On simulation output analysis for generalized semi-markov processes

The usual model for the underlying process of a discrete-event stochastic system is the generalized semi-Markov process (GSMP). A GSMP is defined in terms of a general state space Markov chain that describes the process at successive state-transition times. We provide conditions on the clock-setting distributions and state-transition probabilities of a finite state GSMP under which this underlying chain is O-irreducible and satisfies a drift criterion for stability due to Meyn and Tweedie. If the GSMP also has a single state in which exactly one event is scheduled to occur, then this state is hit infinitely often with probability 1 and the time between successive hits has finite second moment. It follows that the standard regenerative method for analysis of simulation output can be used to obtain strongly consistent point estimates and asymptotic confidence intervals for time-average limits of the process. We also show that, under our conditions, point estimates and confidence intervals for time-average limits can be obtained using methods based on standardized time series. In particular, the method of batch means is applicable. Our results rest on a new functional central limit theorem for GSMP'S together with results of Glynn and Iglehart. The standardized-time-series methods apply even when the GSMP does not have a single state or indeed any type of classical regenerative structure.