Communication networks underpin our modern world, and provide fascinating and challenging examples of large-scale stochastic systems. Randomness arises in communication systems at many levels: for example, the initiation and termination times of calls in a telephone network, or the statistical structure of the arrival streams of packets at routers in the Internet. How can routing, flow control and connection acceptance algorithms be designed to work well in uncertain and random environments? This compact introduction illustrates how stochastic models can be used to shed light on important issues in the design and control of communication networks. It will appeal to readers with a mathematical background wishing to understand this important area of application, and to those with an engineering background who want to grasp the underlying mathematical theory. Each chapter ends with exercises and suggestions for further reading.
Preface 1 (2)
0 Overview 3 (10)
0.1 Queueing and loss networks 4 (2)
0.2 Decentralized optimization 6 (1)
0.3 Random access networks 7 (1)
0.4 Broadband networks 8 (2)
0.5 Internet modelling 10 (3)
Part I 13 (70)
1 Markov chains 15 (9)
1.1 Definitions and notation 15 (3)
1.2 Time reversal 18 (2)
1.3 Erlang's formula 20 (3)
1.4 Further reading 23 (1)
2 Queueing networks 24 (27)
2.1 An M/M/1 queue 24 (2)
2.2 A series of M/M/1 queues 26 (2)
2.3 Closed migration processes 28 (4)
2.4 Open migration processes 32 (6)
2.5 Little's law 38 (3)
2.6 Linear migration processes 41 (5)
2.7 Generalizations 46 (4)
2.8 Further reading 50 (1)
3 Loss networks 51 (32)
3.1 Network model 51 (2)
3.2 Approximation procedure 53 (1)
3.3 Truncating reversible processes 54 (5)
3.4 Maximum probability 59 (4)
3.5 A central limit theorem 63 (6)
3.6 Erlang fixed point 69 (4)
3.7 Diverse routing 73 (9)
3.8 Further reading 82 (1)
Part II 83 (66)
4 Decentralized optimization 85 (23)
4.1 An electrical network 86 (6)
4.2 Road traffic models 92 (9)
4.3 Optimization of queueing and loss 101(6)
networks
4.4 Further reading 107(1)
5 Random access networks 108(25)
5.1 The ALOHA protocol 109(6)
5.2 Estimating backlog 115(4)
5.3 Acknowledgement-based schemes 119(6)
5.4 Distributed random access 125(6)
5.5 Further reading 131(2)
6 Effective bandwidth 133(16)
6.1 Chernoff bound and Cramer's theorem 134(4)
6.2 Effective bandwidth 138(4)
6.3 Large deviations for a queue with 142(6)
many sources
6.4 Further reading 148(1)
Part III 149(68)
7 Internet congestion control 151(35)
7.1 Control of elastic network flows 151(7)
7.2 Notions of fairness 158(4)
7.3 A primal algorithm 162(4)
7.4 Modelling TCP 166(4)
7.5 What is being optimized? 170(2)
7.6 A dual algorithm 172(1)
7.7 Time delays 173(4)
7.8 Modelling a switch 177(7)
7.9 Further reading 184(2)
8 Flow level Internet models 186(31)
8.1 Evolution of flows 186(1)
8.2 α-fair rate allocations 187(2)
8.3 Stability of α-fair rate 189(3)
allocations
8.4 What can go wrong? 192(3)
8.5 Linear network with proportional 195(4)
fairness
8.6 Further reading 199(2)
Appendix A Continuous-time Markov 201(3)
processes
Appendix B Little's law 204(3)
Appendix C Lagrange multipliers 207(3)
Appendix D Foster-Lyapunov criteria 210(7)
Notes 217(1)
References 218(5)
Index 223
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