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

##### S. C. Borst, N. S. Walton, A. P. Zwart

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 at each network resource, it is also natural to want each user to see the same relative benefit, so the bandwidth allocated to each route should remain proportional. We will be interested in bandwidth allocations which scale in this \textit{iso-elastic} manner and, also, maximize a utility function. Utility optimizing bandwidth allocations have been frequently studied, and a popular choice of utility function are the weighted $\alpha$-fair utility functions introduced by Mo and Walrand \cite{MoWa00}. Because weighted $\alpha$-fair utility functions possess this iso-elastic property, they are frequently used to form fluid models of bandwidth sharing networks. In this paper, we present results that show, in many settings, the only utility functions which are iso-elastic are weighted $\alpha$-fair utility functions. Thus, if bandwidth is allocated according to a network utility function which scales with relative network changes then that utility function must be a weighted $\alpha$-fair utility function, and hence, a control protocol that is robust to the future relative changes in network capacity and usage ought to allocate bandwidth inorder to maximize a weighted $\alpha$-fair utility function.

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