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The classical water filling problem is concerned with optimally assigning powers over n independent channels so as to maximize the total transmitted throughput. If each channel is associated with another mobile then it is natural to consider also the problem of fair assignment and to study tradeoffs between fairness and optimality. The object which is allocated is the transmission power, and we are interested in assigning it so as to obtain fairness between either one of three resulting performance measures: the signal to noise ratio, a shifted version of it, or the Shannon capacity. We suggest the generalized α-fairness concept. We obtain explicit solutions for and insight on the fair assignment corresponding to the various performance measures. For the case of a large number of users we consider a variational formulation of the problem. The variational formulation allows us to design distributed resource allocation …