Network iso-elasticity and weighted alpha-fairness

When a communication network's capacity increases, it is natural to want the bandwidth allocated to increase to exploit this capacity. But, if the same relative capacity increase occurs throughout the network, it is also natural to want each user to see the same relative benefit, so the bandwidth allocated remains proportional. We will be interested in bandwidth allocations which scale in this iso-elastic manner and, also, maximize a utility function. In this paper, we present results that show, in many settings, the only iso-elastic utility functions are weighted α-fair utility functions, which have gained wide popularity in the networking literature. Hence, a control protocol that is robust to the relative increases in network capacity and usage ought to allocate bandwidth in order to maximize a weighted α-fair utility function.