An adaptive importance sampling approach for the transient analysis of Markovian queueing networks

We present a new method for the efficient estimation of rare events and small probabilities in Markovian queueing networks. The method uses importance sampling to modify the probability distribution of the events to be observed. In contrast to most other importance sampling approaches, transient instead of steady state analysis is considered and the change of the measure is computed adaptively. The whole approach is based on the combination of discrete event simulation and randomization, a technique to transform a continuous time Markov chain into a discrete time chain and an associated Poisson process. By means of randomization it is possible to derive a simple model describing the relation between the occurrence of the rare event and the probability of different transition types. This model can be used to compute adaptively the change of transition probabilities to make the rare event less rare depending on the observed behavior in some replications and the time horizon of transient analysis. If the time horizon increases, the method is extended by introducing regeneration points in the analysis. By means of small examples it is shown that the method yields satisfactory results even for more complex queueing networks for which optimal or approximatively optimal importance sampling parameters are not known from the theory.