagonal Cell Layout

It is evidently not practical to use the cell conﬁguration shown in Figure 4.3, as this leaves large portions of the service area uncovered. Hence, as explained in Section 4.1, the service area is tessellated with cells. The set of FDM carriers is partitioned into disjoint sets, which are assigned to subsets of the cells, in such a way that cochannel cells respect the D R ratio. In order to analyze such a system, it is convenient to take the cells to be hexagons of equal size. This permits an easy visualization of the tessellation in the two-dimensional plane. It is then useful to recall the simple geometrical concepts shown in Figure 4.5.

4.3

Channel Reuse Analysis: Hexagonal Cell Layout

93

C5 3 R Area 5 3 C 2 5 3 3 R 2 2 2

C R

Figure 4.5 Hexagon geometry: relations between the cell width, C, the cell radius, R, and the area of the hexagon.

4.3.1 Cochannel Cell Groups In Figure 4.6 we show a tessellation of the plane with hexagonal cells. The FDM channels are partitioned into reuse groups. One of these groups is assigned to Cell 0, shown at the center of the cell layout. This will be our reference cell in the following discussion. We next wish to determine which other cells in the layout should use the same group of carriers. For this purpose it is convenient to work with a coordinate system with axes inclined at 60◦ to each other, as shown by the axes u and v in the ﬁgure. For simplicity in our description, we draw a third “axis,” w. The axes pass through the center of the reference cell. There is an angular separation of 60◦ between u and v, and the same between v and w. Notice that moving a cell width, C , along any of the axes takes us to the center of a neighboring cell. Thus, let C be unit length along the axes. Now, starting from the origin of this system (the center of Cell 0), move i units along the u axis and then j units along the v axis. Observe that for i = 3 and j = 2 this brings us to the cell labeled 1. Let the Euclidean distance between the centers of Cell 0 and Cell 1 be D(i, j), the distance between two cells whose relative positions depend on (i, j) in the manner just explained. The following calculation follows from simple geometry. √ 2 3 1 2 D(i, j) = j + i+j 2 2 = =

j2

3 1 + i2 + ij + j2 4 4

i2 + ij + j2

In a similar manner, ﬁxing i = 3 and j = 2, we can identify Cells 2, 3, 4, 5, and 6, as shown in Figure 4.6. These will be the cochannel cells in relation to Cell 0. Observe that if we carry out the same procedure, for i = 3 and j = 2, for Cell 1 in the ﬁgure, then we will obtain Cells 2, 0, and 6, and three other cells, above and to the right of Cell 1; these cells are not shown in the ﬁgure. Thus, this process yields

94

4

Cellular FDM-TDMA

v 2

u

1

w

j D 3

i 0

4

6

5

Cells 0,1,2,3,4,5, and 6 are cochannel cells to locate a cochannel cell w.r.t. to a cell: move i cells along an axis, then turn clockwise and move j cells

Figure 4.6 Tessellation of the coverage area by hexagonal cells. Cells 0, 1, 2, 3, 4, 5, 6 are cochannel cells for (i, j) = (3, 2).

a subset of the hexagons that tessellate the plane. Applying the procedure starting from any element of this subset yields the same set of cells. Notice, however, that if we start from one of the cells adjacent to Cell 0, and use the same (i, j), then we will get a subset of cells that is disjoint from the previous one. In fact, looking at Figure 4.7, for each cell in the large dashed hexagon with Cell 0 at its center, we will obtain a different subset of hexagons, and all these subsets (19 for (i, j) = (3, 2)) are mutually disjoint and together they form a partition of the tessellation. We can call each of these subsets of cells a cochannel group. 4.3.2 Calculating Nreuse The number of cochannel groups (which we had denoted earlier by Nreuse ) thus depends on the choice of (i, j). For example, with (i, j) = (1, 0) there is only one cochannel group. What is the general relation between (i, j) and the number of cochannel groups? This can √ be worked out as follows. In Figure 4.7 the area of the large dashed hexagon is 23 D2 (where we recall that the unit of length is the cell width, C ). There are as many cochannel groups as the number of cells in this large hexagon. Exactly one cell from any of the cochannel groups lies in this large hexagon. Hence, given a large coverage area A, the number of cells in a cochannel group is √ A 2 (see Figure 4.5). The total number of cells is √A . Thus, the ( 3/2)D

3/2

4.3

Channel Reuse Analysis: Hexagonal Cell Layout

2

95

1

A2

D

A1

3

0

6

A3 A0 A6 4

5

A4 A5 The large dashed hexagons are centred at cochannel cells

Figure 4.7 Tessellation of the plane by hexagonal cells. There is one cell from each cochannel group in each of the large dashed hexagonal areas. Notice that Cells A0, A1, A2, A3, A4, A5, and A6 belong to a different cochannel group than Cells 0, 1, 2, 3, 4, 5, and 6.

number of cochannel groups is D2 . Obviously, the number of cochannel groups has to be the same as the number of groups into which we partition the set of FDM carriers (i.e., Nreuse (as deﬁned earlier)). Thus Nreuse = D2 = i2 + ij + j2

We also observe that for a given (i, j), following this procedure for ﬁxing the cochannel cells, we have also ﬁxed the D R ratio to the value (see Figure 4.5): D(i, j) = R =

3(i2 + ij + j2 ) 3Nreuse

Table 4.3 shows the values of Nreuse and D(i,j) R that are obtained for various values of (i, j). Recall that the SIR analysis in Section 4.2 yielded a constraint on the D R ratio. For example, (4.3) provided a constraint on the D ratio when a reference R

96

4

Table 4.3

Cellular FDM-TDMA

i

j

N reuse

D(i,j) R

1 1 2 2 3 2 3 4 3 4 4

0 1 0 1 0 2 1 0 2 1 2

1 3 4 7 9 12 13 16 19 21 28

1.73 3.00 3.46 4.58 5.20 6.00 6.24 6.93 7.55 7.94 9.17

Nreuse, and

D(i,j) ratio, for relative locations of cochannel cells, (i, j ). R

cell is surrounded by NI cochannel cells whose centers are all at distance D from the center of the reference cell. We say that the analysis considered only the ﬁrst tier interferers, the nearest cochannel cells. In general, in large cellular networks, there will be second and third tier interferers and even more beyond. Of course, the interference from second, third, and higher tiers is substantially lower than that from the ﬁrst tier, especially when the path loss exponent, η, is large. In any case, the SIR analysis yields a D R ratio, and then Table 4.3 can be used to determine the value of Nreuse that provides this D R ratio, and the corresponding (i, j) to be used to lay out the cells. For example, if the required D R ratio is 7, then we must take Nreuse = 19, which is achieved with (i, j) = (3, 2). 4.3.3 D R Ratio: Simple Analysis, Cell Sectorization It is instructive to compare various cases of ﬁrst tier cochannel interference while ignoring shadowing, and accounting only for path loss. Such analysis provides quick insight into the comparisons between the various cases. Thus, accounting only for path loss, and taking the transmitter powers, in the worst-case transmitterreceiver conﬁgurations, to be equal (see the discussion in Section 4.2), the following is the general expression for the SIR R−η Ψ = N −η I i=1 Di 1 = −η NI Di i=1

R

where R is the cell radius, NI is the number of ﬁrst tier interferers, and Di , 1 ≤ i ≤ NI , is the distance of the i-th interferer from the receiver in the reference cell.

4.3

Channel Reuse Analysis: Hexagonal Cell Layout

97

Figure 4.8 shows the forward channel (downlink) worst-case situation, where approximations have been made for the various distances between the interferers and the receiver. We see that Ψ= 2

D R

−η

−1

+

1 −η D R

+

D R

−η

+1

Suppose we take Nreuse = 9; then Table 4.3 provides D R = 5.20, from which we ﬁnd that, for η = 4, Ψ = 95.09 = 19.78 dB. In Figure 4.9 we show the reverse channel (uplink) worst case situation. Here we see that Ψ= 6

1 D R

−η

−1

D D1R D2R R D2R D1R D

Figure 4.8 Seven cochannel cells, showing the worst-case conﬁguration of ﬁrst tier downlink interferers. The arrows point at the receiver, and show the direction of the desired signal and the interference. The distances are approximations, and, in general, the cochannel cells may have a different relative orientation from the one shown.

98

4

Cellular FDM-TDMA

D2R D2R D2R

R D2R D2R D2R

Figure 4.9 Seven cochannel cells, showing the worst case conﬁguration of ﬁrst tier uplink interferers. The arrows point at the receiver, and show the direction of the desired signal and the interference. The value D – R is an approximation, and, in general, the cochannel cells may have a different relative orientation from the one shown.

For Nreuse = 9 (i.e., D R = 5.20) we have, for η = 4, Ψ = 51.86 = 17.14 dB. Thus we see that the uplink provides a more than 2.5 dB worse performance for the same D R ratio. In each of these cases, there are six ﬁrst tier interferers at a receiver. If directional antennas are used in the BSs then the number of interferers can be reduced. This is achieved by a technique called sectorization, which is depicted in Figure 4.10. Each cell is shown divided into three 120◦ sectors, each with a directional antenna whose angular coverage is designed to coincide with the angular spread of the sector. Thus, an MS in a given sector of a cell is served by the antenna in that sector. Further, the channels are reused only in the corresponding sectors of the cell reuse groups. This is shown in Figure 4.10, where the carrier f is shown being reused in a particular sector of all the cells in a reuse group of seven cells. To see the advantage of doing sectorization, consider the downlink worstcase situation depicted in the left panel of Figure 4.10. Notice that the MS in Cell 0 sees only two ﬁrst tier interferers, the two BSs in Cells 2 and 3. The corresponding

4.4

Spectrum Efﬁciency

99

1

1 f

f 6

2 f

f

f D1R

6

2

0

f 0 R

Rf 3

5

3 f

D

f

f

D D

5 f

4

4 f

f

Figure 4.10 Seven cochannel cells, with 120◦ sectorization, showing the worst case conﬁguration of ﬁrst tier downlink interferers (left) and uplink interferers (right). The distances shown are approximations.

sectors in Cells 4, 5, and 6 could be using the same channel, but their antenna main lobe is not “visible” to the MS in Cell 0. Making suitable approximations for the distances, the following is the forward channel SIR, ignoring the shadowing. Ψ=

D R

1 −η

+1

+

−η D R

Taking η = 4, Nreuse = 9 (i.e., D R = 5.20) we ﬁnd that Ψ = 489.13 = 26.89 dB, a 7.1 dB improvement over the case without sectorization. The right side of Figure 4.10 shows the worst-case uplink interferers with sectorization. The SIR is given by Ψ= 2

1 −η D R

With Nreuse = 9 (i.e., D R = 5.20), taking η = 4, we ﬁnd that Ψ = 365.58 = 25.63 dB, which is 8.5 dB better than without sectorization. Note, however, that sectorization implies smaller sets of channels in each sector, thus reducing the trunking efﬁciency.

