Using polymatroid structures to provide fairness in multiuser systems

For a wide class of multi-user systems, a subset of capacity region which includes the corner points and the sumcapacity 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 (maxmin 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-spitting 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.