  In this paper we prove the Poisson Hypothesis for the limiting behavior of
the large queueing systems in some simple ("mean-field") cases. We show in
particular that the corresponding dynamical systems, defined by the non-linear
Markov processes, have a line of fixed points which are global attractors. To
do this we derive the corresponding non-linear integral equation and we explore
its self-averaging properties. Our derivation relies on a solution of a
combinatorial problem of rode placements.
