Convex Structure of the Sum Rate on the Boundary of the Feasible Set for Coexisting Radios

The power allocation that maximizes the sum rate of transceivers operating in the same frequency band is a difficult non-convex problem. In our earlier work, we proved that for transceivers operating under a total power constraint, the maximum sum rate occurs at the boundary of the feasible set formed by the hyper plane representing the power constraint. This finding is nontrivial considering that we are dealing with an interference limited system. In this paper, we study the convex structure of the sum rate on the boundary of the power constraint hyper plane. For two transceivers, we prove that the sum rate is always convex on the line created by the power constraint equality. In the case of three transceivers, we identify a region in the middle of the plane created by the power constraint equality, where the sum rate is concave. This is significant because it is in the middle of the boundary plane that the power allocation can be expected to be fair to all three users. We also provide a power allocation protocol and an algorithm that distribute the power among the transceivers with fairness. Simulation results are provided to support the theorems proven in the paper as well as to demonstrate the convergence of the algorithm to the global maximum sum rate. Results of the algorithm are compared with solutions based on Game theory.