PERFORMANCE ANALYSIS OF COMMUNICATIONS NETWORKS AND SYSTEMS PIET VAN MIEGHEM Delft University of Technology cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru,UK First published in print format isbn-13 978-0-521-85515-0 isbn-13 978-0-511-16917-5 © Cambridge University Press 2006 2006 Informationonthistitle:www.cambrid g e.or g /9780521855150 This publication is in copyright. 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Published in the United States of America by Cambridge University Press, New York www.cambridge.org hardback eBook (NetLibrary) eBook (NetLibrary) hardback Waar een wil is, is een weg. to my father to my wife Saskia and my sons Vincent, Nathan and Laurens Contents Preface xi 1 Introduction 1 Part I Probability theory 7 2 Random variables 9 2.1 Probability theory and set theory 9 2.2 Discrete random variables 16 2.3 Continuous random variables 20 2.4 The conditional probability 26 2.5 Several random variables and independence 28 2.6 Conditional expectation 34 3 Basic distributions 37 3.1 Discrete random variables 37 3.2 Continuous random variables 43 3.3 Derived distributions 47 3.4 Functions of random variables 51 3.5 Examples of other distributions 54 3.6 Summary tables of probability distributions 58 3.7 Problems 59 4 Correlation 61 4.1 Generation of correlated Gaussian random variables 61 4.2 Generation of correlated random variables 67 4.3 The non-linear transformation method 68 v vi Contents 4.4 Examples of the non-linear transformation method 74 4.5 Linear combination of independent auxiliary random variables 78 4.6 Problem 82 5 Inequalities 83 5.1 The minimum (maximum) and infimum (supremum) 83 5.2 Continuous convex functions 84 5.3 Inequalities deduced from the Mean Value Theorem 86 5.4 The Markov and Chebyshev inequalities 87 5.5 The Hölder, Minkowski and Young inequalities 90 5.6 The Gauss inequality 92 5.7 The dominant pole approximation and large deviations 94 6 Limit laws 97 6.1 General theorems from analysis 97 6.2 Law of Large Numbers 101 6.3 Central Limit Theorem 103 6.4 Extremal distributions 104 Part II Stochastic processes 113 7 The Poisson process 115 7.1 A stochastic process 115 7.2 The Poisson process 120 7.3 Properties of the Poisson process 122 7.4 The nonhomogeneous Poisson process 129 7.5 The failure rate function 130 7.6 Problems 132 8 Renewal theory 137 8.1 Basic notions 138 8.2 Limit theorems 144 8.3 The residual waiting time 149 8.4 The renewal reward process 153 8.5 Problems 155 9 Discrete-time Markov chains 157 9.1 Definition 157 Contents vii 9.2 Discrete-time Markov chain 158 9.3 The steady-state of a Markov chain 168 9.4 Problems 177 10 Continuous-time Markov chains 179 10.1 Definition 179 10.2 Properties of continuous-time Markov processes 180 10.3 Steady-state 187 10.4 The embedded Markov chain 188 10.5 The transitions in a continuous-time Markov chain 193 10.6 Example: the two-state Markov chain in continuous-time 195 10.7 Time reversibility 196 10.8 Problems 199 11 Applications of Markov chains 201 11.1 Discrete Markov chains and independent random vari- ables 201 11.2 The general random walk 202 11.3 Birth and death process 208 11.4 A random walk on a graph 218 11.5 Slotted Aloha 219 11.6 Ranking of webpages 224 11.7 Problems 228 12 Branching processes 229 12.1 The probability generating function 231 12.2 The limit Z of the scaled random variables Z n 233 12.3 The Probability of Extinction of a Branching Process 237 12.4 Asymptotic behavior of Z 240 12.5 A geometric branching processes 243 13 General queueing theory 247 13.1 A queueing system 247 13.2 The waiting process: Lindley’s approach 252 13.3 The Bene˘s approach to the unfinished work 256 13.4 The counting process 263 13.5 PASTA 266 13.6 Little’s Law 267 14 Queueing models 271 viii Contents 14.1 The M/M/1 queue 271 14.2 Variants of the M/M/1 queue 276 14.3 The M/G/1 queue 283 14.4 The GI/D/m queue 289 14.5 The M/D/1/K queue 296 14.6 The N*D/D/1 queue 300 14.7 The AMS queue 304 14.8 The cell loss ratio 309 14.9 Problems 312 Part III Physics of networks 317 15 General characteristics of graphs 319 15.1 Introduction 319 15.2 The number of paths with m hops 321 15.3 The degree of a node in a graph 322 15.4 Connectivity and robustness 325 15.5 Graph metrics 328 15.6 Random graphs 329 15.7 The hopcount in a large, sparse graph with unit link weights 340 15.8 Problems 346 16 The Shortest Path Problem 347 16.1 The shortest path and the link weight structure 348 16.2 The shortest path tree in N Q with exponential link weights 349 16.3 The hopcount k Q in the URT 354 16.4 The weight of the shortest path 359 16.5 The flooding time W Q 361 16.6 The degree of a node in the URT 366 16.7 The minimum spanning tree 373 16.8 The proof of the degree Theorem 16.6.1 of the URT 380 16.9 Problems 385 17 The e!ciency of multicast 387 17.1 General results for j Q (p) 388 17.2 The random graph J s (Q) 392 17.3 The n-ary tree 401 Contents ix 17.4 The Chuang—Sirbu law 404 17.5 Stability of a multicast shortest path tree 407 17.6 Proof of (17.16): j Q (p) for random graphs 410 17.7 Proof of Theorem 17.3.1: j Q (p) for n-ary trees 414 17.8 Problem 416 18 The hopcount to an anycast group 417 18.1 Introduction 417 18.2 General analysis 419 18.3 The n-ary tree 423 18.4 The uniform recursive tree (URT) 424 18.5 Approximate analysis 431 18.6 The performance measure  in exponentially growing trees 432 Appendix A Stochastic matrices 435 Appendix B Algebraic graph theory 471 Appendix C Solutions of problems 493 Bibliography 523 Index 529 [...]