## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 32, Number 4 (1961), 1243-1259.

### Markov Renewal Processes with Finitely Many States

#### 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.

#### Article information

**Source**

Ann. Math. Statist. Volume 32, Number 4 (1961), 1243-1259.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aoms/1177704864

**Digital Object Identifier**

doi:10.1214/aoms/1177704864

**Mathematical Reviews number (MathSciNet)**

MR154324

**Zentralblatt MATH identifier**

0201.49901

**JSTOR**

links.jstor.org

#### Citation

Pyke, Ronald. Markov Renewal Processes with Finitely Many States. Ann. Math. Statist. 32 (1961), no. 4, 1243--1259. doi:10.1214/aoms/1177704864. http://projecteuclid.org/euclid.aoms/1177704864.