# Markov Chain Models — Rarity and Exponentiality

• Julian Keilson
Book

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

1. Front Matter
Pages i-xiii
2. Julian Keilson
Pages 1-14
3. Julian Keilson
Pages 15-19
4. Julian Keilson
Pages 20-30
5. Julian Keilson
Pages 31-42
6. Julian Keilson
Pages 43-56
7. Julian Keilson
Pages 57-75
8. Julian Keilson
Pages 76-104
9. Julian Keilson
Pages 105-129
10. Julian Keilson
Pages 130-163
11. Julian Keilson
Pages 164-175
12. Back Matter
Pages 176-185

### Introduction

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.

### Keywords

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

### Editors and affiliations

• Julian Keilson
• 1
1. 1.The University of RochesterRochesterUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-6200-8
• Copyright Information Springer-Verlag New York 1979
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-0-387-90405-4
• Online ISBN 978-1-4612-6200-8
• Series Print ISSN 0066-5452
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