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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 112 (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...
Largest Weighted Delay First Scheduling: Large Deviations and Optimality,“ to appear
 Annals of Appl. Prob
"... We consider a single server system with N input flows. We assume that each flow has stationary increments and satisfies a sample path large deviation principle, and that the system is stable. We introduce the largest weighted delay first (LWDF) queueing discipline associated with any given weight ve ..."
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Cited by 70 (5 self)
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We consider a single server system with N input flows. We assume that each flow has stationary increments and satisfies a sample path large deviation principle, and that the system is stable. We introduce the largest weighted delay first (LWDF) queueing discipline associated with any given weight vector α =�α1�����αN�. We show that under the LWDF discipline the sequence of scaled stationary distributions of the delay ˆw i of each flow satisfies a large deviation principle with the rate function given by a finitedimensional optimization problem. We also prove that the LWDF discipline is optimal in the sense that it maximizes the quantity −1 min αi lim i=1 � ��� � N n→ ∞ n log P � ˆw] i>n � � within a large class of work conserving disciplines. 1. Introduction.
Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling, Queueing Systems 13
, 1993
"... We present an introductory review of recent work on the control of open queueing networks. We assume that customers ofdifferent types arrive at a network and pass through the system via one of several possible routes; the set of routes available to a customer depends on its type. A route through th ..."
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Cited by 62 (4 self)
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We present an introductory review of recent work on the control of open queueing networks. We assume that customers ofdifferent types arrive at a network and pass through the system via one of several possible routes; the set of routes available to a customer depends on its type. A route through the network is an ordered set of service stations: a customer queues for service at each station on its route and then leaves the system. The two methods of control we consider are the routing of customers through the network, and the sequencing of service at the stations, and our aim is to minimize the number of customers in the system. We concentrate especially on the insights which can be obtained from heavy traffic analysis, and in particular from Harrison's Brownian etwork models. Our main conclusion is that in many respects dynamic routing simplifies the behaviour of networks, and that under good control policies itmay well be possible to model the aggregate b haviour of a network quite straightforwardly.
Dynamic scheduling with convex delay costs: the generalized cµ rule
, 1995
"... We consider a general singleserver multiclass queueing system that incurs a delay cost Ck ( k) for each class k job that resides k units of time in the system. This paper derives a scheduling policy that minimizes the total cumulative delay cost when the system operates during a nite time horizon. ..."
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Cited by 52 (2 self)
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We consider a general singleserver multiclass queueing system that incurs a delay cost Ck ( k) for each class k job that resides k units of time in the system. This paper derives a scheduling policy that minimizes the total cumulative delay cost when the system operates during a nite time horizon. Denote the marginal delay cost and (instantaneous) service rate functions of class k by ck = C 0 k and k, and let ak(t) be the "age" or time that the oldest class k job has been waiting at time t. Wecall the scheduling policy that at time t serves the oldest waiting job of that class k with the highest index k(t)ck(ak(t)), the Generalized c Rule. As a dynamic priority rule that depends on very little data, the Generalized c Rule is attractive to implement. We show that with nondecreasing convex delay costs, the Generalized c Rule is asymptotically optimal if the system operates in heavy tra c, and give explicit expressions for the associated performance characteristics: the delay (throughput time) process and the minimum cumulative delay cost. The optimality result is robust in that it holds for a countable number of classes and several homogeneous servers in a nonstationary, deterministic or stochastic environment where arrival and service processes can be general and interdependent. 1
Convex Duality and the Skorokhod Problem
 I, II. Probability Theory and Related Fields
, 1998
"... The solution to the Skorokhod Problem defines a deterministic mapping, referred to as the Skorokhod Map, that takes unconstrained paths to paths that are confined to live within a given domain G ae IR n . Given a set of allowed constraint directions for each point of @G, the solution to the Skorok ..."
