Efficient heuristics for the simulation of population overflow in series and parallel queues

Abstract

In this paper we propose state-dependent importance sampling heuristics to estimate the probability of population overflow in Markovian networks of series and parallel queues. These heuristics capture state-dependence along the boundaries (when one or more queues are empty) which is critical for the asymptotic optimality of the change of measure. The approach does not require difficult (and often intractable) mathematical analysis or costly optimization involved in adaptive importance sampling methodologies. Experimental results on tandem and parallel networks with a moderate number of nodes yield asymptotically efficient estimates (often with bounded relative error) where no other state-independent importance sampling techniques are known to be efficient. Insight drawn from simulating basic networks in this paper promises the applicability of the proposed methodology to larger networks with more general topologies.

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

“…The probability measures in Sections 3.1 and 3.2 are (generalizations of) those in Nicola and Zaburnenko [2005a, 2005b, 2006a, 2006b] for tandem and parallel networks, respectively, with any number of nodes. In Sections 3.3 and 3.4 we give heuristic state-dependent probability measures for the efficient simulation of feed-forward and feedback networks, respectively.…”

“…Recently, tandem and parallel networks have been considered in Nicola and Zaburnenko [2005a, 2005b, 2006a, 2006b and Zaburnenko and Nicola [2005] with the aim of formulating a (heuristic) state-dependent change of measure that is sufficiently close to the zero-variance change of measure. The approach does not require complicated mathematical analyses and is not limited by storage and computational requirements of adaptive importance sampling algorithms when applied to large networks (see, e.g., Ahamed et al [2006]; de Boer and ; and Kollman et al [1999]).…”

“…The proposed methodology is dubbed "state-dependent heuristic" or SDH in short. Extensive experimental results from simulations of tandem networks suggest that using the resulting change of measure yields asymptotically efficient estimates having (what empirically appears to be) a bounded relative error for almost any feasible set of network parameters (see Nicola and Zaburnenko [2005a, 2005b, 2006a, 2006b and Zaburnenko and Nicola [2005]). Only when the lowest service rates are equal (i.e., there is more than one bottleneck), the relative error seems to increase (sub-)linearly with the overflow level.…”