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December 2009 Interacting multi-class transmissions in large stochastic networks
Carl Graham, Philippe Robert
Ann. Appl. Probab. 19(6): 2334-2361 (December 2009). DOI: 10.1214/09-AAP614

Abstract

The mean-field limit of a Markovian model describing the interaction of several classes of permanent connections in a network is analyzed. Each of the connections has a self-adaptive behavior in that its transmission rate along its route depends on the level of congestion of the nodes of the route. Since several classes of connections going through the nodes of the network are considered, an original mean-field result in a multi-class context is established. It is shown that, as the number of connections goes to infinity, the behavior of the different classes of connections can be represented by the solution of an unusual nonlinear stochastic differential equation depending not only on the sample paths of the process, but also on its distribution. Existence and uniqueness results for the solutions of these equations are derived. Properties of their invariant distributions are investigated and it is shown that, under some natural assumptions, they are determined by the solutions of a fixed-point equation in a finite-dimensional space.

Citation

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Carl Graham. Philippe Robert. "Interacting multi-class transmissions in large stochastic networks." Ann. Appl. Probab. 19 (6) 2334 - 2361, December 2009. https://doi.org/10.1214/09-AAP614

Information

Published: December 2009
First available in Project Euclid: 25 November 2009

zbMATH: 1179.60067
MathSciNet: MR2588247
Digital Object Identifier: 10.1214/09-AAP614

Subjects:
Primary: 60K25, 60K35, 90B18

Rights: Copyright © 2009 Institute of Mathematical Statistics

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Vol.19 • No. 6 • December 2009
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