Constructing asymptotic series for probability distributions of Markov chains with weak and strong interactions

Abstract

Many applications arise in manufacturing systems, and queueing network problems involve Markov chains having slow and fast components. These components are coupled through weak and strong interactions. The main goal of this work is to study asymptotic properties for the probability distribution of the aforementioned Markov chains. Explicit construction of series expansions, consisting of regular part and boundary layer part or singular part, are developed by means of singular perturbation methods. The regular part is obtained by solving algebraic-differential equations, and the singular part is derived via solution of differential equations. One of the key points in the constructions is to select appropriate initial conditions. This is done by taking into consideration the regular part and the singular part together with their interactions. It is shown that the singular part decays exponentially fast. Analysis of residue is carried out, and the error bound for the remainder terms is ascertained.