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State Space Collapse in ManyServer Diffusion Limits of Parallel Server Systems and Applications
, 2006
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Asymptotically optimal parallel resource assignment with interference. Queueing Systems
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Asymptotic Optimality of Balanced Routing
, 2010
"... Consider a system with K parallel singleservers, each with its own waiting room. Upon arrival, a job is to be routed to the queue of one of the servers. Finding routing policy that minimizes the total workload in the system is a known difficult problem in general. Even if the optimal policy is iden ..."
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Consider a system with K parallel singleservers, each with its own waiting room. Upon arrival, a job is to be routed to the queue of one of the servers. Finding routing policy that minimizes the total workload in the system is a known difficult problem in general. Even if the optimal policy is identified, the policy would require the full queue length information at the arrival of each job; for example, the jointheshortestqueue policy (which is known to be optimal for identical servers with exponentially distributed service times) would require to compare the queue lengths of all the servers. In this paper, we consider a balanced routing control policy that exams only a subset of c servers, with 1 ≤ c ≤ K: specifically, upon the arrival of a job, choose a subset of c servers with a probability proportional to their service rates and then route the job to the one with the shortest queue among the c chosen servers. Under such a balanced policy, we derive the diffusion limits of the queue length processes and the workload processes. We note that the diffusion limits are the same for these processes regardless the choice of c as long as c ≥ 2. We further show that the proposed balanced routing policy for any fixed c ≥ 2 is asymptotically optimal in the sense that it minimizes the workload over all time in the diffusion limit. In addition, the policy helps to distribute works among all the servers evenly.
Controlled Stochastic Networks in Heavy Traffic: Convergence of Value Functions.
, 2010
"... Abstract: Scheduling control problems for a family of unitary networks under heavy traffic with general interarrival and service times, probabilistic routing and an infinite horizon discounted linear holding cost are studied. Diffusion control problems, that have been proposed as approximate models ..."
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Abstract: Scheduling control problems for a family of unitary networks under heavy traffic with general interarrival and service times, probabilistic routing and an infinite horizon discounted linear holding cost are studied. Diffusion control problems, that have been proposed as approximate models for the study of these critically loaded controlled stochastic networks, can be regarded as formal scaling limits of such stochastic systems. However, to date, a rigorous limit theory that justifies the use of such approximations for a general family of controlled networks has been lacking. It is shown that, under broad conditions, the value function of the suitably scaled network control problem converges to that of the associated diffusion control problem. This scaling limit result, in addition to giving a precise mathematical basis for the above approximation approach, suggests a general strategy for constructing near optimal controls for the physical stochastic networks by solving the associated diffusion control problem.
Dynamic Scheduling of Open Multiclass Queueing Networks in a Slowly Changing Environment
, 2004
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CONTROLS∗
"... We consider an inventory system in which inventory level fluctuates as a Brownian motion in the absence of control. The inventory continuously accumulates cost at a rate that is a general convex function of the inventory level, which can be negative when there is a backlog. At any time, the inven ..."
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We consider an inventory system in which inventory level fluctuates as a Brownian motion in the absence of control. The inventory continuously accumulates cost at a rate that is a general convex function of the inventory level, which can be negative when there is a backlog. At any time, the inventory level can be adjusted by a positive or negative amount, which incurs a fixed positive cost and a proportional cost. The challenge is to find an adjustment policy that balances the inventory cost and adjustment cost to minimize the expected total discounted cost. We provide a tutorial on using a threestep lowerbound approach to solving the optimal control problem under a discounted cost criterion. In addition, we prove that a fourparameter control band policy is optimal among all feasible policies. A key step is the constructive proof of the existence of a unique solution to the free boundary problem. The proof leads naturally to an algorithm to compute the four parameters of the optimal control band policy.
On the dynamic control of matching queues
, 2013
"... We consider the optimal control of matching queues with dynamically arriving jobs. In this model, jobs arrive to their dedicated queues, and wait to be matched with jobs from other (possibly many) queues. Given that all jobs in a match are present in the system, the matching itself is instantaneous. ..."
