Perfect sampling with aggregated envelopes

Perfect sampling is a technique that uses coupling arguments to provide a sample from the stationary distribution of a Markov chain. This technique is efficient if the transition function of the Markov chain is monotone. In the non-monotone case, one can construct bounding chains that can detect whether the initial chain has coupled. For instance, if the state space is a lattice, then one such bounding chain can be defined by taking the smallest interval that contains the image of the one step transition function. Here we propose to combine the ideas of bounding processes and the aggregation of Markov chains. We illustrate the proposed approach of aggregated envelope bounding chains on the service tools model proposed by Vliegen and Van Houtum (2009). For this model, the aggregated envelope method allows to reduce exponentially the dimension of the state space and allows effective perfect sampling algorithms. Under some conditions on the transition rates (high service case), the running time of our algorithm is linear in terms of the total capacity in the system.