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.

Optimal Sampling of Overflow Paths in Jackson Networks

“…The fact that κ (N) is the expected number of proposals to obtain an acceptance is a basic fact of acceptance / rejection algorithms (see [3]). The strong efficiency of Z N also follows immediately from inequality (23). The fact that κ (N) = O (1) follows from Proposition 1.…”

“…There are very few algorithms in the rare event simulation literature for stochastic networks that exhibit this type of optimal performance. Efficient rare-event simulation methodology for Jackson networks has been the focus of many papers in the literature in recent years (see [24], [2], [29], [16], [21], [19], [13], [23] [14], [12], [11]; also the survey papers of [20] and [6] for additional references). Most of the development in the literature concentrates on the design and analysis of estimators that are shown to be weakly efficient (for efficiency concepts in rare event simulation see [3], [7]).…”

“…The forward sampling is the one for which a subsolution to the Isaacs equation is required in order to control the behavior of the likelihood ratio precisely on those rogue sample paths. The backward sampling idea has been explored empirically by [23], but without a precise guidance on the selection of the initial distribution in the overflow set and without a complexity analysis of the associated running time. Not surprisingly, given the theory presented here, the empirical performance observed in [23] is excellent whenever the initial distribution imposed in the overflow set happens to be appropriately selected.…”

Control of a tandem queue with a startup cost for the second server

“…Queueing systems in which the departures from one server become the arrivals to a downstream server are often modeled as tandem queues. These models have proven to be very challenging to analyze, [1][2][3][4] and despite decades of research, there are still many open problems without an analytic solution. [5][6][7][8] At the same time, tandem queues abound in applications, and therefore, the study of these systems is also important for practice.…”

Analysis of a Splitting Estimator for Rare Event Probabilities in Jackson Networks

“…A couple of early references on the subject are [20] and [1]. Subsequent work which has also been very influential in the development of efficient algorithms for overflows of Jackson networks include [22], [12], [13], [16], [14], [9], [19], [11] and [8]. The survey papers of [15] and [7] provide additional references on this topic.…”