Efficient importance sampling heuristics for the simulation of population overflow in Jackson networks

Abstract

In this article, we propose state-dependent importance sampling heuristics to estimate the probability of population overflow in Jackson queueing networks. These heuristics capture state-dependence along the boundaries (when one or more queues are empty), which is crucial for the asymptotic efficiency of the change of measure. The approach does not require difficult (and often intractable) mathematical analysis and is not limited by storage and computational requirements involved in adaptive importance sampling methodologies, particularly for a large state space. Experimental results on tandem, parallel, feed-forward, and feedback networks with a moderate number of nodes suggest that the proposed heuristics may yield asymptotically efficient estimators, possibly with bounded relative error, when applied to queueing networks wherein no other state-independent importance sampling techniques are known to be efficient. The heuristics are robust and remain effective for larger networks. Moreover, insights drawn from the basic networks considered in this article help understand sample path behavior along the boundaries, conditional on reaching the rare event of interest. This is key to the application of the methodology to networks of more general topologies. It is hoped that empirical findings and insights in this paper will encourage more research on related practical and theoretical issues.