# Passage times in colored stochastic petri nets

## Abstract

Passage times in colored stochastic Petri nets correspond to delays in discrete-event stochastic systems. Formal definition of a sequence of passage times in a colored stochastic Petri net is in terms of the underlying general state space Markov chain of the marking process. Using symmetry of the net with respect to color, we provide conditions under which a sequence of passage times is a regenerative process in discrete time with finite cycle-length moments. The regenerative property implies time-average convergence, convergence in distribution, and a central limit theorem for sequences of passage times. It follows that strongly consistent point estimates and asymptotic confidence intervals for general characteristics of passage times can be obtained by simulating a finite portion of a single sample path of the underlying general state space Markov chain. Using regenerative structure of the marking process and a version of Little’s Law given by Glynn and Whitt, we also obtain conditions under which a sequence of passage times converges in a time-average sense and the limit can be expressed as a ratio of expected values. The resulting estimation procedure for the limiting average passage time requires no measurement of individual passage times and is valid even if there are no regeneration points for the sequence of passage times. In practice, the most difficult part of the argument required to establish the applicability of regenerative methods for simulation of passage times in colored stochastic Petri nets is to show that certain fixed sets of markings are recurrent. We show that representations for certain conditional distributions of clock readings can be combined with a “geometric trials” criterion to establish recurrence. © 1993, Taylor & Francis Group, LLC. All rights reserved.