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

Abstract

We consider the Poisson equation $(I-P)\mathbf{u}=\mathbf{g}$, where $P$ is the transition matrix of a Quasi-Birth-and-Death (QBD) process with infinitely many levels, $\mathbf{g}$ is a given infinite dimensional vector and $\mathbf{u}$ 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.

Cited by in corpus (3)

• Shift techniques for Quasi-Birth and Death processes: canonical factorizations and matrix equations
• Solving Poisson's equation for birth-death chains: Structure, instability, and accurate approximation
• Inversion of the Toeplitz-block Toeplitz matrices and the structure of the corresponding inverse matrices