4.4

Spectrum Efﬁciency

Let us recall the following system parameters. The RF spectrum allocated to the system is Wsystem , the number of FDM carriers the system bandwidth is partitioned into is C , the number of TDM slots per carrier is s. Assuming equal cell sizes, let

100

4

Cellular FDM-TDMA

a denote the area of each cell. Further, let K denote the number of sectors in each cell (e.g., K = 3 for 120◦ sectors). Recall the deﬁnition of Nreuse , and the function g (n) (see Section 4.1). Let us consider the simplest approach of partitioning the C carriers into Nreuse subsets. Each subset of carriers is then further partitioned into K sets, each

of which is allocated to the same sector in all the cells in a reuse group of cells. Each slot in each carrier in a sector can carry one call. For the present we assume that a call that is initiated in a sector stays in the same sector for its entire duration; sC that is, there are no handovers in the system. Thus, the Nreuse K slots in a sector, along with the call arrivals to or from MSs in that sector, constitute an Erlang blocking model. It follows that, for a target blocking probability of , the number of Erlangs that can be offered to a cell is given by g

sC

×

Nreuse K

sC Nreuse K

×K

where the ﬁrst term is the number of Erlangs per slot in a sector. Let A denote the coverage area of the system. Then the Erlang capacity of the system, denoted by Λ, is given by A Λ = × g a

sC Nreuse K

×

sC Nreuse

Let us deﬁne the spectrum efﬁciency of the system as the Erlang capacity per unit area per Hz of system bandwidth, and denote this by ν. We then have ν := =

Λ A Wsystem 1 sC × g × a Wsystem

Wsystem Wsystem Nreuse K sC

1 Nreuse

(4.5)

sC Notice that Wsystem is ﬁxed for a given system bandwidth, and depends on the FDM-TDM modulation scheme being used. For example, in the GSM system, the FDM carrier spacing is 200 Khz, and there are eight TDM slots per FDM carrier. Thus, given Wsystem , and allowing for some guard bandwidth on either side, the value of C is determined. In Section 4.3 we saw how Nreuse and K can sC Wsystem 1 be chosen to achieve the required SIR. Notice that the term g Wsystem Nreuse K Nreuse decreases with increasing Nreuse or K, but we need to set Nreuse and K so that the SIR requirements are met while keeping this trunking efﬁciency term as large as sC Wsystem 1 possible. Note that g Wsystem Nreuse K Nreuse also increases with Wsystem , but having leased a certain amount of the spectrum, the operator will want to work within this leased amount. Finally, the Erlang capacity of the system can be increased by

4.5

Channel Allocation and Multicell Erlang Models

101

decreasing a; that is, by reducing the cell size. Of course, there are limits to this scaling. As the cell size decreases, there are three issues: a. As the cell size decreases, we need to consider handovers, and the handover rate increases with decreasing cell size. This will impact the Erlang capacity, as resources need to be reserved for handovers. b. The signaling load increases due to the increased handover rate. This means that higher capacity call handling systems need to be installed. c. Reducing cell size requires the installation of more base stations, which can be expensive. Finally, the design of any given system will have to balance these trade-offs.

4.5

Channel Allocation and Multicell Erlang Models

From the expression for spectrum efﬁciency in (4.5), we can infer that, apart from reducing the cell size, another way to increase the efﬁciency is to improve the channel utilization. The earlier analysis assumed a uniform ﬁxed assignment of the FDM carriers to the cells and their sectors. In such an assignment, it is possible that channels are idle in one cell, whereas another cell is overloaded. The trunking efﬁciency can be improved if the channels are viewed as being in various common pools, from which allocations are made as needed. Of course, such dynamic channel allocation must respect the cochannel SIR constraints as the channels are allocated, released, and reallocated to various cells. 4.5.1 Reuse Constraint Graph A simple model that can be used for designing and analyzing dynamic channel allocation strategies is to specify pairwise reuse constraints. Given an array of cells, pairwise reuse constraints specify which pairs of cells cannot use the same FDM carrier at the same time. For example, Figure 4.11 shows a linear array of rectangular cells, such as might be deployed along a highway. The diagrams in the middle and bottom of the ﬁgure depict pairwise reuse constraints as constraint graphs. In a constraint graph, each cell is represented by a vertex. There is an edge between two vertices if an FDM carrier cannot be simultaneously used in both of the corresponding cells. Thus, the constraint graph in the middle of Figure 4.11 only constrains neighboring cells from reusing the same channel; channels can be simultaneously used in alternate cells. The constraint graph at the bottom, however, permits a channel to be reused only in cells that are separated by at least two other cells. We note that, in general, representation of reuse constraints by pairwise constraints is conservative. It is possible that among three cells, any two cells can reuse the same channel, but, if the third cell also uses that channel, then, due to the increased interference, the SIR in all the cells may be at an unacceptable level.

102

4 1

2

3

4

5

6

7

8

9

Cellular FDM-TDMA 10

Figure 4.11 A linear array of 10 cells (top), and two sets of pairwise reuse constraints (middle and bottom), shown as constraint graphs.

The modeling of such, more general, constraints requires hypergraphs, a generalization of graphs in which edges are subsets of nodes with cardinality greater than two. Such models have been studied in the literature, but we will consider only pairwise constraints in this book. Formalizing this discussion, let B = {1, 2, . . . , N} denote the set of cells (or, equivalently, base stations). Let (B, C) denote the constraint graph with C being the edge set; that is, for i ∈ B and j ∈ B, (i, j) ∈ C if the same carrier cannot be used in Cell i and Cell j simultaneously. We note that the constraint graph is undirected; that is, (i, j) ∈ C if and only if (j, i) ∈ C . Let F = {f1 , f2 , . . . , fM } be the set of FDM carriers that need to be assigned to the cells. Suppose that a certain number of calls need to exist in each of the cells. This will require a certain number of carriers xj in each of the cells j, 1 ≤ j ≤ N , in order to be able to carry those calls. For example, if each carrier has eight TDM slots, then in order to carry 9 to 16 calls in a cell, two carriers are needed. Let xj denote the number of carriers required in Cell j, 1 ≤ j ≤ N . We say that the vector x = (x1 , x2 , . . . , xN ) is feasible if there exists an allocation of xk carriers to Cell k such that the reuse constraints are respected. As a simple illustration, if F = {f1 , f2 }, and we have the reuse constraints shown in the middle of Figure 4.11, then x = (2, 1, . . . , 1) is not feasible. Deﬁne X = {x : x feasible}

Recalling some standard concepts from graph theory, we say that a clique of (B, C) is a fully connected subgraph. Thus, a carrier can only be used in exactly one of the cells that form a clique. A maximal clique is one that is not contained in any other clique. We will simply refer to maximal cliques also as cliques. Thus, in the bottom diagram of Figure 4.11, the cliques are {1, 2, 3}, {2, 3, 4}, and so on.

4.5

Channel Allocation and Multicell Erlang Models

103

4.5.2 Feasible Carrier Requirements Let Q be the number of cliques (i.e., maximal cliques) in (B, C). Consider the Q×N matrix A with 1 if cell j is in clique i aij = 0 otherwise We see that a necessary condition for x ∈ X is A·x ≤M 1

where we recall that M is the number of carriers, and 1 is the Q × 1 vector of 1s. Note that this inequality simply says that, for each i, 1 ≤ i ≤ Q, N j=1 aij xj ≤ M, where the expression on the left of this inequality is the number of carriers needed in Clique i in order to achieve the carrier allocation given by x. Let us denote XCPA = {x : A · x ≤ M 1}

where the sufﬁx CPA expands to clique packing allocation. It may appear that XCPA is a convenient characterization of X . Since every carrier allocation must satisfy the clique constraints, we see that X ⊂ XCPA . In general, however, X is a strict subset XCPA ; that is, in general, it can be that x ∈ XCPA , but x ∈ X . An example is shown in Figure 4.12. We can also observe that, if the constraint graph shown in Figure 4.12 is a subgraph of a constraint graph, then X = XCPA .

Exercise 4.2 Consider a linear array of cells (1, 2, . . . , N) (as shown in Figure 4.11) with a constraint graph that has the property that if cells i and j, i ≤ j, are in a clique, then all k such that i < k < j are also in the same clique. Argue that for this situation X = XCPA . Show that if x ∈ XCPA then a feasible carrier assignment is obtained via a greedy algorithm that starts by assigning the required carriers to the clique to which the left-most cell belongs, and then moves across the cells from left to right, reassigning carriers as need arises.

