# Accession Number:

## AD0698295

# Title:

## SOME USEFUL GENERALIZATIONS IN MARKOV RENEWAL PROCESSES.

# Descriptive Note:

## Technical rept.,

# Corporate Author:

## SOUTHERN METHODIST UNIV DALLAS TEX DEPT OF STATISTICS

# Personal Author(s):

# Report Date:

## 1969-11-04

# Pagination or Media Count:

## 89.0

# Abstract:

Consider a particular system which at time t may be in one of a finite number of distinguishable states, labeled for convenience by 1, 2, ..., m. Once the system enters a particular state, say i, it instantaneously selects the next state to be visited, say j, with probability P sub ij. However, transition to state j occurs after holding in state i for a random time sojourn time whose distribution function is F sub ij.. These processes are known as Semi-Markov Process and the associated renewal process is called a Markov Renewal Process. In this paper a new counting process is introduced which at time t counts the number of times the system has made a one-step transition from state i to state j, i, j 1, 2, ..., m. The matrix Nt denotes these counts. The distribution and moments of Nt are derived and cumulative processes associated with Nt are discussed. The results are extended to arbitrary intervals of the form t sub 0, t sub 0 t. Certain limiting results are given and some important special cases discussed. Several open problems are given in the summary chapter. Author

# Descriptors:

# Subject Categories:

- Statistics and Probability
- Operations Research