Max-Min Fairness Rate Control in Wireless Networks: Optimality and Algorithms by Perron-Frobenius Theory

Rate adaptation and power control are two key resource allocation mechanisms in multiuser wireless networks. In the presence of interference, how do we jointly optimize end-to-end source rates and link powers to achieve weighted max-min rate fairness for all sources in the network? This optimization problem is hard to solve as physical layer link rate functions are nonlinear, nonconvex, and coupled in the transmit powers. We show that the weighted max-min rate fairness problem can, in fact, be decoupled into separate fairness problems for flow rate and power control. For a large class of physical layer link rate functions, we characterize the optimal solution analytically by a nonlinear Perron-Frobenius theory through solving a conditional eigenvalue problem that captures the interaction of multiuser interference. We propose an iterative algorithm to compute the optimal flow rate that converges geometrically fast without any parameter configuration. Numerical results demonstrate that our iterative algorithm is computationally fast for the Shannon capacity, CDMA, and piecewise link rate functions.