PSI2: Envelope Perfect Sampling of Non Monotone Systems

The famed perfect sampling method of Propp and Wilson uses a backward coupling scheme to compute unbiased samples of the stationary distribution of Markov chains. It has been implemented in a software tool, called PSI2, that proved very efficient for monotone chains coming from queuing networks. However, when the system includes at least one non-monotone event, the backward simulation scheme has to consider all possible states as starting points. This can be avoided by taking a new point of view that consists in bounding all possible trajectories of the Markov chain by envelopes. The new version of PSI2 presented here implements these latest improvements, including envelope techniques and splitting. Envelopes have been introduced by Bušić et al. As soon as envelopes couple, then all trajectories must have coupled, so that an unbiased sample is obtained. As for splitting, it consists in generating all the trajectories of the Markov chains inside the envelopes to run a classical backward coupling technique from that point on. Combining these two techniques in PSI2 makes it a more efficient tool that covers a wider class of queuing networks than previously. This includes networks with non-monotone events such as negative customers, arrivals by batches, forks and joins as well as Cox-distribution for services.