Perfect simulation and non-monotone Markovian systems

Perfect simulation, or coupling from the past, is an efficient technique for sampling the steady state of monotone discrete time Markov chains. Indeed, one only needs to consider two trajectories corresponding to minimal and maximal state in the system. We show here that even for non-monotone systems one only needs to compute two trajectories: an infimum and supremum envelope. Since the sequence of states obtained by taking infimum (resp. supremum) at each time step does not correspond to a feasible trajectory of the system, the envelopes might not couple or the coupling time might be larger. We show that the envelope approach is efficient for some classes of non-monotone queuing networks, such as networks of queues with batch arrivals, queues with fork and join nodes and/or with negative customers.