4.5.3 Carrier Allocation Strategies Based on the preceding discussion, we can identify various carrier allocation strategies. We recall that, for a system with N cells, x = (x1 , x2 , . . . , xN ) denotes a vector of carrier requirements. Given a set of reuse constraints, a given x may or may not be feasible. We have deﬁned X as the set of all feasible carrier requirement vectors: X = {x : x is feasible}. a. Fixed Carrier Allocation (FCA). The carriers are allocated statically to the cells in such a way that the reuse constraints are satisﬁed. For example, if F = {f1 , f2 }, and we have the reuse constraints shown in the middle of Figure 4.11, then (f1 , f2 , f1 , f2 , . . .) is a valid allocation. With this allocation,

104

4 f1

1

5

2

4

Cellular FDM-TDMA

3

?

f2

f2

f1

Figure 4.12 A pentagon reuse constraint graph for ﬁve nodes is shown on the left. With M = 2, the vector x = (1,1,1,1,1) satisﬁes the clique constraints, but there is no feasible allocation of carriers to cells, as seen in the diagram on the right.

x = (1, 1, 1, . . .) is feasible, and x = (2, 1, 1, . . .) is not. For a given ﬁxed allocation of carriers, let XFCA denote the set of feasible carrier requirements x. Clearly, XFCA ⊂ X .

b. Maximum Packing Allocation (MPA). By deﬁnition, for every x ∈ X there is a carrier assignment that achieves x. When a call arrives to a cell and is accepted, then this will result in a carrier requirement vector y. Under MPA, if y ∈ X , then the call is accepted, even if this requires a rearrangement of the carriers. This is not a practical approach as the rearrangement requires a lot of signaling, and the forced handovers of calls as carriers are being swapped. Writing the set of feasible carrier requirements under MPA by XMPA , we have XMPA = X . c. Clique Packing Assignment (CPA). Since the characterization of XCPA is simple, for theoretical purposes we may assume that each x ∈ XCPA is acceptable. In general, we have XFCA ⊂ X = XMPA ⊂ XCPA

where, as we have seen, the last containment can be strict. Another channel allocation strategy, which can be viewed as a hybrid of FCA and MPA, is that of channel borrowing. Some channels are statically assigned to cells, whereas others are permitted to be borrowed between cells, in order to accommodate local load variations. 4.5.4 Call Blocking Analysis If a carrier allocation respects the SIR constraints, or if it satisﬁes certain reuse constraints that, in turn, assure the SIR constraints, then, with a high probability,

4.5

Channel Allocation and Multicell Erlang Models

105

the accepted calls will experience an acceptable voice quality. This was the purpose of the analysis that we discussed in Section 4.2. Once a particular carrier allocation strategy (denoted CA, generically) is chosen, then the carrier requirement vector x will remain in XCA . Calls will need to be blocked for this to happen; if acceptance of a new call results in a carrier requirement vector x ∈ XCA , then the arriving call is blocked. In addition to a good voice quality during a call, users also are concerned about the probability of their requests being blocked, or accepted requests being dropped because of handover blocking. In this section we show how blocking probabilities can be obtained for carrier allocation strategies. Consider any carrier assignment strategy, and let XCA denote the set of feasible carrier requirements, x, as discussed earlier. We will assume, for simplicity, that each carrier can carry just one call (rather than, for example, eight in the GSM system). In this section, we also assume that calls stay in the cells into which they arrive, that is, that there are no handovers between cells. Let the arrival rate of calls into Cell j be λj , 1 ≤ j ≤ N . The arrival processes are assumed to be Poisson processes that are independent from cell to cell. We assume that the time duration for which a call holds a carrier has mean μ1 , and that the holding times from call to call are i.i.d. We also assume that the carrier holding times are exponentially distributed; this assumption can be relaxed, but we will not dwell on that aspect in this discussion (see, however, Appendix D, Section D.5.1). In this setting, let Xj (t), 1 ≤ j ≤ N , denote the number of carriers utilized in Cell j (equivalently, the number of calls in Cell j) at time t . Then consider the vector random process X(t) = (X1 (t), X2 (t), . . . , XN (t)). If the chosen carrier assignment strategy is used then, for all t , X(t) ∈ XCA . With the assumptions we have made on the arrival processes and carrier holding time distributions, it can easily be seen that the process X(t) is a continuous time Markov chain (CTMC; see Appendix D, Section D.2). For ﬁnite and positive arrival rates and mean holding times, this CTMC is positive recurrent, since it has a ﬁnite number of states. In order to obtain the blocking probabilities we need the stationary distribution π(x), x ∈ XCA . Then the blocking probability of calls arriving into Cell j, denoted by Pb, j is given by Pb,j = π(x) (4.6) {x∈XCA : x+ej ∈XCA }

where ej is the unit vector with a 1 in the j-th position. Note that Pb,j is the fraction of time during which an arrival into Cell j will be blocked. The fact that this is the same as the fraction of calls arriving into Cell j that are blocked (the quantity on the right-hand side of (4.6)) is a consequence of the Poisson Arrivals See Time Averages theorem (PASTA) (see Appendix D, Section D.4.2). The average blocking over all the cells is then given by Pb =

N j=1

λj N

i=1 λi

Pb, j

106

4

Cellular FDM-TDMA

which can be understood by observing that the probability that a call arrival is for λ Cell j is N j . i=1

λi

It remains to determine the stationary distribution π(x), x ∈ XCA . Notice that only the following state transitions are possible in the CTMC X(t). For x ∈ XCA , we can have x → x + ej for some j, 1 ≤ j ≤ N (due to an arrival into Cell j), or x → x − ej (due to a call completion in Cell j; here we require xj > 0 in x). Let λ ρj = μj , 1 ≤ j ≤ N , the Erlang load on Cell j. Deﬁne π(x) ˆ =

xj ρj N Πj=1 xj !

Now consider the transition x → x + ej , and notice that π(x) ˆ × λj = π(x ˆ + ej ) × (xj + 1)μj

Also, for the transition x → x − ej , where xj > 0, we have π(x) ˆ × xj μj = π(x ˆ − ej ) × λ j

With these observations, and deﬁning GCA =

π(x) ˆ

(4.7)

{x:x∈XCA }

it can be shown that (see Exercise 4.3) the stationary distribution is given by xj

ρj 1 π(x) = ΠN j=1 GCA xj !

(4.8)

Exercise 4.3 Use Theorem D.8 in Appendix D to prove that what is being claimed in (4.8) is correct. 4.5.5 Comparison of FCA and MPA Consider the three-cell example, and the corresponding pair-wise constraint graph shown in Figure 4.13. There are M carriers, each of which can handle one call. If we partition the set of carriers into two equal parts, and assign one set to Cells 1 and 3, and the other set to Cell 2, then the reuse constraints are met, and we get the XFCA shown by the dashed box in Figure 4.13. On the other hand, in MPA, any carrier not used in Cell 2 can be used in both Cells 1 and 3; XMPA is also shown in the ﬁgure. Suppose that the arrival rate of calls is the same in all the cells. Let us ﬁrst numerically investigate the blocking probabilities for M = 2. Figure 4.14 shows the set of states in XMPA . The set of states in which calls to

4.5

Channel Allocation and Multicell Erlang Models

107

x3 1

M

2

3

XFCA

M/2

M

M

M/2

XMPA

x2 M/2 M

x1

Figure 4.13 On the top right is shown a 3-cell example and the corresponding reuse constraint graph. There are M carriers. The sets XFCA and XMPA are the points with integer coordinates inside the regions shown.

000

010

001

101

002

102

011

100

020

111

200

110

201

202

Figure 4.14 The set of states for the three-cell network using maximum packing channel allocation with two channels. The downward transitions occur at rate λ and the upward transitions are at rates that are multiples of μ.

108

4

Cellular FDM-TDMA

Cell j are blocked are as follows. For Cell 1: (020), (110), (200), (111), (201), and (202); For Cell 2: (002), (011), (020), (110), (200), (102), (111), (201), and (202); For Cell 3: (020), (011), (002), (111), (102), and (202). Let λj = λ for j = 1, 2, 3. The following blocking probabilities are easy to obtain. GMPA = Pb,1 = Pb,3 = Pb,2 = Pb =

1 4 9 ρ + 2ρ3 + ρ2 + 3ρ + 1 4 2 1 4 4ρ

+ 32 ρ3 + 2ρ2 GMPA

1 4 4ρ

+ 2ρ3 + 32 ρ2 GMPA

2 1 Pb,1 + Pb,2 3 3

Figure 4.15 shows a plot of blocking probability in each of the cells and the overall blocking probability. For comparison, the blocking probability from a ﬁxed channel allocation is also shown; one channel is allocated to Cell 2, and the 1 0.9 0.8 0.7 P1 P2 P Pf

0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5

6

7

8

9

10

Figure 4.15 Plot of the blocking probability for the 3-cell network shown in Figure 4.14, with M = 2, under MPA, in Cell 1 (P1), Cell 2 (P2), and the overall blocking probability (P). Pf is the blocking probability for FCA, with 1 channel allocated to Cell 2, and the other to both Cells 1 and 3.

4.5

Channel Allocation and Multicell Erlang Models

109

other to both Cells 1 and 3. Note that in the middle cell, the blocking probability is worse with MPA than with FCA for large ρ. As can be seen from the set of states that block a call to Cell 2, and also the expression for Pb,2 , there are many more states that affect the blocking in the middle cell. Let us now consider a linear array of N > 3 cells, again with the constraint that neighboring cells cannot reuse the same channel. Extending the two-cell analysis via enumeration for N > 3 is clearly tedious. We therefore do an asymptotic analysis as N → ∞. Before describing the result, let us see what to expect. With increasing N , the number of middle cells increases as N → ∞, and the blocking probability behavior that we saw for Cell 2 in the numerical example earlier should become typical. We will see that this indeed is what happens. Let us consider the blocking probability in an interior Cell i. For the same reuse constraints, we see that the set of states XMPA , is deﬁned by (see Exercise 4.2) XMP = {x : xi + xi+1 ≤ M

for i = 1, . . . , N − 1}

and the blocking states for Cell i are deﬁned by {x : xi−1 + xi = M or xi + xi+1 = M}

or, equivalently, the set of blocking states for Cell i are {x : xi = M, or, (xi−1 + xi = M, 0 ≤ xi < M), or (xi + xi+1 = M, 0 ≤ xi < M)}

Thus, using the union bound, the blocking probability at Cell i with MPA is bounded as follows ⎛ ⎞ M−1 M−1 M k M−k k M−k 1 ⎝ ρ ρ ρ ρ ρ MPA ⎠ Pb,i ≤ G2,i (k) G3,i (k) G1 + + GMPA M! k! (M − k)! k! (M − k)! k=0

k=0

Here G1 , G2,i (k) and G3,i (k) are given by G1 :=

N ρ xj xj ! n=1

x∈X1

G2,i (k) :=

n=i

x∈X2, i (k)

G3,i (k) :=

x∈X3,i (k)

N ρ xj xj ! n=1

n=i−1,i

N ρxj xj ! n=1

n=i,i+1

where X1 is the set of states in which Cell i has M calls, X2,i (k) is the set of states in which Cell i has k calls and Cell (i − 1) has (M − k) calls, and X3,i (k) is the set of states in which Cell i has k calls and Cell (i + 1) has (M − k) calls.

