A particle system in interaction with a rapidly varying environment: Mean field limits and applications

Abstract

We study an interacting particle system whose dynamics depends on aninteracting random environment. As the number of particles growslarge, the transition rate of the particles slows down (perhapsbecause they share a common resource of fixed capacity). Thetransition rate of a particle is determined by its state, by theempirical distribution of all the particles and by a rapidly varyingenvironment. The transitions of the environment are determined bythe empirical distribution of the particles. We prove thepropagation of chaos on the path space of the particles andestablish that the limiting trajectory of the empirical measure ofthe states of the particles satisfies a deterministic differentialequation. This deterministic differential equation involves the timeaverages of the environment process.   We apply the results on particle systems to understand the behavior of computer networks where users access a shared resource using a distributed random Medium Access Control (MAC) algorithm. MAC algorithms are used in all Local Area Network (LAN), and have been notoriously difficult to analyze. Our analysis allows us to provide simple and explicit expressions of the network performance under such algorithms.