On the existence of a proportionally fair operating point for wireless communication systems

This paper investigates the existence of a proportionally fair power allocation for an interference-coupled wireless system. The performance measure is the inverse SIR, and interference functions are assumed to be log-convex. Log-convex interference functions occur in many practical contexts (e.g. linear interference functions, or robust designs). The problem of proportional fairness can be formulated as an infimum over all possible power allocations. Whether or not this infimum exists depends on the coupling between the interference functions. In this paper we provide necessary and sufficient conditions for the existence of a proportionally fair power allocation. It turns out that this problem is fully characterized by the combinatorial structure of the interference coupling. There exist a proportionally fair solution if and only if the dependency matrix can be permuted such that it is block-irreducible and its main diagonal is positive.