Stochastic Networks and Queues

By: Philippe Robert

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Hardcover | 23 June 2003

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424 Pages

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23.39 x 15.6 x 2.39

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Queues and stochastic networks are analyzed in this book with purely probabilistic methods. The purpose of these lectures is to show that general results from Markov processes, martingales or ergodic theory can be used directly to study the corresponding stochastic processes. Recent developments have shown that, instead of having ad-hoc methods, a better understanding of fundamental results on stochastic processes is crucial to study the complex behavior of stochastic networks.

In this book, various aspects of these stochastic models are investigated in depth in an elementary way: Existence of equilibrium, characterization of stationary regimes, transient behaviors (rare events, hitting times) and critical regimes, etc. A simple presentation of stationary point processes and Palm measures is given. Scaling methods and functional limit theorems are a major theme of this book. In particular, a complete chapter is devoted to fluid limits of Markov processes.

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Point Processes
General Definitions
Poisson Processes
Poisson Point Processes on the Real Line
Renewal Point Processes
The GI/GI/1 FIFO Queue and Random Walks
General results on the GI/GI/1 FIFO Queue
Wiener-Hopf Factorization
Applications to the GI/GI/1 Queue
The GI/M/1 and M/GI/1 Queues
The H/G/1 Queue
A Probabilistic Proof
Limit Theorems for the GI/GI/1 Queue
Introduction
The Biased Random Walk
The Tail Distribution of W
The Maximum of a Busy Period
The GI/GI/1 Queue near Saturation
The Random Walk Conditioned to Hit Level a
Stochastic Networks and Reversibility
Introduction
Reversibility of Markov Processes
Local Balance Equations
Queueing Networks with Product Form
The M/M/1 Queue
Introduction
Exponential Martingales
Hitting Times: Downward
Convergence to Equilibrium
Hitting Times: Upward
Rare Events
Fluid Limits
Large Deviations
Appendix
The M/M/infinity Queue
Introduction
Positive Martingales
Hitting Times: Downward
Hitting Times: Upward
Fluid Limits
A Functional Central Limit Theorem
The M/M/N/N Queue
Appendix
Queues with Poisson Arrivals
FIFO Queue
Infinite Server Queue
LIFO Queue with Preemptive Service
Processor-Sharing Queue
The Insensitivity Property
The Distribution Seen by Customers
Recurrence and Transience of Markov Chains
Recurrence of Markov Chains
Ergodicity
Transience
Ergodicity of Markov Processes
Some Applications
The Classical Version of Lyapunov's Theorem
Rescaled Markov Processes and Fluid Limits
Introduction
Rescaled Markov Processes
The Fluid Limits of a Class of Markov Processes
Fluid Limits and Skorohod Problems
Fluid Limits and Ergodicity Properties
Fluid Limits and Local Equilibrium
Bibliographical Notes
Ergodic Theory: Basic Results
Discrete Dynamical Systems
Ergodic Theorems
Continuous Time Dynamical Systems
Markovian Endomorphisms
Stationary Point Processes
Introduction
The Palm Space of the Arrival Process
Construction of a Stationary Point Process
Relations Between the Palm Space and Its Extension
Joint Distribution of the Points Around t=0
Some Properties of Stationary Point Processes
Appendix
The G/G/1 FIFO Queue
The Waiting Time
Virtual Waiting Time
The Number of Customers
The Associated Stationary Point Processes
The Unstable G/G/1 Queue
A Queue with Two Servers, the G/G/2 Queue
Martingales
Discrete Time Parameter Martingales
Continuous Time Martingales
The Stochastic Integral for a Poisson Process
Stochastic Differential Equations with Jumps
Markovian Jump Processes: B.1 Q-Matrices
global Balance Equations
The Associated Martingales
Convergence in Distribution: C.1 The Total Variation Norm on Probability Distributions
Convergence of Stochastic Processes
An Introduction to Skorohod Problems: D.1 Dimension 1
Multi-Dimensional Skorohod Problems
Table of Contents provided by Publisher. All Rights Reserved.

Stochastic Networks and Queues

At a Glance

Hardcover

Shipping

How to return your order

Defective items

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Other Editions and Formats

Paperback

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