Accurate calculations of stationary distributions and mean first passage times in Markov renewal processes and Markov chains

TL;DR

This paper presents a numerically stable and refined method for accurately computing mean first passage times and stationary distributions in finite irreducible Markov chains and renewal processes, avoiding subtraction operations.

Contribution

It introduces a refined, subtraction-free procedure for calculating mean first passage times and stationary distributions, improving numerical stability over previous methods.

Findings

Method is numerically stable and subtraction-free.

Algebraic expressions derived for small state spaces.

Stationary distribution can be obtained from mean first passage times.

Abstract

This article describes an accurate procedure for computing the mean first passage times of a finite irreducible Markov chain and a Markov renewal process. The method is a refinement to the Kohlas, Zeit fur Oper Res, 30,197-207, (1986) procedure. The technique is numerically stable in that it doesn't involve subtractions. Algebraic expressions for the special cases of one, two, three and four states are derived. A consequence of the procedure is that the stationary distribution of the embedded Markov chain does not need to be derived in advance but can be found accurately from the derived mean first passage times. MatLab is utilized to carry out the computations, using some test problems from the literature.