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arXiv · 1604.04420

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

Dario A. Bini·Sarah Dendievel·Guy Latouche·Beatrice Meini
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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.

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Dario A. Bini, Sarah Dendievel, Guy Latouche, Beatrice Meini. 2016-04-15. General solution of the Poisson equation for Quasi-Birth-and-Death processes. https://doi.org/10.1137/16m1065045

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