VARIANCE REDUCTION IN SIMULATIONS OF LOSS MODELS by

We propose a new estimator of steady-state blocking probabilities for simulations of stochastic loss models that can be much more efficient than the natural estimator (ratio of losses to arrivals). The proposed estimator is a convex combination of the natural estimator and an indirect estimator based on the average number of customers in service, obtained from Little’s law (L = λW ). It exploits the known offered load (product of the arrival rate and the mean service time). The variance reduction is dramatic when the blocking probability is high and the service times are highly variable. The advantage of the combination estimator in this regime is partly due to the indirect estimator, which itself is much more efficient than the natural estimator in this regime, and partly due to strong correlation (most often negative) between the natural and indirect estimators. In general, when the variances of two component estimators are very different, the variance reduction from the optimal convex combination is about 1− ρ2, where ρ is the correlation between the component estimators. For loss models, the variances of the natural and indirect estimators are very different under both light and heavy loads. The combination estimator is effective for estimating multiple blocking probabilities in loss networks with multiple traffic classes, some of which are in normal loading while others are in light and heavy loading, because the combination estimator does at least as well as either component estimator, and provides improvement as well. Subject classifications: Simulation, efficiency: variance reduction for estimates of blocking probabilities; Queues, simulation: efficient simulation estimators for loss models; Communications: efficient simulation of loss networks Area of Review: Simulation This paper proposes a method for reducing variance in the estimation of blocking probabilities in simulations of stochastic loss models. A stochastic loss model has one or more arrival processes, modeled as stochastic processes, and has the property that not all of these arrivals are admitted. We are interested in a long-run-average or steady-state blocking probability, i.e., the long-run proportion of arrivals from one arrival process that are not admitted. The mathematical model is quite general; we assume that admitted arrivals each eventually spend some random time in service, possibly after waiting, and then depart. Otherwise, we only assume appropriate long-run averages exist; see (1)–(5) below. In particular, there are no Markov or independence assumptions; very general dependence is allowed among interarrival times and service times. The allowed model generality means that the model can be a complex loss network or resourcesharing model, perhaps with alternative routing, such as a model of a communication network; see Ross (1995). Simulations of large complex loss networks can be very time consuming, often requiring hours or more. Thus, effective variance reduction methods can be very useful. We propose an easily implemented estimator for blocking probabilities that can be remarkably efficient compared to the natural estimator (ratio of losses to arrivals). By “efficient” we mean low variance for given run length or, equivalently, short run length for given variance. The new estimator is a convex combination of the natural estimator and an indirect estimator based on the average number of customers in service, obtained from Little’s law (L = λW ). It turns out that the improvement over the natural estimator provided by the proposed method is especially dramatic when the holding times are highly variable and the blocking probability is relatively high. This is a practically important case for communication networks because, first, multiple services (e.g., voice and computer lines) lead to highly variable holding times and, second, interest in system response to failures leads to considering scenarios with relatively high blocking probabilities. Of course, the response to short-lived failures requires transient analysis, but since serious link failures in telecommunications networks, such as are caused by backhoe accidents, persist for a substantial time compared to call holding times, there is serious interest in the steadystate behavior in the presence of failures. Since continued reliable service is desired, effort is made to provide satisfactory service even in the presence of failures. Hence, simulation experiments are frequently conducted to estimate steady-state blocking probabilities under relatively heavy loads. The proposed procedure is also effective for complex loss networks with multiple traffic classes, some of which are in normal loading while others are in light and heavy loading. The new combination estimator tends to be close to the appropriate component estimator depending on the