March  2010, 5(1): 31-62. doi: 10.3934/nhm.2010.5.31

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

1. 

Institut de Mathématiques de Toulouse, CNRS & Université de Toulouse, 31062 Toulouse cedex 9, France

2. 

Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, Ontario K1N 6M8, Canada

3. 

Microsoft Reasearch, Roger Needham Building, 7 J J Thomson Avenue, Cambridge, CB3 0FB, United Kingdom

Received  February 2009 Revised  November 2009 Published  February 2010

We study an interacting particle system whose dynamics depends on an interacting random environment. As the number of particles grows large, the transition rate of the particles slows down (perhaps because they share a common resource of fixed capacity). The transition rate of a particle is determined by its state, by the empirical distribution of all the particles and by a rapidly varying environment. The transitions of the environment are determined by the empirical distribution of the particles. We prove the propagation of chaos on the path space of the particles and establish that the limiting trajectory of the empirical measure of the states of the particles satisfies a deterministic differential equation. This deterministic differential equation involves the time averages 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.
Citation: Charles Bordenave, David R. McDonald, Alexandre Proutière. A particle system in interaction with a rapidly varying environment: Mean field limits and applications. Networks & Heterogeneous Media, 2010, 5 (1) : 31-62. doi: 10.3934/nhm.2010.5.31
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