Markov Renewal Processes with Finitely Many States

TLDR

The paper studies Markov Renewal processes with finitely many states, introduces general Markov Renewal processes, and determines a related stationary process. The authors introduce general Markov Renewal processes and determine a related stationary process. Explicit expressions for first‑passage‑time distributions, marginal distributions of the associated semi‑Markov process, and double generating functions for N_j‑processes are derived, along with complete stationary probabilities and limiting behavior, and several illustrative examples are provided.

Abstract

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.