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. Often one is interested in steady-state performance measures associated with these networks. Conceptually, under mild conditions, a regenerative structure exists in multiclass networks, making them amenable to regenerative simulation for estimating the steady-state performance measures. However, typically, identification of a regenerative structure in these networks is difficult. A well known exception is when all the interarrival times are exponentially distributed, where the instants corresponding to customer arrivals to an empty network constitute a regenerative structure. In this paper, we consider networks where 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 regenerative structure in the Markov process. We 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. We also show that amongst all such interarrival time decompositions, the one with the largest mean exponential component minimizes the asymptotic variance of the standard deviation estimator.Comment: A preliminary version of this paper will appear in Proceedings of Winter Simulation Conference, Washington, DC, 201

Regenerative simulation for multiclass open queueing networks

Conceptually, under restrictions, multiclass open queueing networks are positive Harris recurrent Markov processes, making them amenable to regenerative simulation for estimating the steady-state performance measures. However, regenerations in such networks are difficult to identify when the interarrival times are generally distributed. We assume that the interarrival times have exponential or heavier tails and show that such distributions can be decomposed into mixture of sums of independent random variables such that at least one of the components is exponentially distributed. This allows an implementable regenerative simulation for these networks. We show that the regenerative mean and standard deviation estimators are consistent and satisfy a joint central limit theorem. We also show that amongst all such interarrival decompositions, the one with largest mean exponential component minimizes the asymptotic variance of the standard deviation estimator. We also propose a regenerative simulation method that is applicable even when the interarrival times have superexponential tails

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

Diffusion parameters of flows in stable multi-class queueing networks

We consider open multi-class queueing networks with general arrival processes, general processing time sequences and Bernoulli routing. The network is assumed to be operating under an arbitrary work-conserving scheduling policy that makes the system stable. We study the variability of flows within the network. Computable expressions for quantifying flow variability have previously been discussed in the literature. However, in this paper, we shed more light on such analysis to justify the use of these expressions in the asymptotic analysis of network flows. Toward that end, we find a simple diffusion limit for the inter-class flows and establish the relation to asymptotic (co-)variance rates

Unbiased estimation of the reciprocal mean for non-negative random variables

In recent years, Monte Carlo estimators have been proposed that can estimate the ratio of two expectations without bias. We investigate the theoretical properties of a Taylor-expansion based estimator of the reciprocal mean of a non-negative random variable. We establish explicit expressions for the computational efficiency of this estimator and obtain optimal choices for its parameters. We also derive corresponding practical confidence intervals and show that they are asymptotically equivalent to the maximum likelihood (biased) ratio estimator as the simulation budget increases