Modeling and Analysis of Telecommunications Networks · MODELING AND ANALYSIS OF TELECOMMUNICATIONS...
Transcript of Modeling and Analysis of Telecommunications Networks · MODELING AND ANALYSIS OF TELECOMMUNICATIONS...
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MODELING ANDANALYSIS OFTELECOMMUNICATIONSNETWORKS
JEREMIAH F. HAYES
THIMMA V. J. GANESH BABU
A JOHN WILEY & SONS, INC., PUBLICATION
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MODELING ANDANALYSIS OFTELECOMMUNICATIONSNETWORKS
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MODELING ANDANALYSIS OFTELECOMMUNICATIONSNETWORKS
JEREMIAH F. HAYES
THIMMA V. J. GANESH BABU
A JOHN WILEY & SONS, INC., PUBLICATION
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Copyright # 2004 by John Wiley & Sons, Inc. All rights reserved
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
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Library of Congress Cataloging-in-Publication Data:
Hayes, Jeremiah F., 1934-
Modeling and analysis of telecommunications networks/Jeremiah F.
Hayes & Thimma V. J. Ganesh Babu.
p. cm.
A Wiley-Interscience Publication.
ISBN 0-471-34845-7 (Cloth)
1. Telecommunication systems. I. Babu, Thimma V. J. Ganesh. II. Title.
TK5101 .H39 2004
621.38201dc222003020806
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
fax 978-646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should
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CONTENTS
Preface xiii
Retrieving Files from the Wiley FTP and Internet Sites xix
1 Performance Evaluation in Telecommunications 1
1.1 Introduction: The Telephone Network, 1
1.1.1 Customer Premises Equipment, 1
1.1.2 The Local Network, 2
1.1.3 Long-Haul Network, 4
1.1.4 Switching, 4
1.1.5 The Functional Organization of Network Protocols, 6
1.2 Approaches to Performance Evaluation, 8
1.3 Queueing Models, 9
1.3.1 Basic Form, 9
1.3.2 A Brief Historical Sketch, 10
1.4 Computational Tools, 13
Further Reading, 14
2 Probability and Random Processes Review 17
2.1 Basic Relations, 17
2.1.1 Set Functions and the Axioms of Probability, 17
2.1.2 Conditional Probability and Independence, 20
2.1.3 The Law of Total Probability and Bayes Rule, 21
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2.2 Random VariablesProbability Distributions and Densities, 22
2.2.1 The Cumulative Distribution Function, 22
2.2.2 Discrete Random Variables, 23
2.2.3 Continuous Random Variables, 31
2.3 Joint Distributions of Random Variables, 38
2.3.1 Probability Distributions, 38
2.3.2 Joint Moments, 40
2.3.3 Autocorrelation and Autocovariance Functions, 41
2.4 Linear Transformations, 42
2.4.1 Single Variable, 42
2.4.2 Sums of Random Variables, 42
2.5 Transformed Distributions, 46
2.6 Inequalities and Bounds, 47
2.7 Markov Chains, 52
2.7.1 The Memoryless Property, 52
2.7.2 State Transition Matrix, 53
2.7.3 Steady-State Distribution, 56
2.8 Random Processes, 61
2.8.1 Defintion: Ensemble of Functions, 61
2.8.2 Stationarity and Ergodicity, 61
2.8.3 Markov Processes, 63
References, 64
Exercises, 64
3 Application of Birth and Death Processes to Queueing Theory 67
3.1 Elements of the Queueing Model, 67
3.2 Littles Formula, 69
3.2.1 A Heuristic, 69
3.2.2 Graphical Proof, 70
3.2.3 Basic Relationship for the Single-Server Queue, 73
3.3 The Poisson Process, 74
3.3.1 Basic Properties, 74
3.3.2 Alternative Characterizations of the Poisson Process, 75
3.3.3 Adding and Splitting Poisson Processes, 78
3.3.4 Pure Birth Processes, 79
3.3.5 Poisson Arrivals See Time Averages (PASTA), 81
3.4 Birth and Death Processes: Application to Queueing, 82
3.4.1 Steady-State Solution, 82
3.4.2 Queueing Models, 85
