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When a communication network's capacity increases, it is natural to want thebandwidth allocated to increase to exploit this capacity. But, if the samerelative capacity increase occurs at each network resource, it is also naturalto want each user to see the same relative benefit, so the bandwidth allocatedto each route should remain proportional. We will be interested in bandwidthallocations which scale in this \textit{iso-elastic} manner and, also, maximizea utility function.  Utility optimizing bandwidth allocations have been frequently studied, and apopular choice of utility function are the weighted $\alpha$-fair utilityfunctions introduced by Mo and Walrand \cite{MoWa00}. Because weighted$\alpha$-fair utility functions possess this iso-elastic property, they arefrequently used to form fluid models of bandwidth sharing networks. In thispaper, we present results that show, in many settings, the only utilityfunctions which are iso-elastic are weighted $\alpha$-fair utility functions.  Thus, if bandwidth is allocated according to a network utility function whichscales with relative network changes then that utility function must be aweighted $\alpha$-fair utility function, and hence, a control protocol that isrobust to the future relative changes in network capacity and usage ought toallocate bandwidth inorder to maximize a weighted $\alpha$-fair utilityfunction.

Network iso-elasticity and weighted $\alpha$-fairness