In this paper, Markov Renewal processes having a finite number of states are studied. Explicit expressions are derived for the distribution functions of first passage times, as well as for the marginal distribution function of the corresponding Semi-Markov process. Double generating functions are obtained for the distribution functions of the $N_j$-processes. The limiting behavior of a Markov Renewal process is discussed, the stationary probabilities being derived completely. General Markov Renewal processes are introduced, and a related stationary process is determined. Several examples are given.
"Markov Renewal Processes with Finitely Many States." Ann. Math. Statist. 32 (4) 1243 - 1259, December, 1961. https://doi.org/10.1214/aoms/1177704864