Simulation run lengths to estimate blocking probabilities

Abstract

We derive formulas approximating the asymptotic variance of four estimators for steady-state blocking probability in a multiserver loss system, exploiting diffusion process limits. These formulas can be used to predict simulation run lengths required to obtain desired statistical precision before the simulation has been run, which can aid in the design of simulation experiments. They also indicate that one estimator can be much better than another, depending on the loading. An indirect estimator based on estimating the mean occupancy is significantly more (less) efficient than a direct estimator for heavy (light) loads. A major concern is the way computational effort scales with system size. For all the estimators, the asymptotic variance tends to be inversely proportional to the system size, so that the computational effort (regarded as proportional to the product of the asymptotic variance and the arrival rate) does not grow as system size increases. Indeed, holding the blocking probability fixed, the computational effort with a good estimator decreases to zero as the system size increases. The asymptotic variance formulas also reveal the impact of the arrival-process and service-time variability on the statistical precision. We validate these formulas by comparing them to exact numerical results for the special case of the classical Erlang M/M/s/0 model and simulation estimates for more general G/GI/s/0 models. It is natural to delete an initial portion of the simulation run to allow the system to approach steady state when it starts out empty. For small to moderately size systems, the time to approach steady state tends to be negligible compared to the time required to obtain good estimates in steady state. However, as the system size increases, the time to approach steady state remains approximately unchanged, or even increases slightly, so that the computational effort associated with letting the system approach steady state becomes a greater portion of the overall computational effort as system size increases.