A Stochastic Network Under Proportional Fair Resource Control - Diffusion Limit with Multiple Bottlenecks

We study a multi-class stochastic processing network operating under the so-called proportional fair allocation scheme, and following the head-of-the-line processor sharing discipline. Specifically, each server’s capacity is shared among the job classes that require its service; and allocated, in every state of the network, among the first waiting job of each class to maximize a log-utility function. We establish the limiting regime of the network under diffusion scaling, allowing multiple bottlenecks in the network, and relaxing some of the conditions required in prior studies. We also examine the asymptotic optimality of the proportional fair allocation, identifying the class of allocation schemes among which the proportional fair allocation minimizes a quadratic cost objective function of the diffusion-scaled queue lengths, and illustrating the limitation of the optimality through a counter-example.