Insensitive, maximum stable allocations converge to proportional fairness

TL;DR

This paper analyzes a queueing model with insensitive, stable allocation policies, showing that in congestion limits, the service allocation converges to a proportional fairness solution.

Contribution

It proves that maximum stable, insensitive allocations in congested limits converge to the proportional fairness optimization.

Findings

Limit allocations are proportional fair solutions.

Insensitive, stable policies achieve maximal stability.

Convergence occurs in highly congested regimes.

Abstract

We describe a queueing model where service is allocated as a function of queue sizes. We consider allocations policies that are insensitive to service requirements and have a maximal stability region. We take a limit where the queueing model become congested. We study how service is allocated under this limit. We demonstrates that the only possible limit allocation is one that maximizes a proportionally fair optimization problem.