Multiple time scale decomposition of discrete time Markov chains

Abstract

The multiple time scale decomposition of discrete time, finite state Markov chains is addressed. In [1, 2], the behavior of a continuous time Markov chain is approximated using a fast time scale, ϵ-independent, continuous time process, and a reduced order perturbed process. The procedure can then be iterated to obtain a complete multiple time scale decomposition. In the discrete time case presented in this paper, the basic approximation has a ‘hybrid’ form. In this form, the fast time scale behavior is approximated using an ϵ-independent, discrete time Markov chain, and the slow behavior is captured by a perturbed, continuous time process. Further time scale decomposition then involves the continuous time procedure in [1, 2]. This extension to discrete time chains bridges previous multiple time scale decomposition results, which have dealt exclusively with either continuous time or discrete time processes, and provides a uniform framework for the analysis of both types of systems.