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Stochastic Networks

Book Description

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.

Table of Contents

  1. Cover
  2. Half title
  3. Series
  4. Title
  5. Copyright
  6. Table of Contents
  7. Preface
  8. Overview
    1. Queueing and loss networks
    2. Decentralized optimization
    3. Random access networks
    4. Broadband networks
    5. Internet modelling
  9. Part I
    1. 1 Markov chains
      1. 1.1 Definitions and notation
      2. 1.2 Time reversal
      3. 1.3 Erlang’s formula
      4. 1.4 Further reading
    2. 2 Queueing networks
      1. 2.1 An M/M/1 queue
      2. 2.2 A series of M/M/1 queues
      3. 2.3 Closed migration processes
      4. 2.4 Open migration processes
      5. 2.5 Little’s law
      6. 2.6 Linear migration processes
      7. 2.7 Generalizations
      8. 2.8 Further reading
    3. 3 Loss networks
      1. 3.1 Network model
      2. 3.2 Approximation procedure
      3. 3.3 Truncating reversible processes
      4. 3.4 Maximum probability
      5. 3.5 A central limit theorem
      6. 3.6 Erlang fixed point
      7. 3.7 Diverse routing
      8. 3.8 Further reading
  10. Part II
    1. 4 Decentralized optimization
      1. 4.1 An electrical network
      2. 4.2 Road traffic models
      3. 4.3 Optimization of queueing and loss networks
      4. 4.4 Further reading
    2. 5 Random access networks
      1. 5.1 The ALOHA protocol
      2. 5.2 Estimating backlog
      3. 5.3 Acknowledgement-based schemes
      4. 5.4 Distributed random access
      5. 5.5 Further reading
    3. 6 Effective bandwidth
      1. 6.1 Chernoff bound and Cramér’s theorem
      2. 6.2 Effective bandwidth
      3. 6.3 Large deviations for a queue with many sources
      4. 6.4 Further reading
  11. Part III
    1. 7 Internet congestion control
      1. 7.1 Control of elastic network flows
      2. 7.2 Notions of fairness
      3. 7.3 A primal algorithm
      4. 7.4 Modelling TCP
      5. 7.5 What is being optimized?
      6. 7.6 A dual algorithm
      7. 7.7 Time delays
      8. 7.8 Modelling a switch
      9. 7.9 Further reading
    2. 8 Flow level Internet models
      1. 8.1 Evolution of flows
      2. 8.2 α-fair rate allocations
      3. 8.3 Stability of α-fair rate allocations
      4. 8.4 What can go wrong?
      5. 8.5 Linear network with proportional fairness
      6. 8.6 Further reading
  12. Appendix A Continuous time Markov processes
  13. Appendix B Little’s law
  14. Appendix C Lagrange multipliers
  15. Appendix D Foster–Lyapunov criteria
  16. References
  17. Index