Fairness in multiuser systems with polymatroid capacity region

For a wide class of multiuser systems, a subset of capacity region which includes the corner points and the sum-capacity facet has a special structure known as polymatroid. Multiple-access channels with fixed input distributions and multiple-antenna broadcast channels are examples of such systems. Any interior point of the sum-capacity facet can be achieved by time-sharing among corner points or by an alternative method known as rate-splitting. The main purpose of this paper is to find a point on the sum-capacity facet which satisfies a notion of fairness among the active users. This problem is addressed in two cases: i) where the complexity of achieving interior points is not feasible, and ii) where the complexity of achieving interior points is feasible. For the first case, the corner point for which the minimum rate of the active users is maximized is desired. A simple greedy algorithm is introduced to find such an optimum corner point. In addition, it is shown for single-antenna Gaussian multiple-access channels, the resulting corner point is leximin maximal with respect to the set of the corner points. For the second case, the properties of the unique leximin maximal rate vector with respect to the polymatroid are reviewed. It is shown that the problems of deriving the time-sharing coefficients or rate-splitting scheme to attain the leximin maximal vector can be solved by decomposing the problem into some lower dimensional subproblems. In addition, a fast algorithm to compute the time-sharing coefficients to attain a general point on the sum-capacity facet is presented.