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Stochastic Modeling and Analysis of Telecoms Networks

Stochastic Modeling and Analysis of Telecoms Networks

by Pascal Moyal, Laurent Decreusefond
Publisher: John Wiley & Sons
Release Date: April 2012
ISBN: 9781118563014
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Book Description

This book addresses the stochastic modeling of telecommunication networks, introducing the main mathematical tools for that purpose, such as Markov processes, real and spatial point processes and stochastic recursions, and presenting a wide list of results on stability, performances and comparison of systems.

The authors propose a comprehensive mathematical construction of the foundations of stochastic network theory: Markov chains, continuous time Markov chains are extensively studied using an original martingale-based approach. A complete presentation of stochastic recursions from an ergodic theoretical perspective is also provided, as well as spatial point processes.

Using these basic tools, stability criteria, performance measures and comparison principles are obtained for a wide class of models, from the canonical M/M/1 and G/G/1 queues to more sophisticated systems, including the current "hot topics" of spatial radio networking, OFDMA and real-time networks.

Contents

1. Introduction.

Part 1: Discrete-time Modeling

2. Stochastic Recursive Sequences.

3. Markov Chains.

4. Stationary Queues.

5. The M/GI/1 Queue.

Part 2: Continuous-time Modeling

6. Poisson Process.

7. Markov Process.

8. Systems with Delay.

9. Loss Systems.

Part 3: Spatial Modeling

10. Spatial Point Processes.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Preface
  5. Chapter 1. Introduction
    1. 1.1. Traffic, load, Erlang, etc.
    2. 1.2. Notations and nomenclature
    3. 1.3. Lindley and Beneš
      1. 1.3.1. Discrete model
      2. 1.3.2. Fluid model
      3. 1.3.3. Reflection problem
    4. 1.4. Notes and comments
  6. Part 1: Discrete-time Modeling
    1. Chapter 2. Stochastic Recursive Sequences
      1. 2.1. Canonical space
      2. 2.2. Loynes’s scheme
      3. 2.3. Coupling
        1. 2.3.1. Definition
        2. 2.3.2. Renovating events
      4. 2.4. Comparison of stochastic recursive sequences
      5. 2.5. Notes and comments
    2. Chapter 3. Markov Chains
      1. 3.1. Definition and examples
        1. 3.1.1. Simulation
      2. 3.2. Strong Markov property
      3. 3.3. Classification of states
      4. 3.4. Invariant measures and invariant probability
      5. 3.5. Effective calculation of the invariant probability
        1. 3.5.1. Iterative method
        2. 3.5.2. Ergodic method
      6. 3.6. Problems
      7. 3.7. Notes and comments
    3. Chapter 4. Stationary Queues
      1. 4.1. Single server queues
        1. 4.1.1. Stability
        2. 4.1.2. Comparisons of G/G/1 queues
        3. 4.1.3. Representation of service disciplines
        4. 4.1.4. Other features at equilibrium
        5. 4.1.5. Optimality of SRPT
        6. 4.1.6. GI/GI/1 queue: optimality of FIFO
        7. 4.1.7. Queues with deadlines: optimality of EDF
      2. 4.2. Processor sharing queue
      3. 4.3. Parallel queues
        1. 4.3.1. Preliminary result
        2. 4.3.2. The service profile
        3. 4.3.3. Stability
        4. 4.3.4. Comparison of systems
        5. 4.3.5. The optimal allocation
      4. 4.4. The queue with S servers
      5. 4.5. Infinite servers queue
        1. 4.5.1. The service profile
        2. 4.5.2. The GI/GI/∞ queue
      6. 4.6. Queues with impatient customers
        1. 4.6.1. The profile of service and patience times
        2. 4.6.2. GI/GI/1/1+GI queue
        3. 4.6.3. Optimality of EDF
        4. 4.6.4. FIFO queues
          1. 4.6.4.1. Single server (b) queue
      7. 4.7. Notes and comments
    4. Chapter 5. The M/GI/1 Queue
      1. 5.1. The number of customers in the queue
      2. 5.2. Pollacek-Khinchin formulas
      3. 5.3. Sojourn time
      4. 5.4. Tail distribution of the waiting time
      5. 5.5. Busy periods
  7. Part 2: Continuous-time Modeling
    1. Chapter 6. Poisson Process
      1. 6.1. Definitions
      2. 6.2. Properties
        1. 6.2.1. Superposition, thinning
        2. 6.2.2. Bus paradox
      3. 6.3. Discrete analog: the Bernoulli process
      4. 6.4. Simulation of the Poisson process
      5. 6.5. Non-homogeneous Poisson process
      6. 6.6. Cox processes
      7. 6.7. Problems
      8. 6.8. Notes and comments
    2. Chapter 7. Markov Process
      1. 7.1. Preliminaries
      2. 7.2. Pathwise construction
      3. 7.3. Markovian semi-group and infinitesimal generator
      4. 7.4. Martingale problem
      5. 7.5. Reversibility and applications
      6. 7.6. Markov Modulated Poisson Processes
      7. 7.7. Problems
      8. 7.8. Notes and comments
    3. Chapter 8. Systems with Delay
      1. 8.1. Little’s formula
      2. 8.2. Single server queue
      3. 8.3. Multiple server queue
        1. 8.3.1. Comparison of systems
      4. 8.4. Processor sharing queue
      5. 8.5. The M/M/∞ queue
      6. 8.6. The departure process
      7. 8.7. Queuing networks
        1. 8.7.1. Open Jackson networks
        2. 8.7.2. Closed Jackson networks
      8. 8.8. Problems
      9. 8.9. Notes and comments
    4. Chapter 9. Loss Systems
      1. 9.1. General
      2. 9.2. Erlang model
      3. 9.3. The M/M/1/1 + C queue
      4. 9.4. The “trunk” effect
      5. 9.5. Engset model
      6. 9.6. IPP/M/S/S queue
      7. 9.7. Generalized Erlang models
        1. 9.7.1. Guard channels
        2. 9.7.2. Multi-class system
      8. 9.8. Hierarchical networks
      9. 9.9. A model with balking
      10. 9.10. A call center with impatient customers
      11. 9.11. Problems
      12. 9.12. Notes and comments
  8. Part 3: Spatial Modeling
    1. Chapter 10. Spatial Point Processes
      1. 10.1. Preliminary
      2. 10.2. Stochastic geometry
      3. 10.3. Poisson process
      4. 10.4. Stochastic analysis
      5. 10.5. Problems
      6. 10.6. Notes and comments
  9. Appendix A. Mathematical Toolbox
    1. A.1. Probability spaces and processes
      1. A.1.1. Countable spaces
      2. A.1.2. Polish spaces
      3. A.1.3. Stochastic processes
      4. A.1.4.σ-fields
    2. A.2. Conditional expectation
      1. A.2.1. Independence and conditioning
      2. A.2.2. Markov property
    3. A.3. Vector spaces and orders
    4. A.4. Bounded variation processes
    5. A.5. Martingales
      1. A.5.1. Discrete time martingales
      2. A.5.2. Continuous time martingales
    6. A.6. Laplace transform
    7. A.7. Notes and comments
  10. Bibliography
  11. Index