  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.
