Randomized approximation scheme and perfect sampler for closed Jackson networks with multiple servers

In this paper, we propose a fully polynomial-time randomized approximation scheme (FPRAS) for the closed Jackson network. Our algorithm is based on Markov chain Monte Carlo (MCMC) method. Thus, our scheme returns an approximate solution, of which the size of error satisfies a given error rate. To our knowledge, the algorithm is the first polynomial time MCMC algorithm for closed Jackson networks with multiple servers. We propose two of new ergodic Markov chains, both of which have a unique stationary distribution that is the product form solution of closed Jackson networks. One of them is for approximate sampler, and we show it mixes rapidly. The other is for perfect sampler based on monotone coupling from the past (CFTP) algorithm proposed by Propp and Wilson, and we show it has a monotone update function.