On the Aloha throughput-fairness tradeoff

TL;DR

This paper analyzes the tradeoff between throughput and fairness in the slotted Aloha protocol, providing a characterization of optimal rates and control strategies under different fairness and throughput constraints.

Contribution

It introduces a new analysis of the throughput-fairness tradeoff in Aloha, using an inner bound of the stability region and exploring optimal control properties.

Findings

Optimal contention probabilities have at most two non-zero values.

Characterization of the throughput-fairness tradeoff under different constraints.

Properties of optimal controls and rates as functions of target throughput.

Abstract

A well-known inner bound of the stability region of the slotted Aloha protocol 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 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 alpha-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 fairness as a function of the target throughput are established. A key fact used in all theorems is the observation that all contention probability vectors that extremize the fairness functions take at most two non-zero values.