Section: New Results

Perfect Sampling for Closed Queueing Networks

In [4] , we investigate coupling from the past (CFTP) algorithms for closed queueing networks. The stationary distribution has a product form only in a very limited number of particular cases when queue capacity is finite, and numerical algorithms are intractable due to the cardinality of the state space. Moreover, closed networks do not exhibit any monotonic property enabling efficient CFTP. We derive a bounding chain for the CFTP algorithm for closed queueing networks. This bounding chain is based on a compact representation of sets of states that enables exact sampling from the stationary distribution without considering all initial conditions in the CFTP. The coupling time of the bounding chain is almost surely finite, and numerical experiments show that it is close to the coupling time of the exact chain.

In [18] , we present Clones, a Matlab toolbox for exact sampling from the stationary distribution of a closed queueing net-work with finite capacities. This toolbox is based on recent results using a compact representation of sets of states that enables exact sampling from the stationary distribu-tion without considering all initial conditions in the coupling from the past (CFTP) scheme. This representation reduces the complexity of the one-step transition in the CFTP al-gorithm to O(KM 2), where K is the number of queues and M the total number of customers; while the cardinality of the state space is exponential in the number of queues. In this paper, we focus on the algorithmic and implementation issues. We propose a new representation, that leads to one-step transition complexity of the CFTP algorithm that is in O(KM). We provide a detailed description of our matrix-based implementation. The toolbox can be downloaded at http://www.di.ens.fr/~rovetta/Clones .