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First, let us see what happens for low values of ρ. For low values of ρ, the G (k) 1 , G2,iMPA , and higher powers of ρ will be insigniﬁcant and we can argue that GGMPA G3,i (k) GMPA

all approach 1 as ρ → 0. Then, as ρ → 0, we can write MPA Pb,i ≤

M−1 ρk ρM−k ρM +2 M! k! (M − k)! k=0

⎛ ⎞ M ρ M ⎝ M! ρM +2 − 1⎠ = M! M! k! (M − k)! k=0

=

ρM M 2M+1 − 1 M ρM +2 (2 − 1) = ρ M! M! M!

For ﬁxed channel allocation, each cell would be allocated M 2 channels and the blocking probability would be (as before, see Appendix D, Section D.5.1) FCA Pb,i

=

ρM/2 (M/2)! M/2 ρk k=0 k!

≈

ρM/2 (M/2)!

where, since ρ is small, in the denominator we just retain the unit term. We can now FCA decreases as ρM/2 , whereas PMPA decreases see that, for large M, and small ρ, Pb,i b,i faster than ρM (for a precise calculation we can use Stirling’s approximation for the factorials). Hence, MPA would perform better at low loads. Let us now see what happens when ρ is large. The stationary probability of there being x active calls in Cell i can be written as πi (x) =

1 GMPA

x∈X3 (x)

M ρ x ρ xj x! j=1 xj !

1

=

GMPA

ρx φ(i, x) x!

j =i

where X3 (x) is the subset of XMPA in which there are x calls active in Cell i and φ(i, x) :=

M ρx j x∈X3 (x)

j=1 j=i

xj !

We can see that the carried load in Cell i is the average number of active calls in Cell i and is given by M x=1 xπi (x). Subtracting the carried load from the offered load (ρ to each cell) and expressing it as a fraction of the offered load, the loss MPA , is probability, Pb,i MPA Pb,i

=

ρ−

1 GMPA

M

ρx x=1 x x! φ(i, x)

ρ

= 1−

1

M−1

GMPA

x=0

ρx φ(i, x + 1) x!

4.5

Channel Allocation and Multicell Erlang Models

111

Obtaining φ(i, x) is involved and we will omit that here. For M = 2, and MPA can be shown to be given by N → ∞, Pb,i MPA Pb,i =

p2 (14 − 10p − 5p2 + 3p3 ) 2(2 + p2 − 2p3 )

Loss probability

where p is the solution in (0, 1) to the cubic equation ρ(1 − p)(2 − p2 ) = 2p. MPA and PFCA as a function of the offered load ρ. Notice Figure 4.16 shows Pb,i b,i at about ρ = 2.6 the ﬁxed channel assignment outperforms the maximum packing dynamic channel assignment! What is more interesting is that it can be shown that as M increases, the crossover happens at lower values of ρ and the crossover point is asymptotically 0! This indicates that for high capacity cellular networks, the ﬁxed channel allocation scheme will outperform the dynamic channel scheme when the load is time and space homogeneous. This result is deﬁnitely counterintuitive; we expect dynamic schemes to be better than static schemes. A heuristic explanation for the effect just described is that the MPA allocation scheme can upset the tight packing of the channels and calls at high loads and spends more time in the many Bad states that are possible with dynamic allocation. A conclusion that we may draw from this analysis is that it might be better to reject some calls, especially at high loads, to be able to improve the overall system performance. The MPA scheme will accept a call if the channels can be rearranged

Crossover probability

Pb,iFCA

MPA

Pb,i

1.0

Figure 4.16 from [73].

MPA and P FCA as a function of Pb,i b,i

2.0

2.6

offered load

ρ for M = 2, and N → ∞. Adapted

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to accommodate it while the FCA will reject a call if all the channels allocated to the cell are busy; that is, it will not borrow channels from other cells to fulﬁll a request. Thus, it is not automatic that a dynamic channel allocation performs better than a ﬁxed allocation scheme. However, we cannot conclude that dynamic channels do not have advantages. Rather that the advantages are realized if the offered load is nonhomogeneous in space and is time varying, in which case the dynamic schemes adapt to the changing load.

4.6

Handovers: Techniques, Models, Analysis

In our discussions thus far, we have essentially assumed that mobiles are conﬁned to the cells in which they initiate their calls. Since typically, neighboring cells do not reuse a carrier, when a mobile moves to a neighboring cell, it must switch over to a different carrier in that cell. This is called a handover. Naturally, for cellular mobile telephony to be a useful service, a handover should be transparent to the user. This imposes two requirements: a. An ongoing call should not experience degradation in service when it is at the fringes of the cell that is handling it. b. A handover should rarely fail due to a channel not being available in the cell into which a mobile call moves. Such a handover failure leads to call dropping, the constraints on which are more stringent than on call blocking. Handovers are performed by the MS making signal strength measurements to neighboring BSs, and conveying this information to the handover management system, which then decides on the need for a handover and the channel to be assigned to the new cell. The transfer of such measurements from the MS to the call handling system became possible in the second generation cellular systems. Note that the handover strategy basically deﬁnes what is meant by a cell’s coverage area. 4.6.1 Analysis of Signal Strength Based Handovers We consider an MS located on the line joining two BSs, BS 0 and BS 1, as shown in Figure 4.17. Let Si (x) = Received signal power from BS i, i ∈ {0, 1}, when the MS is at the distance x from BS 0. Then, recalling the path loss and shadowing model from Section 2.1.4, we have [S0 (x)]dB = S0 (d0 )

dB

− 10η log

x − ξ0 d0

where η is the path loss exponent, and the shadow fading, ξi , i ∈ {0, 1}, is normally distributed with mean 0, and variance σ 2 . Also, 2R − x [S1 (x)]dB = S1 (d0 ) dB − 10η log − ξ1 d0

4.6

Handovers: Techniques, Models, Analysis

113

BS 0

BS 1

MS 0

d 0

2R⫺d 0

R x

2R

2R⫺x

Figure 4.17 Handover: An MS located on the line joining two neighboring BSs that are at the distance 2R. The MS is located at distance x from BS 0. The MS provides signal strength measurements from each of the BSs.

Let us assume that S0 (d0 ) = S1 (d0 ). Then, for d0 ≤ x ≤ 2R − d0 ,

[S0 (x) − S1 (x)]dB

2R − x = 10η log x

+ (ξ1 − ξ0 )

where ξ1 −ξ0 is normally distributed with 0 mean and variance 2σ 2 . In Figure 4.18, we show the variation of [S0 (x) − S1 (x)]dB as the MS moves from BS 0 to BS 1. 2R−x . The two dashed curves above The solid curve shows the mean 10η log x and below the solid curve represent the variability due to shadowing, and can be viewed as the bounds within which [S0 (x) − S1 (x)]dB stays, with a high probability. The√half-width of the curved strip deﬁned by the two dashed curves is proportional to 2σ . Suppose the MS is being served by BS 0. A simple handover approach is to hand over the MS to BS 1 when [S0 (x) − S1 (x)]dB R). There are two issues here: a. If the coverage of either cell extends only up to a distance R, then once the MS is beyond R, the handover should occur with a high probability. b. With this design, if the MS is moving about in the region around the middle of the line joining the two BSs, then it will be repeatedly handed over between the two BSs, thus increasing the chance of the call being dropped, and also increasing the load on the call management processors.

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4

0

Cellular FDM-TDMA

2R2d0

d0 R

+H 2R 2H

h

Figure 4.18 Handover: The difference in signal strengths, S0 (x) − S1 ( x) (in dB) at an MS that is at position x on the line joining BS 0 and BS 1. For an explanation of h, see the text.

These two issues can be addressed by extending the coverage of each BS beyond R, to an additional distance, say, h. Suppose h is chosen so that 10η log

√ 2R − (R + h) + a 2σ < −H R+h

or 10η log

√ R−h + a 2σ < −H R+h

√ where a 2σ is the half width of the dashed strip, and a is chosen from the standard normal tables so that the tail probability of the random variable ξ1 − ξ0 beyond √ a 2σ is small. This choice of h is shown in Figure 4.18, since at x = R + h the upper dashed curve falls below −H . Now, when deciding to hand over from BS 0 to BS 1, we check if both of the following tests are true: [S0 (x)]dB < Sthreshold [S0 (x) − S1 (x)]dB < −H

for a suitably chosen Sthreshold . Both these tests will succeed beyond R + h with a high probability, and, thus, the handover will take place with a high probability. Further, the reverse handover will take place with a very small probability. Thus, this handover strategy has a hysteresis built into it. Although this design solves the problem of repeated handovers from one cell to the other, the extension of the cell coverage into the neighboring cell impacts the earlier SIR analysis. Let h =b R

4.6

Handovers: Techniques, Models, Analysis

115

so that R + h = (1 + b)R

Thus, in the cochannel interference calculations, we now need to use D (1 + b)R

It follows that a larger D R value will need to be used for a given SIR constraint, thus requiring a larger value of Nreuse , and lowering the spectrum efﬁciency. It is thus important to design handover schemes that can reduce the cell expansion factor b. 4.6.2 Handover Blocking, Call Dropping: Channel Reservation Let us consider a cell in an FDM-TDMA cellular system with new call arrival rate λ0 , and handover call arrival rate (from neighboring cells) λh . Deﬁne Pb = new call blocking probability Ph = handover blocking probability Pd = call dropping probability

Note that a call may undergo several handovers, and the call gets dropped at the ﬁrst of its handovers that is blocked. The preceding deﬁnitions can be formally expressed as Ph = lim

t→∞

number of handovers lost in [0, t] number of handovers in [0, t]

and Pd = lim

t→∞

number of accepted calls dropped in [0, t] number of calls accepted in [0, t]