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We consider the optimal control of matching queues with dynamically arriving jobs. In this model, jobs arrive to their dedicated queues, and wait to be matched with jobs from other (possibly many) queues. Given that all jobs in a match are present in the system, the matching itself is instantaneous. The objective is to minimize cumulative job holding costs. In the special case of linear (and equal across classes) holding costs, this is equivalent to maximizing the number of matched jobs. The key control question is whether to match myopically, or to keep “inventory ” of jobs, to facilitate more profitable matches in the future. Problems of holding cost minimization have been wellstudied in the processing networks literature, and, in the traditional parallel server setting, MaxWeight scheduling policies and variants thereof have been proved to perform well. The heavytraffic phenomena of statespace collapse and the ensuing reduction of the problem to a onedimensional Brownian control problem (under appropriate resource pooling conditions), that are the drivers of these wellknown results, do not hold in the matching queue setting that we study here. This difference is driven by the fundamentally different notions of capacity underlying the two settings. We introduce a multidimensional imbalance process, that at each time t, is given by a linear function of the cumulative arrivals to each of the job types. In essence, the imbalance at a time t captures the number of additional jobs required so that some control policy could have matched all the jobs that have arrived by that time (thus leaving the queues empty). The imbalance facilitates the construction of a lower bound that is specified, at each time point, by a solution to a simple constrained optimization problem. Achieving this lower bound requires, in general, “reshuffling ” past matches. Under a socalled matchpooling condition, we are able to devise a discretereview matching policy that asymptotically – as the arrival rates becomes large – achieves the imbalancebased lower bound. Our optimality results hold for both stationary and nonstationary arrivals. 1
Dynamic Scheduling of a Parallel Server System in Heavy Trafficwith Complete Resource Pooling: Asymptotic Optimality of a
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Running Title: Heavy Traffic Analysis of Closed Loop Supply Chain
, 2009
"... We consider a closed loop supply chain where new products are produced to order and returned products are refurbished for reselling. The solution to a pricesetting problem enforces the “heavy traffic ” condition, under which we address the production rate control problem under two types of cost fun ..."
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We consider a closed loop supply chain where new products are produced to order and returned products are refurbished for reselling. The solution to a pricesetting problem enforces the “heavy traffic ” condition, under which we address the production rate control problem under two types of cost functions. We solve a driftcontrol problem for an approximate system driven by a correlated twodimensional Brownian motion. The solutions to this system are then used to obtain asymptotically optimal control policies. We also conduct a numerical study to explore the effects of different parameters on the optimal production rates and the resulting costs.
Dynamic Control in Stochastic Processing Networks
, 2005
"... A stochastic processing network is a system that takes materials of various kinds as inputs, and uses processing resources to produce other materials as outputs. Such a network provides a powerful abstraction of a wide range of real world, complex systems, including semiconductor wafer fabrication f ..."
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A stochastic processing network is a system that takes materials of various kinds as inputs, and uses processing resources to produce other materials as outputs. Such a network provides a powerful abstraction of a wide range of real world, complex systems, including semiconductor wafer fabrication facilities, networks of data switches, and largescale call centers. Key performance measures of a stochastic processing network include throughput, cycle time, and holding cost. The network performance can dramatically be affected by the choice of operational policies. We propose a family of operational policies called maximum pressure policies. The maximum pressure policies are attractive in that their implementation uses minimal state information of the network. The deployment of a resource (server) is decided based on the queue lengths in its serviceable buffers and the queue lengths in their immediate downstream buffers. In particular, the decision does not use arrival rate information that is often difficult or impossible to estimate reliably. We prove that a maximum pressure policy can maximize throughput for a general class of stochastic processing networks. We also establish an asymptotic optimality of maximum pressure policies for stochastic processing networks with a unique bottleneck. The optimality is in terms of minimizing workload process. A key step in the proof of the asymptotic optimality is to show that the network processes under maximum pressure policies exhibit a state space collapse.