Interacting Multi-Class Transmissions in Large Stochastic Networks

Carl Graham (1) , Philippe Robert (2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 RAP - Networks, Algorithms and Probabilities

The mean-field limit of a Markovian model describing the interaction of several classes of permanent connections in a network is analyzed. In the same way as for the TCP algorithm, 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 non-linear 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.

Mots clés: Chaoticity; Non-Linear Stochastic Differential Equations; TCP Algorithm; Equilibrium Distributions; Multi-Class Systems; Mean-Field Interaction; Coupling

Domaines: Probabilités [math.PR]

Carl Graham, Philippe Robert. Interacting Multi-Class Transmissions in Large Stochastic Networks. The Annals of Applied Probability, 2009, 19 (6), pp.2334-2361. ⟨10.1214/09-AAP614⟩. ⟨inria-00326156⟩