A MULTICLASS NETWORK WITH NONLINEAR, NONCONVEX, NONMONOTONIC STABILITY CONDITIONS

We consider a stochastic queueing network with fixed routes and class priorities. The vector of class sizes forms a homogeneous Markov proce ss of countable state space Z(+)(6). The network is said ''stable'' (r esp. ''unstable'') if this Markov process is ergodic (resp. transient) . The parameters are the traffic intensities of the different classes. An unusual condition of stability is obtained thanks to a new argumen t based on the characterization of the ''essential states''. The exact stability conditions are then detected thanks to an associated fluid network: we identify a zone of the parameter space in which diverging, fluid paths appear. In order to show that this is a zone of instabili ty (and that the network is stable outside this zone), we resort to th e criteria of ergodicity and transience proved by Malyshev and Menshik ov for reflected random walks in Z(+)(N) (Malyshev and Menshikov, 1981 ). Their approach allows us to neglect some ''pathological'' fluid pat hs that perturb the dynamics of the fluid model. The stability conditi ons thus determined have especially unusual characteristics: they have a quadratic part, the stability domain is not convex, and increasing all the service rates may provoke instability (Theorem 1.1 and section 7).