This is a text for professional mathematicians. It consists mostly of the mathematical theory--some of which has been developed by the authors--for analyzing the asymptotic behavior of singularly perturbed continuous-time Markov chains. The book’s thesis is that, by using singular perturbations, it is possible to reveal the interrelations of systems modeled as continuous-time Markov chains.

The book is divided into three sections. The first chapter motivates the rest of the book by pointing out the multitude of applications of the method. It suggests that the asymptotic behavior of a variety of applications can be dealt with by considering two time scales. From introducing an epsilon that divides one time scale in order to stretch it out as compared with another, a singularly perturbed system results; then, the faster changing time scale can be averaged out.

Chapter 2 is a short review of the mathematical background needed to understand the rest of the book. These few pages include martingales, irreducibility and quasistationary distributions, and Gaussian processes and diffusions. This chapter is not for the faint-hearted.

Chapter 3, the last chapter of the first section, outlines some of the practical applications of this theory, including queues with finite capacity, random evolutions, seasonal variation models, linear systems with jump Markov disturbances, and decomposition and aggregation of large-scale systems. All sound interesting as applications, but they are not developed, either here or later in the book.

Part 2, the major part of the book, treats the mathematical asymptotic properties of singularly perturbed Markov chains. Chapters cover irreducible generators, normality and exponential bounds, and weak and strong interactions.

Part 3 deals with applications including Markov decision processes under weak and strong interactions, hierarchical production planning, nearly optimal controls of stochastic dynamic systems, and numerical solutions for control and optimization of Markov chains.