The Computation of the Mean First Passage Times for Markov Chains

TL;DR

This paper surveys various methods for computing mean first passage times in Markov chains, introduces a new accurate Extended GTH procedure, and compares it with existing techniques using MATLAB for performance evaluation.

Contribution

The paper presents a new Extended GTH procedure for accurate computation of mean first passage times and compares it with existing methods, highlighting its accuracy and computational trade-offs.

Findings

The new Extended GTH procedure is highly accurate on test problems.

It requires longer computational times compared to some existing methods.

The procedure performs well on large sparse state-space cases.

Abstract

A survey of a variety of computational procedures for finding the mean first passage times in Markov chains is presented. The author recently developed a new accurate computational technique, an Extended GTH Procedure, Hunter (Special Matrices, 2016) similar to that developed by Kohlas (Zeit. fur Oper. Res., 1986). In addition, the author recently developed a variety of new perturbation techniques for finding key properties of Markov chains including finding the mean first passage times, Hunter (Linear Algebra and its Applications, 2016). These recently developed procedures are compared with other procedures including the standard matrix inversion technique using the fundamental matrix (Kemeny and Snell, 1960), some simple generalized matrix inverse techniques developed by Hunter (Asia Pacific J. Oper. Res., 2007), and some modifications to the FUND technique of Heyman (SIAM J Matrix Anal. and Appl., 1995). MATLAB is used to compute errors and estimate computation times when the techniques are used on some test problems that have been used in the literature together with some large sparse state-space cases. For accuracy a preference for the procedure of the author is exhibited for the test problems. However it appears that the procedure, as presented, requires longer computational times.