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Markov Chain Models — Rarity and Exponentiality

  • Book
  • © 1979

Overview

Editors:
  1. Julian Keilson

    1. The University of Rochester, Rochester, USA

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Part of the book series: Applied Mathematical Sciences (AMS, volume 28)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Introduction and Summary

    • Julian Keilson
    Pages 1-14

  3. Discrete Time Markov Chains; Reversibility in Time

    • Julian Keilson
    Pages 15-19

  4. Markov Chains in Continuous Time; Uniformization; Reversibility

    • Julian Keilson
    Pages 20-30

  5. More on Time-Reversibility; Potential Coefficients; Process Modification

    • Julian Keilson
    Pages 31-42

  6. Potential Theory, Replacement, and Compensation

    • Julian Keilson
    Pages 43-56

  7. Passage Time Densities in Birth —Death Processes; Distribution Structure

    • Julian Keilson
    Pages 57-75

  8. Passage Times and Exit Times for More General Chains

    • Julian Keilson
    Pages 76-104

  9. The Fundamental Matrix, and Allied Topics

    • Julian Keilson
    Pages 105-129

  10. Rarity and Exponentiality

    • Julian Keilson
    Pages 130-163

  11. Stochastic Monotonicity

    • Julian Keilson
    Pages 164-175

  12. Back Matter

    Pages 176-185
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Keywords

About this book

in failure time distributions for systems modeled by finite chains. This introductory chapter attempts to provide an over­ view of the material and ideas covered. The presentation is loose and fragmentary, and should be read lightly initially. Subsequent perusal from time to time may help tie the mat­ erial together and provide a unity less readily obtainable otherwise. The detailed presentation begins in Chapter 1, and some readers may prefer to begin there directly. §O.l. Time-Reversibility and Spectral Representation. Continuous time chains may be discussed in terms of discrete time chains by a uniformizing procedure (§2.l) that simplifies and unifies the theory and enables results for discrete and continuous time to be discussed simultaneously. Thus if N(t) is any finite Markov chain in continuous time governed by transition rates vmn one may write for pet) = [Pmn(t)] • P[N(t) = n I N(O) = m] pet) = exp [-vt(I - a )] (0.1.1) v where v > Max r v ' and mn m n law ~ 1 - v-I * Hence N(t) where is governed r vmn Nk = NK(t) n K(t) is a Poisson process of rate v indep- by a ' and v dent of N • k Time-reversibility (§1.3, §2.4, §2.S) is important for many reasons. A) The only broad class of tractable chains suitable for stochastic models is the time-reversible class.

Editors and Affiliations

  • The University of Rochester, Rochester, USA

    Julian Keilson

Bibliographic Information

  • Book Title: Markov Chain Models — Rarity and Exponentiality

  • Editors: Julian Keilson

  • Series Title: Applied Mathematical Sciences

  • DOI: https://doi.org/10.1007/978-1-4612-6200-8

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag New York Inc. 1979

  • Softcover ISBN: 978-0-387-90405-4Published: 23 April 1979

  • eBook ISBN: 978-1-4612-6200-8Published: 06 December 2012

  • Series ISSN: 0066-5452

  • Series E-ISSN: 2196-968X

  • Edition Number: 1

  • Number of Pages: XIV, 184

  • Topics: Probability Theory and Stochastic Processes

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