Continuous-Time Markov Chains and Applications

A Singular Perturbation Approach

  • G. George Yin
  • Qing Zhang

Part of the Applications of Mathematics book series (SMAP, volume 37)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Prologue and Preliminaries

    1. Front Matter
      Pages 1-1
    2. G. George Yin, Qing Zhang
      Pages 3-14
    3. G. George Yin, Qing Zhang
      Pages 15-24
    4. G. George Yin, Qing Zhang
      Pages 25-50
  3. Singularly Perturbed Markov Chains

    1. Front Matter
      Pages 51-51
    2. G. George Yin, Qing Zhang
      Pages 53-78
    3. G. George Yin, Qing Zhang
      Pages 79-109
    4. G. George Yin, Qing Zhang
      Pages 111-166
  4. Control and Numerical Methods

    1. Front Matter
      Pages 219-219
    2. G. George Yin, Qing Zhang
      Pages 221-242
    3. G. George Yin, Qing Zhang
      Pages 243-276
    4. G. George Yin, Qing Zhang
      Pages 277-298
  5. Back Matter
    Pages 299-351

About this book

Introduction

This book is concerned with continuous-time Markov chains. It develops an integrated approach to singularly perturbed Markovian systems, and reveals interrelations of stochastic processes and singular perturbations. In recent years, Markovian formulations have been used routinely for nu­ merous real-world systems under uncertainties. Quite often, the underlying Markov chain is subject to rather frequent fluctuations and the correspond­ ing states are naturally divisible to a number of groups such that the chain fluctuates very rapidly among different states within a group, but jumps less frequently from one group to another. Various applications in engineer­ ing, economics, and biological and physical sciences have posed increasing demands on an in-depth study of such systems. A basic issue common to many different fields is the understanding of the distribution and the struc­ ture of the underlying uncertainty. Such needs become even more pressing when we deal with complex and/or large-scale Markovian models, whose closed-form solutions are usually very difficult to obtain. Markov chain, a well-known subject, has been studied by a host of re­ searchers for many years. While nonstationary cases have been treated in the literature, much emphasis has been on stationary Markov chains and their basic properties such as ergodicity, recurrence, and stability. In contrast, this book focuses on singularly perturbed nonstationary Markov chains and their asymptotic properties. Singular perturbation theory has a long history and is a powerful tool for a wide variety of applications.

Keywords

Markov chain Probability distribution Stochastic processes continuous-time Markov chain operations research optimization queueing theory stochastic process

Authors and affiliations

  • G. George Yin
    • 1
  • Qing Zhang
    • 2
  1. 1.Department of MathematicsWayne State UniversityDetroitUSA
  2. 2.Department of MathematicsUniversity of GeorgiaAthensUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-0627-9
  • Copyright Information Springer-Verlag New York, Inc. 1998
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-6844-4
  • Online ISBN 978-1-4612-0627-9
  • Series Print ISSN 0172-4568
  • About this book