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

Abstract

We study a resource-sharing network where 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" scheme, which allocates the capacity among job classes taking into account both the queue lengths and the shadow prices of link capacity. We show that the usual traffic condition 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. To justify the interchange of two limits, the limit in time and the limit in diffusion scaling, we identify a bounded-workload condition, and show it is a sufficient condition to justify the interchange for both the stationary distributions and their moments. This last result is essential for the validity of the diffusion limit as an approximation to the stationary performance of the original network. We present a set of examples to illustrate justifying the validity of diffusion approximation in resource-sharing networks, and also in a well-known multi-class model, the Kumar-Seidman network.