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23
On Dynamic Scheduling of a Parallel Server System with Complete Resource Pooling
- In Analysis of Communication Networks: Call Centres, Traffic and Performance
, 2000
"... scientific non-commercial use only for individuals, with permission from the authors. We consider a parallel server queueing system consisting of a bank of buffers for holding incoming jobs and a bank of flexible servers for processing these jobs. Incoming jobs are classified into one of several dif ..."
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Cited by 67 (5 self)
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scientific non-commercial use only for individuals, with permission from the authors. We consider a parallel server queueing system consisting of a bank of buffers for holding incoming jobs and a bank of flexible servers for processing these jobs. Incoming jobs are classified into one of several different classes (or buffers). Jobs within a class are processed on a first-in-first-out basis, where the processing of a given job may be performed by any server from a given (class-dependent) subset of the bank of servers. The random service time of a job may depend on both its class and the server providing the service. Each job departs the system after receiving service from one server. The system manager seeks to minimize holding costs by dynamically scheduling waiting jobs to available servers. We consider a parameter regime in which the system satisfies both a heavy traffic and a complete resource pooling condition. Our cost function is an expected cumulative discounted cost of holding jobs in the system, where the (undiscounted) cost per unit time is a linear function of normalized (with heavy traffic scaling) queue length. In a prior work [40], the second author proposed a continuous review threshold control policy for use in such a parallel server system. This policy was advanced as an “interpretation ” of the analytic solution to an associated Brownian control problem (formal heavy
Scheduling control for queueing systems with many servers: Asymptotic optimality in heavy traffic
, 2005
"... A multiclass queueing system is considered, with heterogeneous service stations, each consisting of many servers with identical capabilities. An optimal control problem is formulated, where the control corresponds to scheduling and routing, and the cost is a cumulative discounted functional of the s ..."
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Cited by 43 (6 self)
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A multiclass queueing system is considered, with heterogeneous service stations, each consisting of many servers with identical capabilities. An optimal control problem is formulated, where the control corresponds to scheduling and routing, and the cost is a cumulative discounted functional of the system’s state. We examine two versions of the problem: “nonpreemptive,” where service is uninterruptible, and “preemptive, ” where service to a customer can be interrupted and then resumed, possibly at a different station. We study the problem in the asymptotic heavy traffic regime proposed by Halfin and Whitt, in which the arrival rates and the number of servers at each station grow without bound. The two versions of the problem are not, in general, asymptotically equivalent in this regime, with the preemptive version showing an asymptotic behavior that is, in a sense, much simpler. Under appropriate assumptions on the structure of the system we show: (i) The value function for the preemptive problem converges to V, the value of a related diffusion control problem. (ii) The two versions of the problem are asymptotically equivalent, and in particular nonpreemptive policies can be constructed that asymptotically achieve the value V. The construction of these policies is based on a Hamilton–Jacobi–Bellman equation associated with V.
Asymptotic optimality of maximum pressure policies in stochastic processing networks
- Annals of Applied Probability
, 2008
"... We consider a class of stochastic processing networks. Assume that the networks satisfy a complete resource pooling condition. We prove that each maximum pressure policy asymptotically minimizes the workload process in a stochastic processing network in heavy traffic. We also show that, under each q ..."
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Cited by 43 (4 self)
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We consider a class of stochastic processing networks. Assume that the networks satisfy a complete resource pooling condition. We prove that each maximum pressure policy asymptotically minimizes the workload process in a stochastic processing network in heavy traffic. We also show that, under each quadratic holding cost structure, there is a maximum pressure policy that asymptotically minimizes the holding cost. A key to the optimality proofs is to prove a state space collapse result and a heavy traffic limit theorem for the network processes under a maximum pressure policy. We extend a framework of Bramson [Queueing Systems Theory Appl. 30 (1998) 89–148] and Williams [Queueing Systems Theory Appl. 30 (1998b) 5–25] from the multiclass queueing network setting to the stochastic processing network setting to prove the state space collapse result and the heavy traffic limit theorem. The extension can be adapted to other studies of stochastic processing networks.
Dynamic Control of N-Systems with Many Servers: Asymptotic Optimality of a Static Priority Policy in Heavy Traffic
, 2010
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Optimal control of a high-volume assemble-to-order system
- Mathematics of Operations Research
, 2006
"... For an assemble-to-order system with a high volume of prospective customers arriving per unit time, we show how to set nominal component production rates, quote prices and maxi-mum leadtimes for products, and then, dynamically, sequence orders for assembly and expedite components. (Components must b ..."
