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

TL;DR

This paper derives specific bounds for solutions to the Poisson equation in continuous-time Markov chains, with applications to queueing systems, under certain drift conditions, enhancing understanding of probabilistic solutions.

Contribution

It introduces explicit bounds for the standard solution of the Poisson equation for general Markov chains using $f$-modulated drift conditions, applicable to queueing models.

Findings

Established bounds for the standard solution under $f$-modulated drift conditions.

Applied results to workload processes in MAP/GI/1 and M/GI/1 queues.

Demonstrated the bounds' usefulness in queueing theory contexts.

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.