Proportional Fairness and its Relationship with Multi-class Queueing Networks

A network of single server queues with routing is considered. These networks have a product form stationary distribution. A new limit result proves a sequence of such networks converges weakly to a stochastic flow level model. This stochastic model is insensitive. A large deviation principle for the stationary distribution of these queueing networks is found. Its rate function has a dual formulation that coincides with proportional fairness. It is proven that the stationary throughput of the queueing model converges to a proportionally fair allocation. The queueing models considered have no prescribed optimization structure. Regardless of this, we find a proportionally fair allocation forms an entropy minimizing state of these networks. Proportional fairness occurs as a consequence of a collapse in the state space of the queueing model. 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.