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 $(N^t,\, t{\in}(0,1))$. The main result is that, on this time scale and under appropriate conditions, the state of a node with index $j\geq 1$ is of the order of $N^{a_j(t)}$, with $0{\leq}a_j(t){<}1$, where $t\mapsto a_j(t)$ is a piecewise linear function. Convergence results on the fluid time scale and a stability property are derived as a consequence of this study.