Coupling from the past in hybrid models for file sharing peer to peer systems

International audienceIn this paper we show how file sharing peer to peer systems can be modeled by hybrid systems with a continuous part corresponding to a fluid limit of files and a discrete part corresponding to customers. Then we show that this hybrid system is amenable to perfect simulations (i.e. simulations providing samples of the system states which distributions have no bias from the asymptotic distribution of the system). An experimental study is carried to show the respective influence that the different parameters (such as time-to-live, rate of requests, connection time) play on the behavior of large peer to peer systems, and also to show the effectiveness of this approach for numerical solutions of stochastic hybrid systems

Can we use perfect simulation for non-monotonic Markovian systems ?

International audienceSimulation approaches are alternative methods to estimate the stationary be- havior of stochastic systems by providing samples distributed according to the stationary distribution, even when it is impossible to compute this distribution numerically. Propp and Wilson used a backward coupling to derive a simu- lation algorithm providing perfect sampling (i.e. which distribution is exactly stationary) of the state of discrete time finite Markov chains. Here, we adapt their algorithm by showing that, under mild assumptions, backward coupling can be used over two simulation trajectories only

Perfect Simulation and Non-monotone Markovian Systems

International audiencePerfect 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, envelopes and not feasible trajectories. 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