Poisson's equation for discrete-time quasi-birth-and-death processes

TL;DR

This paper derives solutions to Poisson's equation for quasi-birth-and-death processes using their transition structure, linking it to perturbation analysis and illustrating with queue models.

Contribution

It provides two novel solution forms for Poisson's equation specific to QBDs and connects these solutions to perturbation analysis.

Findings

Derived solutions for Poisson's equation in QBDs

Linked solutions to perturbation analysis for sensitivity assessment

Applied methods to PH/M/1 queue to measure initial state impact

Abstract

We consider Poisson's equation for quasi-birth-and-death processes (QBDs) and we exploit the special transition structure of QBDs to obtain its solutions in two different forms. One is based on a decomposition through first passage times to lower levels, the other is based on a recursive expression for the deviation matrix. We revisit the link between a solution of Poisson's equation and perturbation analysis and we show that it applies to QBDs. We conclude with the PH/M/1 queue as an illustrative example, and we measure the sensitivity of the expected queue size to the initial value.