Efficient simulation of queues in heavy traffic

Abstract

When simulating queues in heavy traffic, estimators of quantities such as average delay in queue d converge slowly to their true values. This problem is exacerbated when interarrival and service distributions are irregular. For the GI/G/ 1 queue, delay moments can be expressed in terms of moments of idle period I . Instead of estimating d directly by a standard regenerative estimator that we call DD, a method we call DI estimates d from estimated moments of I . DI was investigated some time ago and shown to be much more efficient than DD in heavy traffic. We measure efficiency as the factor by which variance is reduced. For the GI/G/ 1 queue, we show how to generate a sequence of realized values of the equilibrium idle period, I e , that are not independent and identically distributed, but have the correct statistical properties in the long run. We show how to use this sequence to construct a new estimator of d , called DE, and of higher moments of delay as well. When arrivals are irregular, we show that DE is more efficient than DI, in some cases by a large factor, independent of the traffic intensity . Comparing DE with DD, these factors multiply . For GI/G/c , we construct a control-variates estimator of average delay in queue d c that is efficient in heavy traffic. It uses DE to estimate the average delay for the corresponding fast single server. We compare the efficiency of this method with another method in the literature. For M/G/c , we use insensitivity to construct another control-variates estimator of d c . We compare the efficiency of this estimator with the two c-server estimators above.