Note that Pb and Pd are user perceived performance measures, whereas Ph is a measure internal to the system. We need Pd to be very small (e.g., 0.1%), whereas Pb is typically 1 to 2 percent. Let us assume that the time that a call spends in a cell is exponentially distributed with mean 1ν . The duration of a call is exponentially distributed with mean μ1 . Then, assuming that whether or not a handover is blocked is independent from handover to handover, we can write Pd =

ν (P + (1 − Ph ) · Pd ) ν+μ h

ν is the probability that a call leaves the cell This can be understood as follows. ν+μ it is in before it ﬁnishes conversation. If it does leave the cell, then the handover

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4

Cellular FDM-TDMA

attempt is blocked with probability Ph , or if the handover is not blocked (with probability 1 − Ph ), then we have a renewal point (see Appendix D) and the remaining call experiences dropping with probability Pd . This expression yields Pd =

=

ν ν+μ Ph νPh μ ν+μ + ν+μ ν μ Ph 1 + μν Ph

The second expression on the right may be approximated by μν Ph when μν Ph is much smaller than 1. Where the approximation works, its interpretation is the mean number of handovers per call multiplied by the handover blocking probability. This calculation yields a target value of Ph , given a target for Pd . We had deﬁned λh as the rate of arrival of handovers into a cell, and we observe that this is not a given. But with the exponential distribution assumptions we have made, we can write the following: ν 1 λh = (λ0 (1 − Pb ) + λh (1 − Ph )) (4.9) · ·6 μ+ν 6 This is obtained as follows. λ0 (1 − Pb ) + λh (1 − Ph ) is the rate of accepted calls into a cell. Each accepted call causes a handover to a neighboring cell with probability ν μ+ν . Each cell is surrounded by six cells, and one-sixth of the handovers of each of its neighbors enters the cell. However, Ph and Pb depend on λh . The approach is to iterate, starting with λ(0) . This will yield Pb(0) and Ph(0) . Given Pb(k−1) and Ph(k−1) h we can obtain λ(k) by using (4.9), and thus the iterations can continue. How Pb(k−1) h

and Ph(k−1) are obtained from λ(k) depends on the way channels are assigned to new h calls and handover calls in a cell, and is the next topic of discussion. The remaining question is whether there is a need to discriminate between new calls and handover calls when assigning channels. If they are handled in the same way, then they will get the same blocking probability (i.e., Pb = Ph ). Since the target value of Ph is much smaller than that of Pb , we will be forced to operate with much too small a value of new call blocking, which will result in a very low Erlang capacity. Hence channel reservation is done for handover calls. The common approach is dynamic channel reservation, which means the following. If there are m carriers in a cell, then a number mh < m is chosen; typically, mh is just 1 or 2. When a call arrives, if the number of busy carriers is less than m − mh , then every call is accepted. However, if the number of busy carriers is ≥ m − mh then only handover calls are accepted. If we assume that the arrival process of new calls and handover calls into a cell are independent Poisson processes, then the number of busy carriers becomes a positive recurrent CTMC. Returning to the iterative calculation, earlier, the analysis of this CTMC will provide Pb(k) and Ph(k) ,

4.7 The GSM System for Mobile Telephony

117

given λ0 and λ(k) . At the k-th iteration, let π(k) (i), 0 ≤ i ≤ m, denote the stationary h probability distribution of the CTMC. Then (k)

Pb =

m

π(k) (i)

i = m−mh

(k)

Ph = π(k) (m)

where, again, the PASTA theorem is used (see Appendix D, Section D.4.2).

Exercise 4.4 a. Show the transition rate diagram of the CTMC with dynamic channel reservation for handovers, and obtain the stationary distribution π(k) (·). b. Write a computer program to carry out the proposed iteration and obtain the new call arrival rate, λ0 , that can be offered when m = 16, and mh = 1, for a target Pd = 0.01. Take the mean call duration to be 100 seconds, and the mean time a call stays in a cell to be 50 seconds. Obtain the new call blocking probability, Pb , that is obtained with this value of λ0 . c. What is the new call arrival rate that can be handled if no special treatment is provided to handovers, but we still require that Pd = 0.01?

4.7 The GSM System for Mobile Telephony After about 15 years of deployment, the FDM-TDMA-based GSM system (Global System for Mobile communications) is the most popular cellular system for mobile telephony and related services. Figure 4.19 shows the components of a GSM cellular network. The wireless links are only between the mobile stations (MSs; shown as cellular phone handsets in Figure 4.19) and the Base Transceiver Stations (BTSs). An MS can be in the vicinity of several BTSs, but at any point in time, an active MS is associated with one BTS, the one with which it is determined that it has the highest probability of reliable communication. Several BTSs are linked to Base Station Controllers (BSCs) by wired links. Together, the BTSs and the associated BSC is called a BSS (Base Station Subsystem). The BTSs provide the ﬁxed ends of the radio links to the MSs; it is the BSC that has the intelligence to participate in the signaling involved in connection handovers. In turn, the BSCs are connected to the Mobile Switching Center (MSC), which connects to the ﬁxed network infrastructure. Worldwide, several bands have been used for the operation of GSM networks. The 900 MHz or 1800 MHz bands are the ones commonly used in most countries. In the 900 MHz band the uplink carriers are in the 890–915 MHz frequency band, and the downlink carriers are in the 935–960 MHz frequency

118

4

Cellular FDM-TDMA

NSS: Network and Switching Subsystem VLR

SS7 Network

HLR GMSC BSC

MSC

fixed network infrastructure BSC

mobile stations

BTS

Figure 4.19 The components of a GSM cellular network.

band. As explained earlier in this chapter, if an operator obtains W MHz of spectrum, actually W MHz is provided from the uplink band and another W MHz is provided from the downlink band. This is for the purpose of frequency division duplexing of bidirectional calls. The bandwidth of a GSM operator, in each direction, is then divided into FDM carriers with a spacing of 200 kHz. These FDM carriers are digitally modulated to create a hierarchical TDM carrier. The basic frame time in this TDM carrier is 4.615 ms, which contains eight slots, each of which can be assigned to a different voice call. Each TDM slot can carry 114 bits of payload. Notice that the coded payload bit rate on each carrier is about 200 Kbps. For one standard GSM voice codec, after channel coding, blocks of 456 bits are emitted, which are accommodated in four TDM slots. Since the resources (i.e., the spectrum) of a cellular wireless network are limited, an MS cannot have permanent access to the network, but has to make a request for a connection. Thus, since an MS is not always connected to the network, there are two problems that need to be addressed: a. Between the time that an MS last accessed the network and the time that it next needs to access, the MS may have moved; hence, it is ﬁrst necessary to locate the MS and associate it with one of the cells of the network. b. Since the MS initially does not have any access bandwidth assigned to it, some mechanism is needed for it to initiate a call or to respond to an incoming call.

4.8

Notes on the Literature

119

Location management and call set up are the major activities that need to be overlaid on the basic cellular wireless infrastructure in order to address the ﬁrst problem. In Figure 4.19 we show the additional components that are needed. Together these are called the Network and Switching Subsystem (NSS), and comprise the MSC, the GMSC (Gateway MSC), the HLR (Home Location Register), the VLR (Visitor Location Register), and the signaling network (standardized as Signaling System 7 (SS7), by the ITU). The SS7 signaling network already exists where there is a modern circuit switched phone network. As their names suggest, the HLR carries the registration of an MS at its home location, and a VLR in an area enters the picture when the MS is roaming in that area. Each operator has a GMSC at which all calls to MSs that are handled by the operator must ﬁrst arrive. The GMSC, HLR, and VLR exchange signaling messages over the SS7 network, and together help in setting up a call to a roaming user. Location management is done as follows. An MS will be registered with an operator in its home area. A roaming mobile that is turned on brieﬂy associates itself with a nearby BTS and provides the network the information that it is now in the area. If this happens to be an area other than where the MS normally is registered, then the MS’s identity is used to determine its home location, and the HLR at this location is informed of the whereabouts of the MS. The VLR at the location that the MS is visiting then receives conﬁrmation from the MS’s HLR that this MS is a valid user. Suppose now that someone somewhere in the world calls this MS. The MS’s number is used to determine the GMSC of its home operator. A signaling message is sent over the SS7 network to this GMSC, which determines the HLR where the MS is registered, and sends a message to this HLR. The HLR then, knowing that the MS is roaming, queries the VLR in the area where the MS is roaming. The VLR knows which local MSC the MS is in the control of, and provides this information to the HLR. The HLR forwards this information to the GMSC, which then directly establishes the call to the MS. Let us now turn to the second of the two problems enumerated. In the GSM system there are several permanent channels deﬁned in each cell. Whenever an MS enters a cell it locks into these channels. One of these channels is called the paging and access grant channel (PAGCH). If a call arrives for an MS, and it is determined that the MS may be in a cell, or in a group of cells, then the MS is paged in all these cells. Another such common channel is basically a slotted Aloha random access channel (RACH) (see Chapter 7), and is shared by all the MSs in the cell. When an MS has to respond to an incoming call (i.e., it is paged on the PAGCH) or has to initiate a call, it contends on the RACH in the cell, and conveys a short message to the network. Subsequently, the network allocates a channel to the MS and call set up signaling starts.

4.8

Notes on the Literature

In this chapter we have discussed concepts and techniques that were researched in the 1970s and 1980s, at a time when the ﬁrst analog cellular telephony systems

120

4

Cellular FDM-TDMA

were being experimented with. Bell System’s Advanced Mobile Phone Service (AMPS) and the cellular concept are described in a seminal article by MacDonald in Bell Systems Technical Journal [96]. There are several textbooks devoted to extensive treatments of cellular telephony, including the classic by Lee, and the more recent book by Garg and Wilkes [39]. The widely adopted text by Rappaport [116] discusses cellular mobile systems in conjunction with a detailed coverage of propagation phenomena in cellular mobile communication systems, physical layer techniques, and speech coding. A rigorous derivation of the formula relating the cochannel cell distance D(i, j) and Nreuse was carried out by Gamst [38] using group and ring theory. The Fenton-Wilkinson approximation, and other similar techniques have been derived in the text on mobile communications by Stuber [123]. The comparison of ﬁxed channel allocation and maximum packing allocation has been adapted from Kelly [73], which also provides some very useful insights into several problems in networking. A very accessible and extensive coverage of the GSM standard has been provided by Mouly and Pautet [104].

