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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.
We consider a processor sharing queue where the number of jobs served at any time is limited to $K$, with the excess jobs waiting in a buffer. We use random counting measures on the positive axis to model this system. The limit of this measure-valued
For his work in the economics of climate change, Professor William Nordhaus was a co-recipient of the 2018 Nobel Memorial Prize for Economic Sciences. A core component of the work undertaken by Nordhaus is the Dynamic Integrated model of Climate and
This article treats two problems dealing with control of linear systems in the presence of a jammer that can sporadically turn off the control signal. The first problem treats the standard reachability problem, and the second treats the standard line
We prove the continuity of the value function of the sparse optimal control problem. The sparse optimal control is a control whose support is minimum among all admissible controls. Under the normality assumption, it is known that a sparse optimal con
Flexible loads, e.g. thermostatically controlled loads (TCLs), are technically feasible to participate in demand response (DR) programs. On the other hand, there is a number of challenges that need to be resolved before it can be implemented in pract