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Analyzing Markov Chains using Kronecker Products

Theory and Applications

  • Book
  • © 2012

Overview

Authors:
  1. Tuğrul Dayar

    1. , Department of Computer Engineering, Bilkent University, Ankara, Turkey

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  • First to provide a solely Kronecker product based treatment of Markov chain analysis
  • The subject matter is interdisciplinary and at the intersection of applied mathematics, specifically numerical linear algebra and computational probability, and computer science
  • The exposition is concise and rigorous, yet it tries to be complete and touches almost all relevant aspects without being too technical.
  • Includes supplementary material: sn.pub/extras
  • Includes supplementary material: sn.pub/extras

Part of the book series: SpringerBriefs in Mathematics (BRIEFSMATH)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-ix
    Download chapter PDF

  2. Introduction

    • Tuğrul Dayar
    Pages 1-7

  3. Preliminaries

    • Tuğrul Dayar
    Pages 9-19

  4. Iterative Methods

    • Tuğrul Dayar
    Pages 21-35

  5. Decompositional Methods

    • Tuğrul Dayar
    Pages 37-56

  6. Matrix-Analytic Methods

    • Tuğrul Dayar
    Pages 57-73

  7. Conclusion

    • Tuğrul Dayar
    Pages 75-75

  8. Back Matter

    Pages 77-86
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Keywords

About this book

Kronecker products are used to define the underlying Markov chain (MC) in various modeling formalisms, including compositional Markovian models, hierarchical Markovian models, and stochastic process algebras. The motivation behind using a Kronecker structured representation rather than a flat one is to alleviate the storage requirements associated with the MC. With this approach, systems that are an order of magnitude larger can be analyzed on the same platform. The developments in the solution of such MCs are reviewed from an algebraic point of view and possible areas for further research are indicated with an emphasis on preprocessing using reordering, grouping, and lumping and numerical analysis using block iterative, preconditioned projection, multilevel, decompositional, and matrix analytic methods. Case studies from closed queueing networks and stochastic chemical kinetics are provided to motivate decompositional and matrix analytic methods, respectively.

Reviews

From the reviews:

“Dayar’s book on Kronecker products for Markov processes addresses an important topic which, up to now, has not been thoroughly discussed as a single entity in the literature. … The book has numerous examples throughout to illustrate the results and methods which the author presents. … the book is a valuable tool for learning to apply Kronecker representation to continuous time Markov chains.” (Myron Hlynka, Mathematical Reviews, May, 2013)

Authors and Affiliations

  • , Department of Computer Engineering, Bilkent University, Ankara, Turkey

    Tuğrul Dayar

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