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.