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Dynamic Scheduling of a System with Two Parallel Servers in Heavy Traffic with Resource Pooling: Asymptotic Optimality of a Threshold Policy
 Annals of Applied Probability
, 1999
"... This paper concerns a dynamic scheduling problem for a queueing system that has two streams of arrivals to infinite capacity buffers and two (nonidentical) servers working in parallel. One server can only process jobs from one buffer, whereas the other server can process jobs from either buffer. Th ..."
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Cited by 118 (6 self)
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This paper concerns a dynamic scheduling problem for a queueing system that has two streams of arrivals to infinite capacity buffers and two (nonidentical) servers working in parallel. One server can only process jobs from one buffer, whereas the other server can process jobs from either buffer. The service time distribution may depend on the buffer being served and the server providing the service. The system manager dynamically schedules waiting jobs onto available servers. We consider a parameter regime in which the system satisfies both a heavy traffic condition and a resource pooling condition. Our cost function is a mean 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. We first review the analytic solution of the Brownian control problem (formal heavy traffic approximation) for this system. We "interpret" this solution by proposing a threshold contro...
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 (5 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.
Scheduling a multiclass queue with many exponential servers: Asymptotic optimality in heavytraffic,” The Annals of Applied Probability
, 2004
"... We consider the problem of scheduling a queueing system in which many statistically identical servers cater to several classes of impatient customers. Service times and impatience clocks are exponential while arrival processes are renewal. Our cost is an expected cumulative discounted function, line ..."
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Cited by 42 (13 self)
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We consider the problem of scheduling a queueing system in which many statistically identical servers cater to several classes of impatient customers. Service times and impatience clocks are exponential while arrival processes are renewal. Our cost is an expected cumulative discounted function, linear or nonlinear, of appropriately normalized performance measures. As a special case, the cost per unit time can be a function of the number of customers waiting to be served in each class, the number actually being served, the abandonment rate, the delay experienced by customers, the number of idling servers, as well as certain combinations thereof. We study the system in an asymptotic heavytraffic regime where the number of servers n and the offered load r are simultaneously scaled up and carefully balanced: n ≈ r + β √ r for some scalar β. This yields an operation that enjoys the benefits of both heavy traffic (high server utilization) and light traffic (high service levels.)
Value Iteration and Optimization of Multiclass Queueing Networks
 Queueing Systems
, 1997
"... . This paper considers in parallel the scheduling problem for multiclass queueing networks, and optimization of Markov decision processes. It is shown that the value iteration algorithm may perform poorly when the algorithm is not initialized properly. The most typical case where the initial value f ..."
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Cited by 41 (12 self)
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. This paper considers in parallel the scheduling problem for multiclass queueing networks, and optimization of Markov decision processes. It is shown that the value iteration algorithm may perform poorly when the algorithm is not initialized properly. The most typical case where the initial value function is taken to be zero may be a particularly bad choice. In contrast, if the value iteration algorithm is initialized with a stochastic Lyapunov function, then the following hold (i): A stochastic Lyapunov function exists for each intermediate policy, and hence each policy is regular (a strong stability condition). (ii): Intermediate costs converge to the optimal cost. (iii): Any limiting policy is average cost optimal. It is argued that a natural choice for the initial value function is the value function for the associated deterministic control problem based upon a fluid model, or the approximate solution to Poisson's equation obtained from the LP of Kumar and Meyn. Numerical studi...
Heavy traffic analysis of open processing networks with complete resource pooling: asymptotic optimality of discrete review policies
 ANN. APPL. PROBAB
, 2005
"... We consider a class of open stochastic processing networks, with feedback routing and overlapping server capabilities, in heavy traffic. The networks ..."
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Cited by 25 (0 self)
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We consider a class of open stochastic processing networks, with feedback routing and overlapping server capabilities, in heavy traffic. The networks
Reliability by Design in Distributed Power Transmission Networks
, 2005
"... The system operator of a large power transmission network must ensure that power is delivered whenever there is demand in order to maintain highly reliable electric service. To fulfill this mandate, the system operator must procure reserve capacity to respond to unforeseen events such as an unexpect ..."
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Cited by 17 (11 self)
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The system operator of a large power transmission network must ensure that power is delivered whenever there is demand in order to maintain highly reliable electric service. To fulfill this mandate, the system operator must procure reserve capacity to respond to unforeseen events such as an unexpected surge in demand. This paper constructs a centralized optimal solution for a power network model by generalizing recent techniques for the centralized optimal control of demanddriven production systems. The optimal solution indicates how reserves must be adjusted according to environmental factors including variability, and the rampingrate constraints on generation. Sensitivity to transmission constraints is addressed through the construction of an effective cost on an aggregate model.