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Cited by 18 (6 self)
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For an assemble-to-order system with a high volume of prospective customers arriving per unit time, we show how to set nominal component production rates, quote prices and maxi-mum leadtimes for products, and then, dynamically, sequence orders for assembly and expedite components. (Components must be expedited if necessary to fill an order within the maximum leadtime). We allow for updating of the prices, maximum leadtimes, and nominal component production rates in response to periodic, random shifts in demand and supply conditions. As-suming expediting costs are large, we prove that our proposed policy maximizes infinite horizon expected discounted profit in the high volume limit. For a more general assemble-to-order sys-tem with arbitrary cost of expediting and the option to salvage excess components, we show how to solve an approximating Brownian control problem and translate its solution into an effective control policy. 1
Stability and asymptotic optimality of generalized maxweight policies
- SIAM Journal on Control and Optimization
"... Abstract It is shown that stability of the celebrated MaxWeight or back pressure policies is a consequence of the following interpretation: either policy is myopic with respect to a surrogate value function of a very special form, in which the "marginal disutility" at a buffer vanishes fo ..."
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Cited by 16 (2 self)
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Abstract It is shown that stability of the celebrated MaxWeight or back pressure policies is a consequence of the following interpretation: either policy is myopic with respect to a surrogate value function of a very special form, in which the "marginal disutility" at a buffer vanishes for vanishingly small buffer population. This observation motivates the h-MaxWeight policy, defined for a wide class of functions h. These policies share many of the attractive properties of the MaxWeight policy: (i) Arrival rate data is not required in the policy. (ii) Under a variety of general conditions, the policy is stabilizing when h is a perturbation of a monotone linear function, a monotone quadratic, or a monotone Lyapunov function for the fluid model. (iii) A perturbation of the relative value function for a workload relaxation gives rise to a myopic policy that is approximately average-cost optimal in heavy traffic, with logarithmic regret. The first results are obtained for a general Markovian network model. Asymptotic optimality is established for a general Markovian scheduling model with a single bottleneck, and homogeneous servers.
Optimal Control of Parallel Server Systems with Many Servers in Heavy Traffic
, 2008
"... We consider a parallel server system that consists of several customer classes and server pools in parallel. We propose a simple robust control policy to minimize the total linear holding and reneging costs. We show that this policy is asymptotically optimal under the many-server heavy traffic regi ..."
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Cited by 16 (7 self)
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We consider a parallel server system that consists of several customer classes and server pools in parallel. We propose a simple robust control policy to minimize the total linear holding and reneging costs. We show that this policy is asymptotically optimal under the many-server heavy traffic regime for parallel server systems when the service times are only server pool dependent and exponentially distributed.
Diffusion approximations for controlled stochastic networks: An asymptotic bound for the value function
- Ann. Appl. Probab
, 2006
"... We consider the scheduling control problem for a family of unitary networks under heavy traffic, with general interarrival and service times, probabilistic routing and infinite horizon discounted linear holding cost. A natural nonanticipativity condition for admissibility of control policies is intr ..."
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Cited by 10 (6 self)
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We consider the scheduling control problem for a family of unitary networks under heavy traffic, with general interarrival and service times, probabilistic routing and infinite horizon discounted linear holding cost. A natural nonanticipativity condition for admissibility of control policies is introduced. The condition is seen to hold for a broad class of problems. Using this formulation of admissible controls and a time-transformation technique, we establish that the infimum of the cost for the network control problem over all admissible sequencing control policies is asymptotically bounded below by the value function of an associated diffusion control problem (the Brownian control problem). This result provides a useful bound on the best achievable performance for any admissible control policy for a wide class of networks.
A little flexibility is all you need: optimality of tailored chaining and pairing. working paper
, 2008
"... Deciding on the appropriate type and amount of flexibility is a classic management problem. The literature has shown that the choice between specialization and flexibility is not an “all-or-nothing ” proposition. It is typically better to use a tailored portfolio of dedicated and flexible resources ..."
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Cited by 9 (0 self)
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Deciding on the appropriate type and amount of flexibility is a classic management problem. The literature has shown that the choice between specialization and flexibility is not an “all-or-nothing ” proposition. It is typically better to use a tailored portfolio of dedicated and flexible resources and a little flexibility goes a long way. Let “level-k ” flexibility refer to a resource’s ability to process k different types of products. Simulations have shown that using only level-2 flexible resources in a special configuration called chaining achieves almost all the benefits of total flexibility. In this paper, we introduce “tailored pairing ” that merges and extends the concepts of chaining and tailoring in dynamic processing systems. We optimize the type and amount of flexibility using a Brownian approximation that is asymptotically correct. We show analytically that for symmetric systems and most practical flexibility cost structures the optimal flexibility configuration invests a lot in dedicated resources, a little in only bi-level flexibility, but nothing in level-k> 2 flexibility, let alone full flexibility. Dedicated resources provide base capacity to serve the majority of the demand while only a small amount of bi-level flexibility is sufficient to serve the variable demand: dedicated capacity is sized roughly proportional to demand while flexible capacity is roughly proportional to the square root of demand and to its coefficient of variation. Our main result can be restated as saying that tailored pairing is optimal for symmetric systems. We investigate the accuracy and robustness of our results in asymmetric systems. It is obvious that the tailored flexible configuration will mirror the asymmetry in the demand. Yet our main result remains: even asymmetric systems do not seem to need k> 2-level flexible resources and complete resource pooling is suboptimal. 1.
