Approximations and Optimal Control for State-dependent Limited Processor Sharing Queues

The paper studies approximations and control of a processor sharing (PS) server where the service rate depends on the number of jobs occupying the server. The control of such a system is implemented by imposing a limit on the number of jobs that can share the server concurrently, with the rest of the jobs waiting in a first-in-first-out (FIFO) buffer. A desirable control scheme should strike the right balance between efficiency (operating at a high service rate) and parallelism (preventing small jobs from getting stuck behind large ones). We employ the framework of heavy-traffic diffusion analysis to devise near optimal control heuristics for such a queueing system. However, while the literature on diffusion control of state-dependent queueing systems begins with a sequence of systems and an exogenously defined drift function, we begin with a finite discrete PS server and propose an axiomatic recipe to explicitly construct a sequence of state-dependent PS servers which then yields a drift function. We establish diffusion approximations and use them to obtain insightful and closed-form approximations for the original system under a static concurrency limit control policy. We extend our study to control policies that dynamically adjust the concurrency limit. We provide two novel numerical algorithms to solve the associated diffusion control problem. Our algorithms can be viewed as average cost iteration: The first algorithm uses binary-search on the average cost and can find an $epsilon$-optimal policy in time $Oleft( log^2 frac{1}{epsilon} right)$; the second algorithm uses the Newton-Raphson method for root-finding and requires $Oleft( log frac{1}{epsilon} loglog frac{1}{epsilon}right)$ time. Numerical experiments demonstrate the accuracy of our approximation for choosing optimal or near-optimal static and dynamic concurrency control heuristics.