Problems 4.1.

The coverage of a cell is ﬁrst obtained by ignoring shadow fading (Rayleigh fading can be assumed to be averaged over). If the shadow fading standard deviation is 8 dB, roughly how much additional power is required so that the outage probability for the same coverage is less than 2%?

4.2.

A fade margin of 20 dB is required to combat shadowing and achieve adequate coverage in a cell. a. If the shadowing standard deviation is 8 dB, what was the target outage probability? b. If the path loss exponent is 4, how much additional coverage would be obtained if there is no shadowing?

4.3.

A GSM operator leases 7 MHz of spectrum (i.e., 7 MHz each in the uplink and the downlink), and estimates that a D R of at least 4 is required. If the cell radius, R, is 2 km (assume hexagonal cells), determine the Erlangs per square kilometer for the network, for a target blocking probability of 1%.

4.4.

A GSM operator leases 7 MHz of spectrum. Assuming that the uplink constrains performance, a path loss exponent of 4, and ignoring shadowing and additive noise, and given that an SIR of 14 dB is required, determine the Erlang capacity per cell for a blocking probability of 1%. Do not consider sectorization. Assume a hexagonal cell geometry.

4.5.

Consider a highway cellular system. Assume that the highway is exactly linear, the cells are of length 2R, and the cell width (i.e., the width of

Problems

121

the highway) can be ignored. Frequencies can be reused in cells whose centers are D units apart. The base station in each cell is at its center, and has two directional antennas, one covering each half of the cell (i.e., the cells are “sectorized” into two sectors). a. Relate D, R, and the number of reuse groups N . b. Accounting only for ﬁrst tier interferers, assuming that Rayleigh fading is averaged out, assuming independent log-normal shadowing for all the received signals, determine the minimum D R value so that the SIR falls below 12 dB with a probability of 1%. You must analyze both the forward and reverse channels. The standard deviation of log-normal shadowing is 8 dB. Take the power law path loss exponent to be 4. c. Explain why the SIR analysis is greatly simpliﬁed in this problem by assuming directional antennas; that is, by sectorization. d. Given that there are 200 trafﬁc channels available (assume single channel per carrier) determine the maximum number of Erlangs that each cell can be offered. 4.6.

Consider a channelized cellular system with a total of 320 trafﬁc channels. Denote the cell radius (center to apex) by R, and the minimum distance between cochannel cells by D. Assume that we can average over Rayleigh fading. Take the lognormal shadowing to have a standard deviation of σ = 8 dB, and the path loss component to be 4. Considering only the uplink channel answer the following. a. Obtain the channel reuse ratio for an uplink channel target SIR of 6 dB and an outage probability of 10%. Use the Fenton-Wilkinson method, and a table of the normal distribution. You may assume that the worst case interferer distance is D − R. b. List two assumptions that this analysis makes. In your solution in (a), where is Rayleigh fading being accounted for (even though it is being averaged over)? c. For this reuse ratio and the given number of channels, obtain the Erlang capacity per cell assuming a ﬁxed channel allocation, and a call blocking probability of 2%. Use an Erlang blocking table.

4.7.

Consider a TDM/TDMA cellular system in which each carrier handles eight calls. Voice activity detection (VAD) is used to reduce cochannel interference; an MS does not transmit when there is no speech activity. The probability of an MS being active is p. Consider a hexagonal cell layout; ignore shadowing and Rayleigh fading; take the path loss exponent to be η. In the following, use the standard approximations for the hexagonal geometry. Use tables of the standard normal distribution and Erlang blocking tables.

122

4

Cellular FDM-TDMA

a. Considering only the uplink, and accounting for voice activity, determine the minimum D/R ratio required for a SIR γ, if the probability of SIR falling below γ is allowed to be 2.3%. (Hint: consider the total power at the reference BS, and individual powers from each of the interfering MSs.) b. For γ = 20 dB, η = 4 and p = 0.4 determine the reuse ratio without and with VAD. Show that the effect of VAD is equivalent roughly to reducing the value of γ by 3 dB. 4.8.

In the ﬁgure are shown ﬁve cochannel cells each with four 90◦ sectors, oriented as shown. cell radius 5 R

D

5 cochannel cells, showing the 90 degree vectors

a. Copy the diagram and mark the cochannel sectors with g1 , g2 , g3 , and g4 . b. Ignoring Rayleigh fading and log-normal shadowing, obtain the value of D/R for a reverse channel worst-case S/I of 20 dB. Take the path loss exponent to be 4. 4.9.

a. The ﬁgure shows seven cells and pairwise reuse constraints between them. Show that for these constraints, and two channels, XCPA is strictly larger than XMPA . 1

7

2

6

3

5

4

Problems

123

b. Consider three cells with a triangular pairwise reuse constraint graph. There are N channels and calls arrive to each cell at rate λ. The calls have a mean channel holding time of b. i. Sketch the set of possible vectors of the numbers of calls that can be present in each of the cells (i.e., X ). ii. Find the probability that a call is blocked. 4.10.

Consider a linear array of K cells, with reuse constraint graph given. Let the J × K clique incidence matrix be denoted by A. Assume that the maximum number of frequency channels that can be used simultaneously in the j−th maximum clique is given to be nj , 1 ≤ j ≤ J (nj ≤ M, where M is the total number of frequency channels in the cellular system). Let N = (n1 , n2 , . . . , nJ ). a. Find the set S(N) of feasible cell occupancy vectors x = (x1 , x2 , . . . , xK ). b. Ignoring mobility, assume Poisson call arrivals with trafﬁc intensity ρk in cell k, 1 ≤ k ≤ K, and assume that the cell occupancy vector has the steady-state distribution x

π(x) = G(N)ΠK k=1

ρk k xk !

, x ∈ S(N),

where ⎛ G(N) = ⎝

x∈S(N)

x

ΠK k=1

ρk k xk !

⎞−1 ⎠

.

Show that the steady-state probability that a call arrival in cell k is blocked is Bk = 1 −

G(N) G N − AekT

where ek is the length-K vector (0, . . . , 0, 1, 0, . . . , 0) with the only non-zero element 1 appearing at the k-th position. 4.11.

Consider two neighboring base stations BS 1 and BS 2, a distance 2R apart (where R = 10d0 ), and an MS on the line joining them. Assuming that Rayleigh fading is averaged over, the minimum SINR required for acceptable communication is 10 dB. Let Ψ0 denote the average SNR at

124

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Cellular FDM-TDMA

a distance d0 from a transmitter. Assume that the power law path loss exponent is 4; the shadowing standard deviation is σ = 8 dB. a. With hand-off (at the cell boundary) obtain the (Ψ0 )dB required for an outage probability of 10%. b. With hand-off at 10% beyond the half-way point between the BSs (i.e., 1.1R), repeat part (a).

CHAPTER 5 Cellular CDMA

W

e discussed the CDMA concept in Chapter 2, Section 2.4.1. One of the two major technologies for second-generation cellular systems is based on CDMA. Third-generation (3G) cellular access systems that provide high speed data and multimedia access are also based on CDMA. In this chapter we will study various resource allocation problems in cellular CDMA systems, basing our discussion mainly on SINR (signal-to-interference plus noise ratio) analysis.

Overview Unlike the FDM-TDMA cellular systems discussed in Chapter 4, CDMA cellular systems are based on the principle of universal frequency reuse; that is, the same portion of the spectrum is reused at every BS. These systems employ frequency division duplexing; each system is assigned a pair of bands, one for the uplink and the other for the downlink. These two bands then are used at every BS. Thus, every uplink transmission interferes, in principle, with every other uplink transmission in the system; the same holds for the downlink. As discussed in Chapter 2, Section 2.4.1, the performance of an instance of communication between an MS and a BS depends on the SINR achieved at the receiver (see (2.24)). Second-generation CDMA systems were designed mainly for carrying telephone quality voice. CDMA systems have been evolving so as to be able to efﬁciently carry other guaranteed QoS services, such as interactive video, and also elastic services, such as ﬁle transfer and web access. We consider resource allocation for both these types of services. Each guaranteed QoS connection needs to achieve an SINR target. In Section 5.1, we write down general inequalities that need to be satisﬁed by the transmission powers used at all the uplink transmitters in the system. An important question that we then ask is about the existence of a set of transmit power levels at all the MSs so that the inequalities are satisﬁed. This leads to the concept of admission control; arbitrary collections of MSs, each with its SINR target cannot be handled by the system. Hence, some call requests need to be blocked. Focusing on the uplink problem, we begin by assuming a spatially homogeneous system in which the interference at a BS, from MSs associated with other BSs, can be taken to be just a multiple of the total received power at a BS. We develop the case of a single call class (say, voice) in Section 5.2. We ﬁnd that each call can be characterized by a resource requirement expressed in terms of the

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target SINR. We ﬁnd that the resource requirements of the calls just add up, and the admission control ensures that a certain measure of total system resource is not exceeded. This measure of system resource depends on the other-cell interference. We show how this is calculated, for hard handover and soft handover of calls between BSs. In Section 5.3, we expand our discussion to multiclass calls. The Chernoff bound is used to develop an admission control that again treats each call as having a resource requirement, and the requirement due to a set of calls is the sum of the individual resource requirements. In Section 5.4 we abandon the spatial homogeneity assumption and consider a general conﬁguration of MSs scattered among several BSs. For a given association of MSs and BSs, we develop a necessary and sufﬁcient condition for there to exist a feasible transmit power allocation. The condition is in terms of the PerronFrobenius eigenvalue of a matrix derived from the channel gains. This also leads to an iterative power control algorithm. Finally, in Section 5.5, we consider the scheduling of downlink elastic transfers. Depending on the channel power gains between the BSs and the MSs, there is a convex set of transfer rates that can be achieved to the MSs. There is a trade-off between maximizing the total transfer rate over all the MSs (which leads to maximization of operator revenue), and fairness between the rates assigned to the MSs. We use the sum-utility maximization formulation to compare various approaches. One such formulation leads to the idea of proportional fairness, for which we then show how the ﬁle transfer delay can be analyzed in terms of the M/G/1 processor sharing model. A brief overview of 2G and 3G CDMA cellular standards is then provided in Section 5.6.

