Abstract

A network of single server queues with routing is considered. It is proven a sequence of such networks can converge weakly to a stochastic flow level model. A large deviations principle for the stationary distribution of these networks is found. Its rate function has a dual formulation that coincides with proportional fairness. It is proven the throughput of the original queueing model behaves as a proportionally fair allocation as the number of customers across routes increases.

The queueing networks considered have no prescribed optimization structure. Regardless of this, we find proportional fairness forms the entropy minimizing state of these networks. From this it is found proportionally fair optimization occurs as a consequence of state space collapse behaviour in these non-optimizing queueing models.

This work merges classical results on queueing networks and recent work on proportional fairness. One could view these seemingly different models as the same system described at different levels of granularity: a microscopic packet/queueing level description; a macroscopic flow level description and a teleological optimization description.