Matrix-analytic methods for solving Poisson's equation with applications to Markov chains of GI/G/1-type

TL;DR

This paper develops matrix-analytic techniques to solve Poisson's equation for Markov chains, focusing on deviation and functional matrices, with applications to GI/G/1-type chains and queues, including extensions to continuous-time models.

Contribution

It introduces novel matrix-analytic methods for Poisson's equation solutions, specifically for GI/G/1-type Markov chains and related queueing models.

Findings

Derived explicit solutions for deviation and functional matrices.

Applied methods to GI/G/1-type Markov chains and queues with negative customers.

Extended techniques to continuous-time Markov chains.

Abstract

In this paper, we are devoted to developing matrix-analytic methods for solving Poisson's equation for irreducible and positive recurrent discrete-time Markov chains (DTMCs). Two special solutions, including the deviation matrix D and the expected additive-type functional matrix K, will be considered. The results are applied to Markov chains of GI/G/1-type and MAP/G/1 queues with negative customers. Further extensions to continuous-time Markov chains (CTMCs) are also investigated.