...Preface Performance analysis belongs to the domain of applied mathematics The major domain of application in this book concerns telecommunications systems and networks We will mainly use stochastic analysis and probability theory to address problems in the performance evaluation of telecommunications systems and networks The rst chapter will provide a motivation and a statement of several problems... self-similar and long range dependent processes turns out to be fairly complex and beyond the scope of this book Finally, we mention the current interest in understanding and modeling complex networks such as the Internet, biological networks, social networks and utility infrastructures for water, gas, electricity and transport (cars, goods, trains) Since these networks consists of a huge number of nodes and. .. Introduction The aim of this rst chapter is to motivate why stochastic processes and probability theory are useful to solve problems in the domain of telecommunications systems and networks In any system, or for any transmission of information, there is always a non-zero probability of failure or of error penetration A lot of problems in quantifying the failure rate, bit error rate or the computation of redundancy... concept of entropy He introduced (see e.g Walrand, 1998) the notion of the Shannon capacity of a channel, the maximum rate at which bits can be transmitted with arbitrary small (but non zero) probability of errors, and the concept of the entropy rate of a source which is the minimum average number of bits per symbol required to encode the output of a source Many others have extended his basic ideas and. .. of chance1 and the likelihood of winning a game In most of these games, there was a nite number of possible outcomes and each of them was equally likely The 1 La rốgle des partis, a chapter in Pascals mathematical work (Pascal, 1954), consists of a series of letters to Fermat that discuss the following problem (together with a more complex question that is essentially a variant of the probability of. .. 1 ? 2 9 10 Random variables probability of the event of interest was dened as Pr [ ] = is the number of favorable outcomes (samples points of ) If the where number of outcomes of an experiment is not nite, this classical denition of probability does not su ce anymore In order to establish a coherent and precise theory, probability theory employs concepts of group or set theory The set of all possible... outcomes of an experiment is called the sample space A possible outcome of an experiment is called a sample point that is an element of the sample space An event consists of a set of sample points An event is thus a subset of the sample space The complement of an event consists of all sample points of the sample = \ Clearly, ( ) = space that are not in (the set) , thus and the complement of the sample... the sieve of Eratosthenes (Hardy and Wright, 1968, p 4), a procedure to construct the table of prime numbers4 up to Consider the increasing sequence of integers = {2 3 4 } and remove successively all multiples of 2 (even numbers starting from 4, 6, ), all multiples of 3 (starting from 32 and not yet removed previously), all multiples of 5, all multiples of the next number larger than 5 and still in... = A family F of events is a set of events and thus a subset of the sample space that possesses particular events as elements More precisely, a family F of events satises the three conditions that dene a -eld2 : (a) F, (b) if 1 2 F, then =1 F and (c) if F, then F These conditions guarantee that F is closed under countable unions and intersections of events Events and the probability of these events... than 1 and itself The sequence of the rst primes are 2, 3, 5, 7, 11, 13, etc If and are divisors of , then = from which it follows that and cannot exceed both Hence, any composite number is divisible by a prime that does not exceed 16 Random variables The number of primes smaller than a real number is ( ) and, evidently, denotes the -th prime, then ( ) = Let denote the set of the if multiples of the . The majordomainofapplicationinthisbookconcernstelecommunicationssys- tems and networks. We will mainly use stochastic analysis and probability theory to address problems in the performance evaluation of telecommuni- cations systems and networks. The first. PERFORMANCE ANALYSIS OF COMMUNICATIONS NETWORKS AND SYSTEMS PIET VAN MIEGHEM Delft University of Technology cambridge university press Cambridge,. biological networks, social networks and utility infrastructures for water, gas, electricity and transport (cars, goods, trains). Since these networks consists of a huge number of nodes Q and links