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 $(N t , t∈(0, 1))$. 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 $N^{a_j(t)}$ , with $0≤a_j(t)<1$, where $t →a_j(t)$ is a piecewise linear function. Convergence results on the fluid time scale and a stability property are derived as a consequence of this study.