Proportional fairness and its relationship with multi-class queueing networks

TL;DR

This paper establishes a theoretical connection between multi-class queueing networks, stochastic flow models, and proportional fairness, proving convergence of throughput to fair allocations and deriving a large deviation principle.

Contribution

It introduces a new limit theorem linking queueing networks to flow models and proves that stationary throughput converges to a proportional fairness solution.

Findings

Weak convergence of queueing networks to stochastic flow models

Insensitivity of the flow level model

Convergence of throughput to proportional fairness

Abstract

We consider multi-class single-server queueing networks that have a product form stationary distribution. A new limit result proves a sequence of such networks converges weakly to a stochastic flow level model. The stochastic flow level model found is insensitive. A large deviation principle for the stationary distribution of these multi-class queueing networks is also found. Its rate function has a dual form that coincides with proportional fairness. We then give the first rigorous proof that the stationary throughput of a multi-class single-server queueing network converges to a proportionally fair allocation. This work combines classical queueing networks with more recent work on stochastic flow level models and proportional fairness. One could view these seemingly different models as the same system described at different levels of granularity: a microscopic, queueing level description; a macroscopic, flow level description and a teleological, optimization description.