Maximizing Spatial <inline-formula> <tex-math notation="LaTeX">$\alpha$ </tex-math> </inline-formula>-Fairness in Multi-Tier Multi-Rate Spatial Aloha Networks

Abstract

We consider the maximization of $\alpha $ -fair utility in a generalized spatial Aloha network consisting of multiple tiers of transmitter–receiver (T–R) pairs each forming a Poisson bipolar process. The tiers are distinguished by transmission power and the T–R distance. Multi-rate communication between the T–R pairs is facilitated by multiple received signal-to-interference ratio thresholds. We aim to optimize the transmission probability of each tier. This results in a complex non-convex optimization problem due to intra-tier and cross-tier interference. We propose a solution termed minorize–maximization with tier separation (MMTS), through designing an iterative sequence of lower bound problems that can be decomposed into tier-separable one-dimensional convex optimization problems and solved efficiently. Specific solutions are derived for the cases $0\leq \alpha , $\alpha =1$ , and $\alpha >1$ . We show the convergence of MMTS to the objective value of a Karush–Kuhn–Tucker (KKT) point of the original problem and further identifies several conditions under which it finds the global optimum. Numerical results demonstrate the near optimality of MMTS and substantial performance advantage over existing alternatives.