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 equations and we explore its self-averaging properties. We also argue that in cases of heavy-tail service times the PH can be violated.
We present the main combinatorial step of our proof. That combinatorial statement deals with the rod placements on the line.