Interaction of TCP flows as billiards

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.) when taking synchronization into account. It is shown that in the small buffer case, the dynamics of such a network 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 class of billiards exhibits both periodic and nonperiodic asymptotic oscillations, the characteristics of which are extremely sensitive to the parameters of the network. It is also shown that for large populations and in the presence of synchronization, aggregated throughputs exhibit fluctuations that are due to the network as a whole, that follow some complex fractal patterns, and that come on top of other and more classical flow or packet level fluctuations. The consequences on TCP's fairness are exemplified on a few typical cases of small dimension.