General solution of the Poisson equation for Quasi-Birth-and-Death processes

TL;DR

This paper provides a comprehensive solution to the Poisson equation for Quasi-Birth-and-Death processes with infinite levels, leveraging matrix structures to derive explicit solutions.

Contribution

It introduces a general method to solve the Poisson equation for QBD processes using block matrix techniques and resolvent triples, extending previous approaches.

Findings

Explicit solutions for the Poisson equation in QBD processes.

Use of block tridiagonal and Toeplitz structures to simplify the problem.

Development of a systematic approach applicable to infinite-level QBDs.

Abstract

We consider the Poisson equation $(I-P)\boldsymbol{u}=\boldsymbol{g}$, where $P$ is the transition matrix of a Quasi-Birth-and-Death (QBD) process with infinitely many levels, $\bm g$ is a given infinite dimensional vector and $\bm u$ is the unknown. Our main result is to provide the general solution of this equation. To this purpose we use the block tridiagonal and block Toeplitz structure of the matrix $P$ to obtain a set of matrix difference equations, which are solved by constructing suitable resolvent triples.