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Using Polymatroid Structures to Provide Fairness in Multiuser Systems

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. 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 active users. In one case, the corner point for which the minimum rate of the active users is maximized (max-min corner point) is computed for signaling. In another case, the polymatroid properties are exploited to locate a rate-vector on the sum-capacity facet which is optimally fair in the sense that the minimum rate among all users is maximized (max-min rate). It is shown that the problems of deriving the time-sharing coefficients or rate-splitting scheme can be solved by decomposing the problem to 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 proposed