Introduction and Overview

Abstract

This book presents asymptotic analyses of singularly perturbed Markov chains and reveals the interrelations of Markov chains and singular perturbations. Treating a wide variety of real-world systems under uncertainties, one frequently uses Markovian models. Quite often, the formulations lead to singularly perturbed Markov chains. In many applications, various factors change at different rates: Some evolve slowly, whereas others vary rapidly. As a result, the separation of fast and slow time scales arises. The phenomena are often described by introducing a small parameter ε>0, which leads to a singularly perturbed system involving two-time scales, namely, the actual time t and the stretched time t/ε To analyze such systems, one seeks to “average out” the fast variables and to consider only certain averaged characteristics via asymptotic methods.