Large deviations of the stationary measure of networks under proportional fair allocations

TL;DR

This paper proves that in bandwidth-sharing networks under proportional fair allocation, the stationary measure exhibits large deviations properties similar to reversible allocations, supporting the idea of fairness insensitivity.

Contribution

It establishes large deviations results for proportional fair allocations in Markovian and monotone networks, using Lyapunov functions and stochastic comparisons.

Findings

Proportional fair and reversible allocations are geometrically ergodic.

Both allocations share the same large deviations characteristics.

Results support the insensitivity of proportional fairness to service time requirements.

Abstract

We address a conjecture introduced by Massouli\'e (2007), concerning the large deviations of the stationary measure of bandwidth-sharing networks functioning under the Proportional fair allocation. For Markovian networks, we prove that Proportional fair and an associated reversible allocation are geometrically ergodic and have the same large deviations characteristics using Lyapunov functions and martingale arguments. For monotone networks, we give a more direct proof of the same result relying on stochastic comparisons that hold for general service requirement distribution. These results comfort the intuition that Proportional fairness is 'close' to allocations of service being insensitive to the service time requirement.