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A family of explicit Lyapunov function for positive recurrent MarkovianJackson network is constructed. With this result we obtain explicit estimatesof the tail distribution of the first time, when the process returns to largecompact sets, and some explicit estimates of the essential spectral radius ofthe process. The essential spectral radius of the process provides the bestgeometric convergence rate to equilibrium that one can get by changing thetransitions of the process in a finite set.

Explicit Lyapunov functions and estimates of the essential spectral radius for Jackson networks