Probabilistic models for computer networks : tools and solved problems

Probabilistic models for computer networks: Tools and solved problems overviews the main probabilistic tools and theory used in the performance analysis and modelling of modern computer networks and communication systems. With over one hundred examples and solved problems, the reader will be introduced to Poisson processes, Markov chains and queueing theory in an intuitive manner. Important theorems and tools are followed by easy network-based examples, showing the reader their applicability in real scenarios. This book is highly recommended for students in their final years of a degree in engineering, and for Ph.D. students willing to strengthen their skills in the evaluation of network performance. Probability review 13 Random variables 13 Probability Density Functions 14 Expectation and variance 15 Some useful PDFs 17 Conditional and joint probability 18 Conditional expectation 20 The Bayes' theorem 21 The Central Limit Theorem 22 Further problems 23 The exponential random variable 37 Definition 37 Simulation and estimation 38 The memoryless property 40 Minimum of several exponential random variables 42 Comparison of several exponential random variables 43 Further problems 45 The Poisson process 55 Counting processes 55 Stationary and independent increments 56 The Poisson process 57 Inter-arrival times of events 58 Waiting times 60 Conditional distribution of arrivals 62 Aggregation and sampling of Poisson processes 63 The theorem of Palm-Khintchine 64 Further problems 66 Discrete-Time Markov Chains 73 Introduction by example 73 States and transitions between states 74 Problem formulation 76 The Markov property and the Chapman-Kolmogorov equations 79 Sojourn times 80 Reachability and types of states 82 Steady-state analysis 84 First-passage times 85 Further problems 86 Continuous-Time Markov Chains 101 CTMCs by example 101 The infinitesimal generator 105 Definition and Chapman-Kolmogorov equations 107 The transient behaviour of a CTMC 109 Steady-state analysis and balance equations 111 First-passage times 114 Further problems 116 Classical queueing theory 129 Introduction by example 129 Definition and Kendall's notation 132 Little's theorem 136 The classical M/M/1 queue 138 The M/M/1/K queueing system 143 The M/M/c queueing system 145 Other non-classical Markovian queueing systems 151 Further problems 154 Open queueing networks 177 Introduction 177 Burke's theorem 178 Analysis of two M/M/1 queues in tandem 180 Types of networks and Jackson's theorem 181 Extended analysis for other open networks and Kleinrock's approximation 184 Further problems 187 Index 199