## 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.
The coexistence of numerous connections in a network with a general number of nodes has
been analyzed in a previous work through a mean-field limit of a Markovian model
describing the interaction of several classes of *permanent connections*.

In [19] , this line of work has been generalized to the case when connections are not permanent but can be either active (ON) when it is transmitting data along its route, or idle (OFF). A Markovian model is provided by the states (OFF, or ON with some transmission rate) of the connections. Each connection is assumed to have a self-adaptive behavior in that its transmission rate along its route depends on the level of congestion of the nodes of the route. The number of connections in each class being potentially huge, a mean-field limit result is proved with an appropriate scaling so as to reduce the dimensionality. In the limit, the evolution of the states of the connections can be represented by a non-linear system of stochastic differential equations, of dimension the number of classes. Additionally, it is shown that the corresponding stationary distribution can be expressed by the solution of a fixed-point equation of finite dimension involving the resolvent of a Markov process describing the evolution of an isolated connection.