Kemeny's Function for Markov Chains and Markov Renewal Processes

TL;DR

This paper extends Kemeny's constant to Markov renewal processes and continuous-time Markov chains, identifying conditions under which these functions are constant and deriving explicit formulas for various cases.

Contribution

It introduces three variants of Kemeny's functions for continuous-time processes and establishes when they are constant, generalizing the discrete-time Markov chain results.

Findings

One variant of Kemeny's function is constant if mean holding times are constant.

Weighted sum of mean first passage times is constant in finite state space.

Explicit formulas for Kemeny's functions in continuous-time Markov chains and birth-death processes.

Abstract

Extensions of Kemeny's constant, as derived for irreducible finite Markov chains in discrete time, to Markov renewal processes and Markov chains in continuous time are discussed. Three alternative Kemeny's functions and their variants are considered. Typically, they lead to a constant if and only if the mean holding times between the states in the Markov renewal process are constant. However one particular variant leads to a constant, analogous to the discrete time Markov chain result. Specifically, if the state space is finite, the weighted sum of the mean first passage times (omitting the mean return time) with the stationary probabilities associated with the continuous time semi-Markov process is a constant for any Markov renewal process. Expressions for the Kemeny's functions and the relevant constants are derived for Markov renewal processes and special cases involving continuous time Markov chains and birth and death processes.