3. Markovian Point Processes

Renewal processes provide simple models of point processes which may describe an ordered set of points (arrival instants, service completion epochs, equipment failures, etc.) on [0, ∞). Their main simplifying feature is the independence and equidistribution of successive interrenewal intervals. We study in some detail renewal processes for which the intervals have a PH distribution. In the second part of the chapter, we describe a Markovian point process which generalizes and unifies the Markovian processes commonly used in applied probability. The results presented here illustrate anew the simplicity and the algorithmic tractability offered by the underlying Markovian structure in the method of phases. 3.1 The PH Renewal Process We consider a renewal process with a PH distribution for the interrenewal intervals. We call such a process a PH renewal process. We shall primarily deal with the continuous case, stating the parallel results for discrete time only in summary form. As with PH distributions, we may associate a Markov process with a PH renewal process.