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 $L$ requests to transmit and is idle, it tries to access the channel at a rate proportional to $log(1+L)$. 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 behavior 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 $j\ge 1$ is of the order of $N^{a_j\left(t\right)}$, with $0\le a_j\left(t\right)<1$, where $t?a_j\left(t\right)$ is a piecewise linear function. Convergence results on the fluid time scale and a stability property are derived as a consequence of this study.