Efficient heuristics for simulating rare events in queuing networks

In this thesis we propose state-dependent importance sampling heuristics to estimate the probability of population overflow in queuing networks. These heuristics capture state-dependence along the boundaries (when one or more queues are almost empty) which is crucial for the asymptotic efficiency 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, parallel, feed-forward, and a 2-node feedback Jackson queuing networks as well as a 2-node tandem non-Markovian network suggest that the proposed heuristics yield asymptotically efficient estimators, sometimes with bounded relative error. For these queuing networks no state-independent importance sampling techniques are known to be efficient

State-dependent Importance Sampling for a Slow-down Tandem Queue

In this paper we investigate an advanced variant of the classical (Jackson) tandem queue, viz. a two-node system with server slow-down. The slow-down mechanism has the primary objective to protect the downstream queue from frequent overflows, and it does so by reducing the service speed of the upstream queue as soon as the number of jobs in the downstream queue reaches some pre-specified threshold. To assess the efficacy of such a policy, techniques are needed for evaluating overflow metrics of the second queue. We focus on the estimation of the probability of the following rare event: overflow in the downstream queue before exhausting the system, starting from any given state in the state space.\ud Due to the rarity of the event under consideration, naive, direct Monte Carlo simulation is often infeasible. We therefore rely on the application of importance sampling to obtain variance reduction. The principal contribution of this paper is that we construct an importance sampling scheme that is asymptotically efficient. In more detail, the paper addresses the following issues. (i) We rely on powerful heuristics to identify the exponential decay rate of the probability under consideration, and verify this result by applying sample-path large deviations techniques. (2) Immediately from these heuristics, we develop a proposal for a change of measure to be used in importance sampling. (3) We prove that the resulting algorithm is asymptotically efficient, which effectively means that the number of runs required to obtain an estimate with fixed precision grows subexponentially in the buffer size. We stress that our method to prove asymptotic efficiency is substantially shorter and more straightforward than those usually provided in the literature. Also our setting is more general than the situations analyzed so far, as we allow the process to start off at any state of the state space, and in addition we do not impose any conditions on the values of the arrival rate and service rates, as long as the underlying queueing system is stable

Importance Sampling Simulation of Population Overflow in Two-node Tandem Networks

In this paper we consider the application of importance sampling in simulations of Markovian tandem networks in order to estimate the probability of rare events, such as network population overflow. We propose a heuristic methodology to obtain a good approximation to the 'optimal' state-dependent change of measure (importance sampling distribution). Extensive experimental results on 2-node tandem networks are very encouraging, yielding asymptotically efficient estimates (with bounded relative error) where no other state-independent importance sampling techniques are known to be efficient The methodology avoids the costly optimization involved in other recently proposed approaches to approximate the 'optimal' state-dependent change of measure. Moreover, the insight drawn from the heuristic promises its applicability to larger networks and more general topologies

Analysis of State-Independent Importance-Sampling Measures for the Two-Node Tandem Queue

We investigate the simulation of overflow of the total population of a Markovian two-node tandem queue model during a busy cycle, using importance sampling with a state-independent change of measure. We show that the only such change of measure that may possibly result in asymptotically efficient simulation for large overflow levels is exchanging the arrival rate with the smallest service rate. For this change of measure, we classify the model's parameter space into regions of asymptotic efficiency, exponential growth of the relative error, and infinite variance, using both analytical and numerical techniques