Interaction of TCP flows as billiards

F. Baccelli and D. Hong


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.

HTTP reference