On the Aloha Throughput-Fairness Tradeoff

A well-known inner bound of the stability region of the finite-user slotted Aloha protocol (with fixed contention probabilities) on the collision channel with n users assumes worst case service rates (all user queues non-empty). Using this inner bound as a feasible set of achievable rates, a characterization of the throughput-fairness tradeoff over this set is obtained, where the throughput is defined as the sum of the individual user rates, and two definitions of fairness are considered: the Jain-Chiu-Hawe function and the sum-user α-fair (isoelastic) utility function. This characterization is obtained using both an equality constraint and an inequality constraint on the throughput, and properties of the optimal controls, the optimal rates, and the maximum fairness as a function of the target throughput are established. A key structural property underpinning all theorems is the observation that the vector of contention probabilities that extremizes both fairness objectives has its nonzero components taking at most two distinct values.