Connections Between Spectral Properties Of Asymptotic Mappings And Solutions To Wireless Network Problems

We establish connections between asymptotic functions and properties of solutions to problems in wireless networks. We start by introducing self-mappings (called asymptotic mappings) constructed with asymptotic functions, and we show that their spectral properties explain the behavior of solutions to max-min utility optimization problems. For example, in some max-min utility power control problems, we prove that the optimal utility as a function of the power available to transmitters is approximately linear in the low power regime. However, as we move away from this regime, there exists a transition point, easily computed from the spectral radius of an asymptotic mapping, from which gains in utility become increasingly marginal. From these results we derive analogous properties of the transmit energy efficiency. In this study we also generalize and unify existing approaches for feasibility analysis in wireless networks. Feasibility problems often reduce to determining the existence of the fixed point of a standard interference mapping, and we show that the spectral radius of an asymptotic mapping provides a necessary and sufficient condition for the existence of such a fixed point. We further present a result that determines whether the fixed point satisfies a constraint given in terms of a monotone norm.