Markov Chain Models — Rarity and Exponentiality

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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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.

• Markov
• Markov chain
• Markowsche Kette
• Random Walk
• Sage
• Variance
• birth-death process
• ergodicity
• regenerative process
• uniformization

• The University of Rochester, Rochester, USA

  Julian Keilson

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