5.1 The Uplink SINR Inequalities In CDMA cellular systems, each active mobile station (MS) is associated with one of the base stations (BSs) in its vicinity. When an MS is involved in a conversation, then it is assigned a power level with which it should transmit. As explained in Section 2.4.1, in CDMA access networks the link performance obtained by each mobile station (MS) is governed by the strength of its signal and the interference experienced by the MS’s signal at the intended receiver. For each radio link between an MS and a BS, a SINR target needs to be met. Hence it is important to associate MSs with BSs, and to assign them transmit powers in such a way that signal strengths of intended signals are high and interference from unintended signals is low. It is evident that increasing the transmit power to help one MS may not solve the overall problem, as this increase may cause unacceptably high interference at the intended receiver (i.e., a BS) of another MS. We will say that an association of MSs with BSs, and an allocation of transmit powers, is feasible if all SINR targets are achieved. In some situations there may be no feasible power allocation. The analysis of CDMA systems is performed via certain SINR inequalities. We will begin our discussion by setting up these inequalities in general.

5.1 The Uplink SINR Inequalities

127

Consider a CDMA system with multiple interfering cells (see Figure 5.1). The system bandwidth is W (e.g., 1.25 MHz in the IS-95 standard), and the chip rate is Rc ≤ W (e.g., 1.2288 Mcps (Mega chips per second)) in IS-95. There are M MSs and N BSs, with B = {1, 2, 3, . . . , N} denoting the set of BSs. Let hi, j , 1 ≤ i ≤ M, 1 ≤ j ≤ N , denote the power “gains” (i.e., attenuations) from MS i to BS j. Let A = (a1 , a2 , . . . , aM ), ai ∈ B, denote an association of MSs with the BSs; thus, in the association A, MS i is associated with BS ai . Let pi be the transmit signal power used by MS i, 1 ≤ i ≤ M. For the most part of the following discussion, we will assume that the power gains and the association are ﬁxed. With these deﬁnitions we can write the uplink received signal power to interference plus noise ratio for MS k as (SINR)k =

hk, ak pk {i : 1≤i≤M, i=k} hi, ak pi + N0 W

BS j

hk,j MS k

BS 1

BSN

hi,j

MS i

Figure 5.1 A depiction of the power allocation problem for several MSs in the vicinity of some BSs. The solid lines indicate signals from MSs to the BSs with which they are associated. MS k is associated with BS j, and its signal (solid line labeled hk,j ) is interfered with by all the other MSs associated with the other BSs (dashed lines), and also by the other MSs associated with BS j. The signal from MS k has a channel “gain” of hk,j to BS j.

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where N0 is the power spectral density of the additive noise, and W is the radio spectrum bandwidth. Assume that the interference plus noise is well modeled by a white Gaussian noise process. Various types of calls may be carried on the system; for example, there could be different types of voice telephony calls that use various codecs. Suppose that a call requires a bit rate Rk . In order to ensure a target bit-error-rate (BER) (which is governed by the required QoS for the application being carried; see the discussion to follow later in Section 5.2.2), we need to lower bound the product of the SINRk and the processing gain Lk := RRc (see (2.24)). For example, with Rk = 9.6 Kbps k and Rc = 1.2288 Mcps, we obtain L = 128. If the desired lower bound is γk , then, deﬁning Γk := γk RRk , we obtain (see (2.24)), for MS k, c

hk, ak pk

hi, a pi + N0 W k

≥ Γk

(5.1)

i : 1 ≤ i ≤ M, i = k

For a given association and given channel gains, we thus obtain M linear inequalities in the M uplink powers of the M MSs. Suppose ak = j; then if we deﬁne

Ij :=

hi, j pi

{i : 1 ≤ i ≤ M, ai =j}

the total power received at BS j from MSs associated with it, and deﬁne Io, j :=

hi, j pi

{i : 1 ≤ i ≤ M, ai =j}

the total interference power at BS j from MSs associated with other BSs, then we can write the SINR inequalities as hk, ak pk ≥ Γk (Iak − hk, ak pk ) + Io, ak + N0 W

(5.2)

for each k, 1 ≤ k ≤ M. Let us understand this derivation by looking at the geometry of the twouser case. Both users are associated with the same BS and there is no interference from any other cell. In Figure 5.2 we depict the analysis for two users. The SINR inequalities are (since there is only one BS we write hi,1 as hi ): h1 p1 − Γ1 h2 p2 ≥ Γ1 N0 W −Γ2 h1 p1 + h2 p2 ≥ Γ2 N0 W

5.1 The Uplink SINR Inequalities p2

129

1

p2

2

feasible power controls 2 1 p1

p1

p*

Figure 5.2 Power control feasibility for two users. The left panel shows the situation in which there are feasible power controls; then there is a power control that achieves the SINR targets with equality. The right panel shows a situation in which there is no feasible power control.

with p1 ≥ 0, p2 ≥ 0. These inequalities are depicted in Figure 5.2 by the lines labeled 1 and 2, for MS 1 and MS 2, respectively. The region to the right of, and below, the line labeled 1 is feasible for MS 1, and the region to the left of, and above, the line labeled 2 is feasible for MS 2. It is easy to see that there is a nonempty feasible region if Γ2 h1 h1 > Γ1 h2 h2

equivalently, if Γ1 Γ2 < 1. It can easily be checked that this is equivalent to Γ1 Γ2 + 0:

BS 1

BS 2 h

h h9

h9

Figure 5.3 The uplink power control problem for two cells with each of which there are M MSs associated. The MSs associated with each BS are collocated, and h and h are the channel power gains, as shown.

1 Note that, for simplicity, we are only considering path loss, and not shadowing, so that, for the geometry shown in the picture, it is plausible that the two groups of MSs have the same gains to the BSs.

5.2 A Simple Case: One Call Class

Q≥M >M

131

Γ ((1 + ν)Q + N0 W) 1+Γ Γ (1 + ν)Q 1+Γ

where the strict inequality arises because N0 W > 0. Thus, we ﬁnd that a necessary condition is M

Γ 1 < 1+Γ 1+ν

(5.5)

or, in other words, the number of admitted calls M should satisfy M

0. For the power allocation obtained, the value of Q is given by Q=

Γ (N0 W) M 1+Γ

Γ 1 − M(1 + ν) 1+Γ

and the interference at a BS from MSs associated with the other BS is given by Io = νQ. 5.2.2 Multiple BSs and Uniformly Distributed MSs We now assume that the MSs are uniformly distributed, and the radio propagation is spatially homogeneous, and, thus, that the BSs are uniformly loaded; that is, each BS receives the same total power Q from the MSs associated with it. Then we can continue to assume that at each BS the interference received from MSs not

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associated with it is some factor ν times Q; that is, Io = νQ for every BS. The SINR inequalities become hk pk ≥

Γ ((1 + ν)Q + N0 W) (1 + Γ)

(5.6)

for each k, 1 ≤ k ≤ M, where hk is the channel gain of MS k to the BS with which it is associated. As before, we sum these inequalities over the MSs associated with a BS to obtain the following necessary condition for a set of powers pk , 1 ≤ k ≤ M, to exist: Γ Q ≥M ((1 + ν)Q + N0 W) 1+Γ But then, noting that all the terms on the right are positive, and hence lower bounding this expression, we see that it is necessary that Γ (1 + ν)Q Q >M 1+Γ It follows that a necessary condition for the existence of a set of powers pk , 1 ≤ k ≤ M, that satisfy the SINR inequalities (5.6) is 1 Γ < M (5.7) 1+Γ 1+ν Now suppose that this condition holds, by associating new calls with a BS in such a way as to ensure that the condition is not violated. Taking equalities in (5.6) and summing, we obtain the following power allocation. For each k, 1 ≤ k ≤ M, ⎞ ⎛ Γ N0 W ⎝ 1+Γ ⎠ pk = (5.8) hk 1 − M(1 + ν) Γ 1+Γ

These powers are all positive when (5.7) holds, and, hence, we have a feasible power allocation (which meets the SINR constraints with equality). For this power allocation, by setting Q = M k=1 hk pk , we see that Γ M 1+Γ (N0 W) Q= Γ 1 − M (1 + ν) 1+Γ Thus, the condition expressed by (5.7) is found to be necessary and sufﬁcient for the existence of a feasible power control, in the present setting (i.e., at a BS, the uplink interference from MSs associated with other BSs can be modeled as a factor ν times the total power received at the BS from the MSs associated with it).

