A Scaling Analysis of a Star Network with Logarithmic Weights

The paper investigates the properties of a class of resource allocation algorithms for communication networks: if a node of this network has x requests to transmit, then it receives a fraction of the capacity proportional to log(1+L), the logarithm of its current load L. A stochastic model of such an algorithm is investigated in the case of the star network, in which J nodes can transmit simultaneously, but interfere with a central node 0 in such a way that node 0 cannot transmit while one of the other nodes does. One studies the impact of the log policy on these J + 1 interacting communication nodes. A fluid scaling analysis of the network is derived with the scaling parameter N being the norm of the initial state. It is shown that the asymptotic fluid behaviour of the system is a consequence of the evolution of the state of the network on a specific time scale (Nt, t∈(0, 1)). The main result is that, on this time scale and under appropriate conditions, the state of a node with index j ≥ 1 is of the order of Nj, with 0≤aj(t)<1, where t 7→ aj(t) is a piecewise linear function. Convergence results on the fluid time scale and a stability property are derived as a consequence of this study.