## Section: New Results

### Scaling Methods: Interaction of TCP Flows

Participant : Philippe Robert.

This is a collaboration with Carl Graham (CMAP, École Polytechnique). Mathematical modeling of data transmission in communication networks has been the subject of intense activity for some time now. For data transmission, the Internet can be described as a very large distributed system with self-adaptive capabilities to the different congestion events that regularly occur at its numerous nodes. Various approaches have been used in this respect: control theory, ordinary differential equations, Markov processes, optimization techniques, ...

The coexistence of numerous connections in a network with a general number of nodes is
analyzed in this work. The mean-field limit of a Markovian model describing the
interaction of several classes of permanent connections in a network is analyzed.
As with TCP, each connection 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
set of non-linear stochastic differential equations 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. See Graham and
Robert [10] .

The uniqueness of the solutions of the associated fixed point equation have been
investigated in Graham *et al.* [17] . Uniqueness results of such solutions are
proved for different topologies: rings, trees and a linear network and with various
configurations for routes through nodes.