This paper contains the definition of and some preliminary results on Markov Renewal processes and Semi-Markov processes. The close relationship between these two types of processes is described. The concept of regularity is introduced and characterized. A classification of the states of a Markov Renewal process is described and studied.
"Markov Renewal Processes: Definitions and Preliminary Properties." Ann. Math. Statist. 32 (4) 1231 - 1242, December, 1961. https://doi.org/10.1214/aoms/1177704863