Specific bounds for a probabilistically interpretable solution of the Poisson equation for general state-space Markov chains with queueing applications

Abstract

This paper considers the Poisson equation for general state-space Markov chains in continuous time. The main purpose of this paper is to present specific bounds for the solutions of the Poisson equation for general state-space Markov chains. The solutions of the Poisson equation are unique in the sense that they are expressed in terms of a certain probabilistically interpretable solution (called the {\it standard solution}). Thus, we establish some specific bounds for the standard solution under the $f$-modulated drift condition (which is a kind of Foster-Lyapunov-type condition) and some moderate conditions. To demonstrate the applicability of our results, we consider the workload processes in two queues: MAP/GI/1 queue, and M/GI/1 queue with workload capacity limit.