Diffusion Limit of Fair Resource Control - Stationarity and Interchange of Limits

We study a stochastic processing network, in which each job requires the concurrent occupancy of a subset of links (servers/resources), and each link’s capacity is shared among job classes that require its service. The real-time allocation of the service capacity among job classes is determined by the so-called “proportional fair” allocation scheme, which allocates the capacity among job classes taking into account both the queue lengths and the shadow prices of link capacity. We demonstrate that the usual traffic condition of the diffusion limit is necessary and sufficient for the diffusion limit to have a stationary distribution. We also establish the uniform stability of the pre-limit networks — and hence, the existence of their stationary distributions — under the usual traffic condition. Furthermore, we identify a bounded workload condition, and show it is a sufficient condition for justifying the interchange of the two limits — the limit in time and the limit in diffusion scaling — for both the stationary distributions and the p-th moment. This last result is essential for the validity of the diffusion limit as an approximation to the stationary performance of the original network. Our study has provided a streamlined and systematic approach to developing diffusion approximations, which has the potential to extend to a broader range of multi-class stochastic network.