5.2 A Simple Case: One Call Class

133

Discussion a. From the previous derivation, we conclude that, in the single class case (with the spatial homogeneity assumptions we made), the following admission control will permit a feasible power allocation. A connection request is characterized by its “effective” resource requirement 1 +Γ Γ . The connection is added to the existing calls at a BS if and only if the following inequality is satisﬁed: Number of existing connections ×

Γ Γ 1 + < 1+Γ 1+Γ 1+ν

(5.9)

where ν is a spatial parameter that captures other cell interference. We will discuss how ν can be derived later in this chapter. b. We notice that a large value of Γ reduces the number of calls we can carry. How is the value of Γ determined? Suppose we wish to carry a new enhanced quality voice call, streaming audio call, or streaming video call using the CDMA access system just described. The source coding scheme that is used will determine the aggregate bit rate R that needs to carried. Also, sophisticated source coders will encode the source into bit streams of varying degrees of importance (called Class A, B, and C bits in some speech coders). When bit errors occur, a radio link layer protocol can recover the CDMA bursts containing the errored bits, but this recovery takes time, which adds to the end-to-end delay for the connection. After some number of attempts, bits may need to be discarded, in the hope that the decoder can reconstruct the speech or audio with some desirable quality using the received bits. It is thus clear that, for each coder, there will be a threshold bit error rate above which the speech (or audio or video) quality will not be acceptable. Finally, the physical layer (PHY) techniques employed (e.g., exploitation of multipath diversity (via a Rake receiver), interference cancellation, multiuser Eb detection) will determine the N , γ, required to provide the desired bit error 0 rate to the connection (see the discussions in Chapter 2). More sophisticated PHY techniques will result in a lower value of γ, hence a lower value of Γ Γ = γ RRc , and thus a lower resource requirement 1 + Γ for the connection. c. To get a feel for the numbers, let us consider telephone quality voice over the IS 95 CDMA system. A commonly used speech coder has R = 9.6 Kbps. The system bandwidth is 1.25 MHz, and the chip rate is 1.2288 Mcps. Thus 6 6 the processing gain is 1.2288×10 = 128 ≈ 21 dB (i.e., 10 log 1.2288×10 ≈ 21). 9.6×103 9.6×103 It turns out that, for the PHY techniques employed in the IS 95 standard, Eb for this speech coder is 6 dB. It follows that the target SINR, the target N 0 1 Γ = Rγc , is 6 − 21 = −15 dB (in fact, Γ = 32 ). The target SINR of −15 dB R

should be contrasted with narrowband systems such as FDM-TDMA (see Chapter 4) where the target SINR could be as high as 8 to 10 dB.

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d. We can see that the interference Io = νQ can be reduced by exploiting voice activity detection; the voice call transmits only when carrying actual speech, and turns off during silence periods, thereby reducing the other cell interference for a given number of accepted calls. Note that, roughly, this will result in the factor 1 + ν getting multiplied by the voice activity factor (note that even the intracell interference reduces by the voice activity factor, hence we multiply 1 + ν by this factor). The voice activity factor is typically 0.4 to 0.5, and thus this technique results in the capacity being increased by a multiplicative factor of 2 to 2.5. 5.2.3 Other Cell Interference: Hard and Soft Handover Let us examine the form of the power allocation proposed in (5.8). We will now refer to a BS and the region in which MSs will normally associate with this BS as a cell. Notice that, in the one class case, with homogeneous interference at each cell, the received powers hk pk are all equal at every BS. Thus, when the entire system carries just one type of call (as is the case in the early deployment of all cellular telephony systems), then the powers of all MSs, in any cell, need to be controlled in such a way that the received powers at their respective BSs are all equal. If the power to be received at each BS from any MS has to be the same, then, in order that an MS uses the least transmit power it should associate with the geographically nearest BS (assuming only deterministic path loss proportional to an inverse power of the distance). For a location with coordinates (x, y) let rj (x, y) denote the distance of BS j from the location (x, y). We can say that the default coverage area of BS j is all (x, y) such that rj (x, y) < rk (x, y) for every other BS k. If this is done, then the coverage areas are actually so-called Voronoi cells, which are uniquely determined by the BS locations. We obtained the power allocation shown in (5.8), assuming that the other cell interference factor ν was somehow given. The power allocation actions in one cell, however, affect the other-cell interference seen by other cells. For example, if MS k is at the fringe of the coverage area of the BS with which it is associated then the value of hk will be small, thus requiring a large value of pk (see (5.8)). But this large value of pk will result in a higher level of other-cell interference at neighboring BSs. In fact, it is possible that the MS may have a better channel to a neighboring BS than to the one with which it is associated. If on the basis of this better channel to the neighboring BS the MS is handed over to that BS, then we say that we are performing soft handovers. On the other hand if the region is demarcated into coverage areas on the basis of path loss measurements, and MSs are associated with a BS so long as they are in its coverage area, then we say that we are performing hard handovers. We will carry out an interference analysis, assuming that all calls are of the same type, and hence (for a spatially homogeneous system, as assumed in our simple analysis earlier) the target received power from an MS is the same at every BS. This analysis will yield the value of ν for hard hand-off and for soft hand-off.

5.2 A Simple Case: One Call Class

135

With this we will have all the ingredients to perform a quantitative evaluation of the system capacity as given by (5.7). Let us ﬁrst consider hard hand-off. Let Sr denote the target uplink received power at a BS from any MS associated with it. In Figure 5.4 we show an MS at the location (x, y) in the coverage area of BS 1. The distance of the MS (located at (x, y)) to BS 1 is r1 (x, y), and to BS 0 is r0 (x, y). Modeling the power law path loss and shadowing, it can be seen that the interference power, say, S0 , at BS 0 due to the MS at location (x, y) is given by S0 = Sr

r1 (x, y) r0 (x, y)

η

10(ξ1 (x, y)+ζ(x, y))/10 10(ξ0 (x, y)+ζ(x, y))/10

where η is the path loss exponent, ξ1 (x, y), ξ0 (x, y), and ζ(x, y) are i.i.d. normally 2 distributed random variables with mean 0, and variance σ2 . Here, (ξ1 (x, y)+ζ(x, y)) correspond to the log-normal shadowing on the path to BS 1, and (ξ0 (x, y)+ζ(x, y)) to the log-normal shadowing on the path to BS 2. The shadowing is modeled as being composed of local shadowing around the MS, ζ(x, y), and the shadowing on the two different paths, ξ1 (x, y), ξ2 (x, y). The total shadowing standard deviation over each path is σ .

2

(x, y ) r1(x, y ) 1

3 r0(x, y)

0

6

4

5

Figure 5.4 Other-cell interference with hard hand-off. An MS at the location (x, y) is power controlled by BS 1, and the power it radiates causes uplink interference at BS 0.

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The previous expression can be understood as follows. Starting with the target power Sr at BS 1, we trace back to the MS to obtain its transmission power. This gives the numerator of the expression multiplying Sr . Then we obtain the interference power seen at BS 0. This is obtained by dividing by the channel attenuation along the path from (x, y) to BS 0. Note that the local shadowing terms cancel out, and, further, we assume that the distributions of ξ1 (x, y) and ξ0 (x, y) do not depend on the MS location (x, y). Then denoting these generic random variables by ξ1 and ξ0 , we get S0 = Sr

r1 (x, y) r0 (x, y)

η

10ξ1 /10 10ξ0 /10

Then the total expected other-cell interference at BS 0 is obtained by adding up the interference from all the other cell MSs and taking the expectation of this sum. This computation is done by assuming a uniform distribution of MSs over the coverage area, with density d MSs per unit area, and then integrating over the area outside of the cell covered by BS 0. This yields

ξ1 −ξ0 Io = Sr E 10 10

{(x, y)∈Cell / 0}

rBS (x, y) r0 (x, y)

η d dxdy

(5.10)

where rBS (x, y) denotes the distance of the location (x, y) from the BS in whose cell (x, y) lies. Clearly, the total power received at BS 0 from MSs associated with it is Q = Sr dA, where A is the area covered by a BS. It follows that ν=

ξ1 −ξ0 I0 rBS (x, y) η 1 dxdy = E 10 10 Q r0 (x, y) A {(x, y)∈Cell / 0}

It can be seen that the integral in the right-hand side of this expression does not vary with the cell radius, R. This integral can be numerically evaluated to approximately 0.44 for η = 4. Further, we observe that ln 10 ξ1 −ξ0 = E e 10 (ξ1 −ξ0 ) E 10 10

=e

σ2 2

ln 10 10

2

where we use the fact that ξ1 − ξ0 is normally distributed with mean 0 and variance σ 2 . For σ = 8 dB and η = 4, we then ﬁnd that ν = e(

σ 2 ln 10 2 2 ( 10 ) )

× 0.44 = 2.38. Thus, with an 8 dB standard deviation for the shadowing, and a path loss exponent of 4, the other-cell interference is 2.38 times the power received from MSs within the cell. We notice that with σ = 0 we have ν = 0.44, for η = 4.

5.2 A Simple Case: One Call Class

137

Let us now turn to the same analysis with soft handovers. Figure 5.5 depicts the concept. An MS at location (x, y) is power controlled by either BS 1 or BS 0. What this means is that the MS will use a transmit power that is the smaller of the two values required to achieve a received signal power of Sr at either of the two BSs. In the situation of random shadowing, this will result in the MS causing less interference than if it was dedicated to the more proximate of the two BSs. Thus, with random shadowing, an MS may get power controlled by a geographically farther away BS. For two neighboring BSs i and j (e.g., BS 1 and BS 0), and for a location (x, y) in the region where an MS chooses between either of them (e.g., (x, y) in Figure 5.5 is power controlled by BS 1 or BS 0), deﬁne αi, j (x, y) =

(ri (x, y))η 10ξi (x, y)/10 (rj (x, y))η 10ξj (x, y)/10

r1(x, y) 2

3

1 (x, y)

0 6

r0(x, y)

4

5

Figure 5.5 In soft hand-off, an MS is power controlled by the best of two or more BSs. This diagram shows an MS located at position (x, y) being power controlled by the best of BS 1 or BS 0. Each diamond shaped area, with a BS at each end of its long diagonal, shows the area in which an MS would be power controlled by either of those two BSs. By ♦i,j we will mean the diamond between BS i and BS j; as an illustration, ♦0,3 is shown shaded.

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where ri (x, y) and rj (x, y) are the distances of (x, y) from BS i and BS j, respectively, and ξi (x, y) (resp. ξj (x, y)) corresponds to log-normal shadowing near BS i (resp. BS j). As before, the distributions of these shadowing random variables will be taken to be independent of the location (x, y). Also ξi (x, y) and ξj (x, y) are assumed statistically independent in the following analysis. We can see that αi,j (x, y) is the relative attenuation from (x, y) to the BSs i and j; αi,j (x, y) > 1 implies that the power attenuation from the location (x, y) to BS i is larger (than that to BS j) and hence an MS located at the position (x, y) should be power controlled by j, since this will require the MS to use less transmission power. As before, let d be the density of mobiles per unit of the system coverage area. It can then be seen that the total power received at BS 0 (i.e., intracell power and other-cell interference) is given by 6

♦0, 1

Sr 1{α0, 1 (x,y)≤1} + Sr α1, 0 (x, y) 1{α1, 0 (x, y)

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