Interacting multi-class transmissions in large stochastic networks

TL;DR

This paper analyzes the mean-field limit of a multi-class stochastic network model with adaptive transmission rates, deriving nonlinear stochastic differential equations and studying their invariant distributions.

Contribution

It introduces a novel mean-field approach for multi-class networks with adaptive behaviors, including existence, uniqueness, and invariant distribution analysis.

Findings

Mean-field limit characterized by nonlinear stochastic differential equations

Existence and uniqueness of solutions established

Invariant distributions determined by fixed-point equations

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.