Poisson Hypothesis for information networks (A study in non-linear Markov processes)

TL;DR

This paper proves the Poisson Hypothesis for large queueing systems modeled by non-linear Markov processes, showing their dynamics converge to fixed points and exploring self-averaging properties through integral equations.

Contribution

It introduces a proof of the Poisson Hypothesis for mean-field queueing systems using non-linear integral equations and combinatorial analysis.

Findings

Large queueing systems exhibit convergence to fixed points.

The dynamical systems have a line of global attractors.

Self-averaging properties are confirmed through integral equations.

Abstract

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.