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.Comment: Published at http://dx.doi.org/10.1214/14-AAP1057 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org