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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...
Scheduling flexible servers with convex delay costs: Heavytraffic optimality of the generalized cμrule
 OPER. RES
, 2004
"... ..."
On Dynamic Scheduling of a Parallel Server System with Complete Resource Pooling
 In Analysis of Communication Networks: Call Centres, Traffic and Performance
, 2000
"... scientific noncommercial 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 71 (5 self)
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scientific noncommercial 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 firstinfirstout basis, where the processing of a given job may be performed by any server from a given (classdependent) 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
State space collapse and diffusion approximation for a network operating under a fair bandwidthsharing policy, in preparation
, 2004
"... We consider a connectionlevel model of Internet congestion control, introduced by Massoulié and Roberts [36], that represents the randomly varying number of flows present in a network. Here bandwidth is shared fairly amongst elastic document transfers according to a weighted αfair bandwidth sharin ..."
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Cited by 46 (8 self)
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We consider a connectionlevel model of Internet congestion control, introduced by Massoulié and Roberts [36], that represents the randomly varying number of flows present in a network. Here bandwidth is shared fairly amongst elastic document transfers according to a weighted αfair bandwidth sharing policy introduced by Mo and Walrand [37] (α ∈ (0,∞)). Assuming Poisson arrivals and exponentially distributed document sizes, we focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. A fluid model (or functional law of large numbers approximation) for this stochastic model was derived and analyzed in a prior work [29] by two of the authors. Here we use the long time behavior of the solutions of this fluid model established in [29] to derive a property called multiplicative state space collapse, which loosely speaking shows that in diffusion scale the flow count process for the stochastic model can be approximately recovered as a continuous lifting of the workload process. Under weighted proportional fair sharing of bandwidth (α = 1) and a mild
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.
Heavy Traffic Limits for Some Queueing Networks
 Annals of Applied Probability
, 2001
"... Using a slight modification of the framework in Bramson [7] and Williams [52], we prove heavy traffic limit theorems for six families of multiclass queueing networks. The first three families are single station systems operating under firstin firstout (FIFO), generalized headoftheline proportio ..."
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Cited by 33 (4 self)
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Using a slight modification of the framework in Bramson [7] and Williams [52], we prove heavy traffic limit theorems for six families of multiclass queueing networks. The first three families are single station systems operating under firstin firstout (FIFO), generalized headoftheline proportional processor sharing (GHLPPS) and static buffer priority (SBP) service disciplines. The next two families are reentrant lines operating under firstbufferfirstserve (FBFS) and lastbufferfirstserve (LBFS) service disciplines; the last family consists of certain 2station, 5class networks operating under an SBP service discipline. Some of these heavy traffic limits have appeared earlier in the literature; our new proofs demonstrate the significant simplifications that can be achieved in the present setting.
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
A Heavy Traffic Limit Theorem for a Class of Open Queueing Networks with Finite Buffers
, 1997
"... We consider a queueing network of d single server stations. Each station has a finite capacity waiting buffer, and all customers served at a station are homogeneous in terms of service requirements and routing. The routing is assumed to be deterministic and hence feedforward. A server stops working ..."
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Cited by 17 (4 self)
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We consider a queueing network of d single server stations. Each station has a finite capacity waiting buffer, and all customers served at a station are homogeneous in terms of service requirements and routing. The routing is assumed to be deterministic and hence feedforward. A server stops working when the downstream buffer is full. We show that a properly normalized ddimensional queue length process converges in distribution to a ddimensional semimartingale reflecting Brownian motion (RBM) in a ddimensional box under a heavy traffic condition. The conventional continuous mapping approach does not apply here because the solution to our Skorohod problem may not be unique. Our proof relies heavily on a uniform oscillation result for solutions to a family of Skorohod problems. The oscillation result is proved in a general form that may be of independent interest. It has the potential to be used as an important ingredient in establishing heavy traffic limit theorems for general finite buffer networks. Key words and phrases: Finite capacity network, blocking probabilities, loss network, semimartingale reflecting Brownian motion, RBM, heavy traffic, limit theorems, oscillation estimates.
An invariance principle for semimartingale reflecting Brownian motions in cones with piecewise constant reflection fields, in preparation
, 2004
"... Semimartingale Reflecting Brownian Motions (SRBMs) living in the closures of domains with piecewise smooth boundaries are of interest in applied probability because of their role as heavy traffic approximations for some stochastic networks. In this paper, assuming certain conditions on the domains a ..."
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Cited by 12 (6 self)
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Semimartingale Reflecting Brownian Motions (SRBMs) living in the closures of domains with piecewise smooth boundaries are of interest in applied probability because of their role as heavy traffic approximations for some stochastic networks. In this paper, assuming certain conditions on the domains and directions of reflection, a perturbation result, or invariance principle, for SRBMs is proved. This provides sufficient conditions for a process that satisfies the definition of an SRBM, except for small random perturbations in the defining conditions, to be close in distribution to an SRBM. A crucial ingredient in the proof of this result is an oscillation inequality for solutions of a perturbed Skorokhod problem. We use the invariance principle to show weak existence of SRBMs under mild conditions. We also use the invariance principle, in conjunction with known uniqueness results for SRBMs, to give some sufficient conditions for validating approximations involving (i) SRBMs in convex polyhedrons with a constant reflection vector field on each face of the polyhedron, and (ii) SRBMs in bounded domains with piecewise smooth boundaries and possibly nonconstant reflection vector fields on the boundary surfaces. Copyright c○W. Kang and R. J. Williams, 2006. All print and electronic rights reserved.
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