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

Kemeny's Function for Markov Chains and Markov Renewal Processes