Superposition of Markov Renewal Processes.
Mark P. Becker · Deep Blue (University of Michigan) · 1981
The superposition of Markov renewal processes is investigated to derive the probability that such a pooled process is in a given state at a given time or that such a process will go next to one state given that it is now in another. In addition, the distribution of the waiting time in going from one state to another state is found for the superposed process. These results are made possible by the derivation of the joint distribution of forward and backward recurrence time in an ordinary Markov renewal process. Certain results of Cox and Smith for superposed renewal processes are extended. For the superposed Markov renewal process, the distribution of recurrence time of a given state is obtained along with the asymptotic variance of the number of visits to a given state. Application is made to a human reproduction model and a predator-prey model. A FORTRAN source program is provided for the simulation of superposed Markov renewal processes. Waiting time distributions can be exponential, chi-square, or lognormal. A goodness-of-fit statistic for Markov renewal processes, due to Brock and Kshirsagar, is included. Terminating Markov renewal processes are also examined. Various modes of termination are defined, and the distributions of the time to termination and the number of transitions until termination are obtained.