We study a subset of a data network consisting of N data sources and a router. Data sources are controlled by the additive-increase/multiplicative-decrease (AIMD) mechanism which is the principal congestion-avoidance algorithm in the Internet. We propose a mathematical model based on an N-dimensional non-Markov stochastic process. We prove that after the proper space-time rescaling this stochastic process converges to some dynamical system in a bounded domain of R^N. Trajectories of the dynamical system are piecewise linear and contain random jumps from boundary points to the interior of the domain. We find an invariant measure for the limiting dynamical system in two-dimensional case N=2. The invariant measure is related with some distribution belonging to the class of infinite Bernoulli convolutions. We calculate some performance characteristics such as fairness and smoothness which are of special interest for data networks.

multidimensional stochastic models, TCP NewReno algorithm, congestion control, scalings, dynamical systems, invariant measures

Mathematical problems and theory of numerical methods