Shift techniques for Quasi-Birth and Death processes: canonical factorizations and matrix equations

TL;DR

This paper explores shift techniques for Quasi-Birth and Death processes, focusing on canonical factorizations and matrix equations, and introduces new theoretical results with applications to solving the Poisson equation.

Contribution

It provides new insights into the existence and properties of canonical factorizations for QBDs and advances the understanding of associated quadratic matrix equations.

Findings

New results on solutions of quadratic matrix equations for QBDs

Applications to solving the Poisson equation for QBDs

Enhanced theoretical framework for shift techniques in QBDs

Abstract

We revisit the shift technique applied to Quasi-Birth and Death (QBD) processes (He, Meini, Rhee, SIAM J. Matrix Anal. Appl., 2001) by bringing the attention to the existence and properties of canonical factorizations. To this regard, we prove new results concerning the solutions of the quadratic matrix equations associated with the QBD. These results find applications to the solution of the Poisson equation for QBDs.