3.4.3 The M/M/1 QueueInfinite Waiting Room, 863.4.4 The M/M/1/L QueueFinite Waiting Room, 893.4.5 The M/M/S QueueInfinite Waiting Room, 913.4.6 The M/M/S/L QueueFinite Waiting Room, 953.4.7 Finite Sources, 97
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3.5 Method of Stages, 98
3.5.1 Laplace Transform and Averages, 98
3.5.2 Insensitivity Property of Erlang B, 100
3.5.3 The Erlang B Blocking Formula: N Lines, Homogeneous Traffic, 103
References, 106
Exercises, 106
4 Networks of Queues: Product Form Solution 113
4.1 Introduction: Jackson Networks, 113
4.2 Reversibility: Burkes Theorem, 114
4.2.1 Reversibility Defined, 114
4.2.2 Reversibility and Birth and Death Processes, 116
4.2.3 Departure Process from the M/M/S Queue: Burkes Theorem, 118
4.3 Feedforward Networks, 119
4.3.1 A Two-Node Example, 119
4.3.2 Feedforward Networks: Application of Burkes Theorem, 120
4.3.3 The Traffic Equation, 121
4.4 Product Form Solution for Open Networks, 123
4.4.1 Flows Within Feedback Paths, 123
4.4.2 Detailed Derivation for a Two-Node Network, 124
4.4.3 N-Node Open Jackson Networks, 127
4.4.4 Average Message Delay in Open Networks, 132
4.4.5 Store-and-Forward Message-Switched Networks, 134
4.4.6 Capacity Allocation, 138
4.5 Closed Jackson Networks, 139
4.5.1 Traffic Equation, 139
4.5.2 Global Balance EquationSolution, 141
4.5.3 Normalization ConstantConvolution Algorithm, 142
4.5.4 Extension to the Infinite Server Case, 146
4.5.5 Mean Value Analysis of Closed Chains, 147
4.5.6 Application to General Networks, 149
4.6 BCMP Networks, 150
4.6.1 Overview of BCMP Networks, 150
4.6.2 Single NodeExponential Server, 151
4.6.3 Single NodeInfinite Server, 152
4.6.4 Single NodeProcessor Sharing, 156
4.6.5 Single NodeLast Come First Served (LCFS), 158
4.7 Networks of BCMP Queues, 161
4.7.1 Store-and-Forward Message-Switched Nodes, 163
4.7.2 Example: Window Flow ControlA Closed Network Model, 170
4.7.3 Cellular Radio, 175
References, 178
Exercises, 179
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5 Markov Chains: Application to Multiplexing and Access 187
5.1 Time-Division Multiplexing, 187
5.2 The Arrival Process, 188
5.2.1 Packetization, 188
5.2.2 Compound Arrivals, 189
5.3 Asynchronous Time-Division Multiplexing, 190
5.3.1 Finite Buffer, 192
5.3.2 Infinite Buffer, 195
5.4 Synchronous Time-Division Multiplexing, 197
5.4.1 Application of Rouches Theorem, 199
5.4.2 Calculations Involving Rouches Theorem, 201
5.4.3 Message Delay, 203
5.5 Random Access Techniques, 207
5.5.1 Introduction to ALOHA, 207
5.5.2 Analysis of Delay, 210
References, 215
Exercises, 216
6 The M/G/1 Queue: Imbedded Markov Chains 2196.1 The M/G/1 Queue, 219
6.1.1 Imbedded Markov Chains, 220
6.1.2 Distribution of Message Delay: FCFS, 222
6.1.3 Residual Life Distribution: Alternate Derivation of
the PollaczekKhinchin Formula, 231
6.1.4 Variation for the Initiator of a Busy Period, 234
6.1.5 Busy Period of the M/G/1 Queue, 2376.2 The G/M/1 Queue, 2416.3 Priority Queues, 244
6.3.1 Preemptive Resume Discipline, 245
6.3.2 L-Priority Classes, 252
6.3.3 Nonpreemptive Priorities, 256
6.4 Polling, 265
6.4.1 Basic Model: Applications, 265
6.4.2 Average Cycle Time, 267
6.4.3 Average Delay: Exhaustive, Gated, and Limited Service, 267
References, 274
Exercises, 275
7 Fluid Flow Analysis 281
7.1 OnOff Sources, 281
7.1.1 Single Source, 281
7.1.2 Multiple Sources, 284
7.2 Infinite Buffers, 286
7.2.1 The Differential Equation for Buffer Occupancy, 286
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7.2.2 Derivation of Eigenvalues, 289
7.2.3 Derivation of the Eigenvectors, 292
7.2.4 Derivation of Coefficients, 295
7.3 Finite Buffers, 298
7.4 More General Sources, 300
7.5 Analysis: Leaky Bucket, 300
7.6 Equivalent Bandwidth, 303
7.7 Long-Range-Dependent Traffic, 304
7.7.1 Definitions, 304
7.7.2 AMatching Technique for LRD Traffic Using the Fluid FlowModel, 306
