Rare event simulation of non-Markovian queueing networks using RESTART method

RESTART is an accelerated simulation technique that allows the probabilities of rare events to be evaluated. In this method, a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of a rare event of interest is higher. These regions are defined by means of a function of the system state called the importance function. An appropriate choice of the importance function is crucial for the effective application of RESTART because, although the rare event estimator is unbiased for any importance function, the acceleration achieved is closely dependent on the selected function. Formulas for obtaining suitable importance functions to estimate overflow probabilities, previously provided for Jackson networks, are extended here to non-Markovian queueing networks. This extension is made by introducing an innovative concept, the effective load of a node, defined as the actual load of a node of a Jackson network which has a similar queue length distribution. The formulas are tested in four network topologies, ranging from a two-node network with strong feedback to a 15-node network with multiple feedbacks, with different interarrival and service time distributions. The paper shows how probabilities of rare events are accurately estimated in all the tested cases with short computational time. The large variety of cases simulated suggests that the proposed importance function may be suitable for many other queueing networks.