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Cited by 52 (16 self)
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The solution to the Skorokhod Problem defines a deterministic mapping, referred to as the Skorokhod Map, that takes unconstrained paths to paths that are confined to live within a given domain G ae IR n . Given a set of allowed constraint directions for each point of @G, the solution to the Skorokhod Problem defines the constrained version OE of /, where the constraining force acts along one of the given boundary directions using the "least effort" required to keep OE in G. The Skorokhod Map is one of the main tools used in the analysis and construction of constrained deterministic and stochastic processes. Examples of these processes include stochastic differential equations with reflection, a related class of constrained ordinary differential equations, queueing models, and constrained stochastic approximation schemes. When the Skorokhod Map is sufficiently regular, and in particular when it is Lipschitz continuous on path space, the study of many problems involving these constrai...
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 23 (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
Numerical methods for stochastic singular control problems
 SIAM J. Control Optim
, 1991
"... Key words. Markov chain approximation methods, numerical methods, Singular stochastic control, reflected diffusions The Markov chain approximation method is a widely used numerical approach to computing optimal controls and value functions for general nonlinear jump diffusions, with a possible refle ..."
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Cited by 19 (6 self)
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Key words. Markov chain approximation methods, numerical methods, Singular stochastic control, reflected diffusions The Markov chain approximation method is a widely used numerical approach to computing optimal controls and value functions for general nonlinear jump diffusions, with a possible reflecting boundary. We extend the method to models with singular controls, where the control increment has the form g(x(t−))dH(t), which we call state dependent owing to the multiplier g(x). For the most part, past work concerned the case where g(·) is constant. There are major differences in the properties of and treatment of the two cases. Owing to the possibility of “multiple simultaneous impulses, ” H(·) must be interpreted in a generalized sense, and the analysis done in a “stretchedout ” time scale, analogously to that previously used by the author and colleagues. 1 Introduction: The Model
Optimal Control of Assignment of Jobs to Processors under Heavy Traffic
 Stochastics and Stochastic Reports
, 1999
"... The paper is concerned with the optimal control of the assignment of jobs from several arriving random streams to one of a bank of processors. Owing to the difficulty of the general problem, a heavy traffic approach is used. The required work depends on the processor to which it is assigned. The inf ..."
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Cited by 13 (3 self)
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The paper is concerned with the optimal control of the assignment of jobs from several arriving random streams to one of a bank of processors. Owing to the difficulty of the general problem, a heavy traffic approach is used. The required work depends on the processor to which it is assigned. The information that the assignment can be based on is quite flexible, and several information structures (data on which the control is based) are considered. The assignment can be made on arrival or when the job is to be processed. There can be bursty arrivals (the bursts depending on randomly varying environmental factors), rather general nonlinear cost functions and other complications. It is shown, under reasonably general conditions, that the optimal costs for the physical systems converge to the optimal cost for the heavy traffic limit problem, as the heavy traffic parameter goes to its limit. Numerical data is presented to illustrate some of the potential uses of the limit process for obtain...
Heavy Traffic Analysis of Controlled Multiplexing Systems
 SIAM J. Control and Optimization
, 1997
"... The paper develops the mathematics of the heavy traffic approach to the control and optimal control problem for multiplexing systems, where there are many mutually independent sources which feed into a single channel via a multiplexer (or of networks composed of such subsystems). Due to the widely v ..."
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Cited by 12 (2 self)
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The paper develops the mathematics of the heavy traffic approach to the control and optimal control problem for multiplexing systems, where there are many mutually independent sources which feed into a single channel via a multiplexer (or of networks composed of such subsystems). Due to the widely varying bit rates over all sources, control over admission, bandwidth, etc., is needed to assure good performance. Optimal control and heavy traffic analysis has been shown to yield systems with greatly improved performance. Indeed, the heavy traffic approach covers many cases of great current interest, and provides a useful and practical approach to problems of analysis and control arising in modern high speed telecommunications. Past works on the heavy traffic approach to the multiplexing problem concentrated on the uncontrolled system or on the use of the heavy traffic limit control problem for applications, and did not provide details of the proofs. This is done in the current paper. The ...