References, 309
Exercises, 310
8 The Matrix Geometric Techniques 313
8.1 Introduction, 313
8.2 Arrival Processes, 313
8.2.1 The Markov Modulated Poisson Process (MMPP), 314
8.2.2 The Batch Markov Arrival Process, 316
8.2.3 Further Extensions, 319
8.2.4 Solutions of Forward Equation for the Arrival Process, 319
8.3 Imbedded Markov Chain Analysis, 321
8.3.1 Revisiting the M/G/1 Queue, 3218.3.2 The Multidimensional Case, 323
8.3.3 Application of Renewal Theory, 328
8.3.4 Moments at Message Departure, 334
8.3.5 Steady-State Queue Length at Arbitrary Points in Time, 335
8.3.6 Moments of the Queue Length at Arbitrary Points in Time, 336
8.3.7 Virtual Waiting Time, 336
8.4 A Matching Technique for LRD Traffic, 337
8.4.1 d MMPPs and Equivalents, 337
8.4.2 A Fitting Algorithm, 339
Appendix 8A: Derivation of Several Basic Equations Used in Text, 343
Appendix 8B: Derivation of Variance and Covariance Functions of Two-State
MMPP, 347
References, 355
Exercises, 355
9 Monte Carlo Simulation 359
9.1 Simulation and Statistics, 359
9.1.1 Introduction, 359
9.1.2 Sample Mean and Sample Variance, 359
9.1.3 Confidence Intervals, 361
9.1.4 Sample Sizes and Run Times, 362
9.1.5 Histograms, 364
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9.1.6 Hypothesis Testing and the Chi-Square Test, 368
9.2 Random-Number Generation, 370
9.2.1 Pseudorandom Numbers, 370
9.2.2 Generation of Continuous Random Variables, 371
9.2.3 Discrete Random VariablesGeneral Case, 375
9.2.4 Generating Specific Discrete Random Variables, 377
9.2.5 The Chi-Square Test Revisited, 379
9.3 Discrete-Event Simulation, 380
9.3.1 Time-Driven Simulation, 380
9.3.2 Event-Driven Simulation, 381
9.4 Variance Reduction Techniques, 382
9.4.1 Common Random-Number Technique, 383
9.4.2 Antithetic Variates, 384
9.4.3 Control Variates, 385
9.4.4 Importance Sampling, 386
References, 387
Exercises, 387
Index 389
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PREFACE
BACKGROUND
The insinuation of telecommunications into the daily fabric of our lives has been
arguably the most important and surprising development of the last 25 years. Before
this revolution, telephone service and its place in our lives had been largely stable
for more than a generation. The growth was, so to speak, lateral, as the global reach
of telecommunications extended and more people got telephone service. The
distinction between oversea and domestic calls blurred with the advances in
switching and transmission, undersea cable, and communication satellites. Traffic
on the network remained overwhelmingly voice, largely in analog format with
facsimile (Fax) beginning to make inroads. A relatively small amount of data traffic
was carried by modems operating at rates up to 9600 bits per second over voice
connections. Multiplexing of signals was rudimentarymost connections were
point-to-point business applications.
The contrast with todays network is overwhelming. The conversion from analog
to digital has long since been completed. A wide range of services, each with its
unique set of traffic characteristics and performance requirements, are available. At
the core of the change is the Internet, which is becoming accepted to handle all
telecommunications traffic and functions.
In order to effect such a far-reaching change, many streams converged. At the
most basic level was the explosive growth of the technology. The digital switching
and processing that are intrinsic to the modern network are possible only through
integrated-circuit technology. There are any number of examples of this technology
at work. For one who has worked in data communications, the most striking is the
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