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In this paper we investigate an advanced variant of the classical (Jackson) tandem queue, viz. a two-node system with server slowdown. By this mechanism, the service speed of the upstream queue is reduced as soon as the number of jobs in the downstream queue reaches some pre-specified threshold. We focus on the estimation of the probability of overflow in the downstream queue before the system becomes empty, starting from any given state in the state space. The principal contribution of this paper is that we construct importance sampling schemes to estimate these probabilities in case they are small; in particular: (1) We use 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) Based on these heuristics we develop a change of measure to be used in importance sampling. (3) We prove that this scheme is asymptotically efficient, using a shorter and more straightforward method than usually provided in the literature. Unfortunately, this scheme is difficult to use in practice, therefore (4) we propose an algorithm that offers considerable computational advantage over the first scheme. For this scheme we provide a proof of asymptotic efficiency for certain parameter settings, as well as numerical results showing that the scheme works well for all parameters.

This article focuses on estimating rare events using multilevel splitting schemes. The event of interest is that a Markov process enters some rare set before another (“tabu”) set. It is known that in this setting a large deviations analysis is not always sufficient for constructing asymptotically efficient importance sampling schemes; additional modifications to the change of measure suggested by large deviations are needed. As an alternative, we design an asymptotically efficient multilevel splitting scheme that relies on the large deviations analysis only. This property makes it more flexible and easier to implement than corresponding importance sampling schemes.

This paper considers importance sampling as a tool for rare-event simulation. The focus is on estimating the probability of overflow in the downstream queue of a Jackson two-node tandem queue. It is known that in this setting 'traditional' state-independent importance-sampling distributions perform poorly. We therefore concentrate on developing a state-dependent change of measure that is provably asymptotically efficient. More specific contributions are the following. (i) We concentrate on the probability of the second queue exceeding a certain predefined threshold before the system empties. Importantly, we identify an asymptotically efficient importance-sampling distribution for any initial state of the system. (ii) The choice of the importance-sampling distribution is backed up by appealing heuristics that are rooted in large-deviations theory. (iii) Our method for proving asymptotic efficiency is substantially more straightforward than some that have been used earlier. Keywords: Rare event simulation, importance sampling, state-dependent change of measure, asymptotic optimality, tandem queue

This article considers importance sampling as a tool for rare-event simulation. The focus is on estimating the probability of overflow in the downstream queue of a Jacksonian two-node tandem queue; it is known that in this setting “traditional” state-independent importance-sampling distributions perform poorly. We therefore concentrate on developing a state-dependent change of measure, that we prove to be asymptotically efficient. More specific contributions are the following. (i) We concentrate on the probability of the second queue exceeding a certain predefined threshold before the system empties. Importantly, we identify an asymptotically efficient importance-sampling distribution for any initial state of the system. (ii) The choice of the importance-sampling distribution is backed up by appealing heuristics that are rooted in large-deviations theory. (iii) The method for proving asymptotic efficiency relies on probabilistic arguments only. The article is concluded by simulation experiments that show a considerable speedup.