Markov Processes for Stochastic Modeling

This book presents an algebraic development of the theory of countable state space Markov chains with discrete- and continuous-time parameters. A Markov chain is a stochastic process characterized by the Markov prop­ erty that the distribution of future depends only on the current state, not on the whole history. Despite its simple form of dependency, the Markov property has enabled us to develop a rich system of concepts and theorems and to derive many results that are useful in applications. In fact, the areas that can be modeled, with varying degrees of success, by Markov chains are vast and are still expanding. The aim of this book is a discussion of the time-dependent behavior, called the transient behavior, of Markov chains. From the practical point of view, when modeling a stochastic system by a Markov chain, there are many instances in which time-limiting results such as stationary distributions have no meaning. Or, even when the stationary distribution is of some importance, it is often dangerous to use the stationary result alone without knowing the transient behavior of the Markov chain. Not many books have paid much attention to this topic, despite its obvious importance.

Table of Contents:
Introduction: Stochastic processes; the Markov property; some examples; transition probabilities; the strong Markov property; exercises. Discrete-time Markov chains: First time passages; classification of states; recurrent Markov chains; finite Markov chains; time-reversable Markov chains; the rate of convergence to stationary; absorbing Markov chanins and their applications; Lossy Markov chains; exercises. Monotone Markov chains: Preliminaries; distribution classes of interest; stochastic ordering relations; monotone Markov chains; unimodality of transition probabilities; first-passage-time distributions; bounds for quasi-stationary distributions; renewal processes in discrete time; comparability of Markov chains; exercises. Continuous-time Markov chains: transition probability functions; finite Markov chains in continuous time; uniformization; more on finite Markov chains; absorbing Markov chains in continuous time; calculation of transition probability functions; stochastic monotonicity; semi-Markov processes; exercises. Birth-death processes: Boundary classifications; birth-death polynomials; finite birth-death processes; the Karlin-McGregor representation theorem; asymptotics of birth-death polynomials; quasi-stationary distributions; the decay parameter; the M/M/1 queue; exercises. Appendix A Review of matrix theory: Nonnegative matrices; ML-matrices; infinite matrices. Appendix B Generating functions and Laplace transforms: Generating functions; Laplace transforms. Appendix C Total positivity: TP functions; the variation-diminishing property.