  In this paper we study the Poisson Hypothesis, which is a device to analyze
approximately the behavior of large queueing networks. We prove it in some
simple limiting cases. We show in particular that the corresponding dynamical
system, defined by the non-linear Markov process, has a line of fixed points
which are global attractors. To do this we derive the corresponding non-linear
equation and we explore its self-averaging properties. We also argue that in
cases of havy-tail service times the PH can be violated.