A multiclass queue in heavy traffic with throughput time constraints: Asymptotically optimal dynamic controls. Queueing Syst
 Theory Appl
, 2001
"... Abstract. Consider a singleserver queueing system with K job classes, each having its own renewal input process and its own general service time distribution. Further suppose the queue is in heavy traffic, meaning that its traffic intensity parameter is near the critical value of one. A system mana ..."
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Cited by 12 (4 self)
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Abstract. Consider a singleserver queueing system with K job classes, each having its own renewal input process and its own general service time distribution. Further suppose the queue is in heavy traffic, meaning that its traffic intensity parameter is near the critical value of one. A system manager must decide whether or not to accept new jobs as they arrive, and also the order in which to serve jobs that are accepted. The goal is to minimize penalties associated with rejected jobs, subject to upper bound constraints on the throughput times for accepted jobs; both the penalty for rejecting a job and the bound on the throughput time may depend on job class. This problem formulation does not make sense in a conventional queueing model, because throughput times are random variables, but we show that the formulation is meaningful in an asymptotic sense, as one approaches the heavy traffic limit under diffusion scaling. Moreover, using a method developed recently by Bramson and Williams, we prove that a relatively simple dynamic control policy is asymptotically optimal in this framework. Our proposed policy rejects jobs from one particular class when the server’s nominal workload is above a threshold value, accepting all other arrivals; and the sequencing rule for accepted jobs is one that maintains near equality of the relative backlogs for different classes, defined in a natural sense. 1. Introduction and
A large deviations approach to asymptotically optimal control of crisscross network in heavy traffic
 The Annals of Applied Probability
, 2005
"... In this work we study the problem of asymptotically optimal control of a wellknown multiclass queuing network, referred to as the “crisscross network, ” in heavy traffic. We consider exponential interarrival and service times, linear holding cost and an infinite horizon discounted cost criterion. ..."
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Cited by 11 (5 self)
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In this work we study the problem of asymptotically optimal control of a wellknown multiclass queuing network, referred to as the “crisscross network, ” in heavy traffic. We consider exponential interarrival and service times, linear holding cost and an infinite horizon discounted cost criterion. In a suitable parameter regime, this problem has been studied in detail by Martins, Shreve and Soner [SIAM J. Control Optim. 34 (1996) 2133–2171] using viscosity solution methods. In this work, using the pathwise solution of the Brownian control problem, we present an elementary and transparent treatment of the problem (with the identical parameter regime as in [SIAM J. Control Optim. 34 (1996) 2133–2171]) using large deviation ideas introduced
Singular Control with State Constraints on Unbounded Domain
 Annals of Probability
"... We study a class of stochastic control problems where a cost of the form E e [0,∞) −βs [ℓ(Xs)ds + h(Y ◦ s)dY s] is to be minimized over control processes Y whose increments take values in a cone Y of R p, keeping the state process X = x + B + GY in a cone X of R k, k ≤ p. Here, x ∈ X, B is a Brown ..."
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Cited by 9 (3 self)
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We study a class of stochastic control problems where a cost of the form E e [0,∞) −βs [ℓ(Xs)ds + h(Y ◦ s)dY s] is to be minimized over control processes Y whose increments take values in a cone Y of R p, keeping the state process X = x + B + GY in a cone X of R k, k ≤ p. Here, x ∈ X, B is a Brownian motion with drift b and covariance Σ, G is a fixed matrix, and Y ◦ is the Radon–Nikodym derivative dY/dY . Let L = −(1/2)trace(ΣD 2) − b · D where D denotes the gradient. Solutions to the corresponding dynamic programming PDE, [(L + β)f − ℓ] ∨ sup y∈Y:Gy=1 [−Gy · Df − h(y)] = 0, on X o are considered with a polynomial growth condition and are required to be supersolution up to the boundary (corresponding to a “state constraint ” boundary condition on ∂X). Under suitable conditions on the problem data, including continuity and nonnegativity of ℓ and h, and polynomial growth of ℓ, our main result is the unique viscositysense solvability of the PDE by the control problem’s value function in appropriate classes of functions. In some cases where uniqueness generally fails to hold in the class of functions that grow at most polynomially (e.g., when h = 0), our methods provide uniqueness within the class of functions that, in addition, have compact level sets. The results are new even in the following special cases: (1) The onedimensional case k = p = 1, X = Y = R+; (2) The firstorder case Σ = 0; (3) The case where ℓ and h are linear. The proofs