Kemeny's function for Markov chains and Markov renewal processes

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, that arises under expected time to hitting of a state chosen according the continuous-time stationary distribution, leads to a constant, analogous to the discrete-time Markov chain result. Specifically, if the state space S is finite, the sum, j∈S of each mean first passage time mij (omitting the mean return time mjj when i=j) weighted by the stationary probability ϖj associated with the continuous-time semi-Markov process, is a constant, independent of i, 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-Death Processes.