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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
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
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
Scheduling and control of manufacturing systems — a fluid approach
 Proceedings of the 37 Allerton Conference
, 1999
"... We suggest a fluid framework for solving scheduling and control problems of manufacturing systems. We formulate the fluid approximation, show how to solve it, give an important graphical display of the fluid solution, construct a schedule from the fluid solution, and provide probabilistic bound of i ..."
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Cited by 16 (9 self)
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We suggest a fluid framework for solving scheduling and control problems of manufacturing systems. We formulate the fluid approximation, show how to solve it, give an important graphical display of the fluid solution, construct a schedule from the fluid solution, and provide probabilistic bound of its suboptimality. 1
Control of polling in presence of vacations in heavy traffic with applications to satellite and mobile radio systems
 SIAM J. on Control and Optimization
, 1999
"... Consider a queueing system with many queues, each with its own input stream, but with only one server. The server must allocate its time among the queues to minimize or nearly minimize some cost criterion. The allocation of time among the queues is often called polling and is the subject of a large ..."
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Cited by 14 (6 self)
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Consider a queueing system with many queues, each with its own input stream, but with only one server. The server must allocate its time among the queues to minimize or nearly minimize some cost criterion. The allocation of time among the queues is often called polling and is the subject of a large literature. Usually, it is assumed that the queues are always available, and the server can allocate at will. We consider the case where the queues are not always available due to disruption of the connection between them and the server. Such occurrences are common in wireless communications, where any of the mobile sources might become unavailable to the server from time to time due to obstacles, atmospheric or other effects. The possibility of such "vacations" complicates the polling problem enormously. Due to the complexity of the basic problem we analyze it in the heavy traffic regime where the server has little idle time over the average requirements. It is shown that the suitable scaled total workloads converge to a controlled limit diusion process with jumps. The jumps
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...
Admission control for combined guaranteed performance and best effort communications systems under heavy traffic
 SIAM J. Control and Optimization
, 1999
"... Communications systems often have many types of users. Since they share the same resource, there is a conflict in their needs. This conflict leads to the imposition of controls on admission or elsewhere. In this paper, there are two types of customers, GP (Guaranteed Performance) and BE (Best Effort ..."
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Cited by 13 (8 self)
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Communications systems often have many types of users. Since they share the same resource, there is a conflict in their needs. This conflict leads to the imposition of controls on admission or elsewhere. In this paper, there are two types of customers, GP (Guaranteed Performance) and BE (Best Effort). We consider an admission control of GP customer which has two roles. First, to guarantee the performance of the existing GP customers, and second, to regulate the congestion for the BE users. The optimal control problem for the actual physical system is difficult. A heavy traffic approximation is used, with optimal or nearly optimal controls. It is shown that the optimal values for the physical system converge to that for the limit system and that good controls for the limit system are also good for the physical system. This is done for both the discounted and average cost per unit time cost criteria. Additionally, asymptotically, the pathwise average (not mean) costs for the physical system are nearly
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 ...
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 timetransformation 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.