A Computational Framework for the Mixing Times in the QBD Processes with Infinitely-Many Levels

TL;DR

This paper introduces a computational framework for analyzing the mixing times of Markov chains with infinitely-many levels, especially QBD processes, using matrix Poisson equations and UL-type $RG$-factorization.

Contribution

It develops matrix Poisson equations for mean and variance of mixing times and applies UL-type $RG$-factorization to solve them, specifically for level-dependent QBD processes.

Findings

Derived matrix Poisson equations for mixing times.

Provided detailed computation methods for QBD processes.

Enhanced matrix-analytic techniques for infinite-level Markov chains.

Abstract

In this paper, we develop some matrix Poisson's equations satisfied by the mean and variance of the mixing time in an irreducible positive-recurrent discrete-time Markov chain with infinitely-many levels, and provide a computational framework for the solution to the matrix Poisson's equations by means of the UL-type of $RG$-factorization as well as the generalized inverses. In an important special case: the level-dependent QBD processes, we provide a detailed computation for the mean and variance of the mixing time. Based on this, we give new highlight on computation of the mixing time in the block-structured Markov chains with infinitely-many levels through the matrix-analytic method.