A note on the invariant distribution of a quasi-birth-and-death process

TL;DR

This paper derives an explicit formula for the invariant distribution of a quasi-birth-and-death process using matrix-valued orthogonal polynomials, enabling computation from the transition matrix's block entries.

Contribution

It introduces a novel approach to compute the invariant distribution via squared norms of matrix-valued orthogonal polynomials, regardless of their diagonal structure.

Findings

Explicit formula for invariant distribution derived

Method applies even when squared norms are non-diagonal

Example demonstrates practical computation of invariant distribution

Abstract

The aim of this paper is to give an explicit formula of the invariant distribution of a quasi-birth-and-death process in terms of the block entries of the transition probability matrix using a matrix-valued orthogonal polynomials approach. We will show that the invariant distribution can be computed using the squared norms of the corresponding matrix-valued orthogonal polynomials, no matter if they are or not diagonal matrices. We will give an example where the squared norms are not diagonal matrices, but nevertheless we can compute its invariant distribution.