Regenerative Simulation for Queueing Networks with Exponential or Heavier Tail Arrival Distributions

Multiclass open queueing networks find wide applications in communication, computer, and fabrication networks. Steady-state performance measures associated with these networks is often a topic of interset. Conceptually, under mild conditions, a sequence of regeneration times exists in multiclass networks, making them amenable to regenerative simulation for estimating steady-state performance measures. However, typically, identification of such a sequence in these networks is difficult. A well-known exception is when all interarrival times are exponentially distributed, where the instants corresponding to customer arrivals to an empty network constitute a sequence of regeneration times. In this article, we consider networks in which the interarrival times are generally distributed but have exponential or heavier tails. We show that these distributions can be decomposed into a mixture of sums of independent random variables such that at least one of the components is exponentially distributed. This allows an easily implementable embedded sequence of regeneration times in the underlying Markov process. We show that among all such interarrival time decompositions, the one with an exponential component that has the largest mean minimizes the asymptotic variance of the standard deviation estimator. We also show that under mild conditions on the network primitives, the regenerative mean and standard deviation estimators are consistent and satisfy a joint central limit theorem useful for constructing asymptotically valid confidence intervals.