**Interaction of TCP flows as billiards**

**F. Baccelli and D. Hong**

**Abstract:**

The aim of this paper is to analyze the performance of a large number of
long lived TCP controlled flows sharing many routers (or links), from
the knowledge of the network parameters (capacity, buffer size, topology)
and of the characteristics of each TCP flow (RTT, route etc.) in the
presence of synchronization. This work is based on the AIMD model which
describes the joint evolution of the window sizes of all flows in the
congestion avoidance phase over a single bottleneck router, in terms of
iterates of random affine maps. It is shown that the generalization of
this dynamics to a network composed of several routers can be described
in terms of iterate of random piecewise affine maps, or geometrically
as a billiards in the Euclidean space with as many dimensions as the
number of flow classes and as many reflection facets as there are routers.
This can first be used as a simulation tool allowing one to emulate the
interaction of millions of flows on tens of thousands of routers. This
representation also leads to results of mathematical nature: this class of
billiards exhibits both periodic and non-periodic asymptotic oscillations
(to be interpreted as network level fluctuations for traffic aggregates),
the characteristics of which are extremely sensitive to the parameters of
the network; the consequences on TCP's fairness are exemplified on a few
typical cases of small dimension. Finally, we also show that aggregated
traffic generated by this billiards representation exhibits the same short
time scale statistical properties as those observed on real traces.

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