  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 $\log(1+x)$, the logarithm of its current load. A detailed fluid scaling
analysis of such a network with two nodes is presented. It is shown that the
interaction of several time scales plays an important role in the evolution of
such a system, in particular its coordinates may live on very different time
and space scales. As a consequence, the associated stochastic processes turn
out to have unusual scaling behaviors. A heavy traffic limit theorem for the
invariant distribution is also proved. Finally, we present a generalization to
the resource sharing algorithm for which the $\log$ function is replaced by an
increasing function. Possible generalizations of these results with $J>2$ nodes
or with the function $\log$ replaced by another slowly increasing function are
discussed.
