Discounted-rate utility maximization (DRUM): A framework for delay-sensitive fair resource allocation

We introduce a new optimization framework, built over a discounted-rate metric, that captures the sensitivities of wireless users to time-variations in their fairness measure of rate allocations. The resulting, so-called, Discounted-Rate Utility Maximization (DRUM) formulation not only accommodates traditional long-term and less-explored instant fairness concepts in its extremes, but also encompasses all intermediate degrees of sensitivity to fluctuations in the users' rate allocations. After introducing the versatile DRUM formulation, we fully characterize its solution in the instantly-fair and long-term-fair extremes for the general class of ω-weighted α-fair utility functions. These solutions reveal the non-trivial impact of fading channel statistics and the utility function parameters on the rate allocations, even under perfectly symmetric network conditions. In particular, we demonstrate that the rate allocations lie between the maximum and the harmonic mean of the fading-channel rates. To achieve rates in-between these extremes, we also address the general solution of DRUM by proposing a novel low-complexity dynamic rate allocation algorithm that does not require the knowledge of the channel statistics. This algorithm is proven to achieve the optimal performance of the instantly-fair and long-term-fair solutions as the discount parameter approaches its lower and upper limits, respectively. We also study the fairness and rate allocation characteristics of our algorithm for intermediate values of the discount parameter in a Rayleigh-Fading environment.