## Section: New Results

### Stochastic Networks: Jackson Networks

Participant : Danielle Tibi.

Lyapunov functions and essential spectral radius of Jackson networks, joint work with I. Ignatiouk-Robert (University of Cergy-Pontoise). A family of explicit multiplicative Lyapunov functions is constructed for any stable Jackson network. Optimizing the multiplicative factor over this family provides an upper bound for the essential spectral radius of the associated Markov process. For some particular classes of Jackson networks, this upper bound coincides with a lower bound derived from large deviations arguments, thus providing the exact value of the essential spectral radius. The main example is given by Jackson networks with routing matrix having a tree structure (in the sense that for any node i, at most one other node can route its customers to i). The result also holds for other types of routing matrices (e.g. completely symmetrical), under some conditions over the different arrival and service rates.