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Lyapounov Functions for Jackson Networks

We construct explicitly Lyapounov functions for Markovian Jackson networks. Two direct corollaries are obtained: first a proof of the necessary and sufficient conditions for ergodicity, without using the famous Jackson's product form; secondly, an exponential convergence rate to the stationary distribution. We also consider small perturbations of the transition probabilities yielding thus non-Jackson networks and prove that the corresponding stationary distribution is an analytic function of these perturbations.