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132
On Positive Harris Recurrence of Multiclass Queueing Networks: A Unified Approach Via Fluid Limit Models
 Annals of Applied Probability
, 1995
"... It is now known that the usual traffic condition (the nominal load being less than one at each station) is not sufficient for stability for a multiclass open queueing network. Although there has been some progress in establishing the stability conditions for a multiclass network, there is no unified ..."
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Cited by 349 (27 self)
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It is now known that the usual traffic condition (the nominal load being less than one at each station) is not sufficient for stability for a multiclass open queueing network. Although there has been some progress in establishing the stability conditions for a multiclass network, there is no unified approach to this problem. In this paper, we prove that a queueing network is positive Harris recurrent if the corresponding fluid limit model eventually reaches zero and stays there regardless of the initial system configuration. As an application of the result, we prove that single class networks, multiclass feedforward networks and firstbufferfirstserved preemptive resume discipline in a reentrant line are positive Harris recurrent under the usual traffic condition. AMS 1991 subject classification: Primary 60K25, 90B22; Secondary 60K20, 90B35. Key words and phrases: multiclass queueing networks, Harris positive recurrent, stability, fluid approximation Running title: Stability of mu...
Stability and Convergence of Moments for Multiclass Queueing Networks via Fluid Limit Models
 IEEE Transactions on Automatic Control
, 1995
"... The subject of this paper is open multiclass queueing networks, which are common models of communication networks, and complex manufacturing systems such as wafer fabrication facilities. We provide sufficient conditions for the existence of bounds on longrun average moments of the queue lengths at ..."
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Cited by 113 (37 self)
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The subject of this paper is open multiclass queueing networks, which are common models of communication networks, and complex manufacturing systems such as wafer fabrication facilities. We provide sufficient conditions for the existence of bounds on longrun average moments of the queue lengths at the various stations, and we bound the rate of convergence of the mean queue length to its steady state value. Our work provides a solid foundation for performance analysis either by analytical methods or by simulation. These results are applied to several examples including reentrant lines, generalized Jackson networks, and a general polling model as found in computer networks applications. Keywords: Multiclass queueing networks, ergodicity, general state space Markov processes, polling models, generalized Jackson networks, stability, performance analysis. 1 Introduction The subject of this paper is open multiclass queueing networks, which are models of complex systems such as wafer fabri...
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...
Scheduling Networks of Queues: Heavy Traffic Analysis of a Multistation Network With Controllable Inputs
, 1989
"... Motivated by a factory scheduling problem, we consider the problem of input control (subject to a specified input mix) and priority sequencing in a multistation, multiclass queueing network with general service time distributions and a general routing structure. The objective is to minimize the long ..."
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Cited by 89 (6 self)
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Motivated by a factory scheduling problem, we consider the problem of input control (subject to a specified input mix) and priority sequencing in a multistation, multiclass queueing network with general service time distributions and a general routing structure. The objective is to minimize the longrun expected average number of customers in the system subject to a constraint on the longrun expected average output rate. Under balanced heavy loading conditions, this scheduling problem can be approximated by a control problem involving Brownian motion. Linear programming is used to reduce the
Fluid Approximations And Stability Of Multiclass Queueing Networks: WorkConserving Disciplines
, 1995
"... This paper studies the fluid approximation (also known as the functional strong lawoflargenumbers) and the stability (positive Harris recurrent) for a multiclass queueing network. Both of these are related to the stabilities of a linear fluid model, constructed from the firstorder parameters (i. ..."
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Cited by 86 (9 self)
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This paper studies the fluid approximation (also known as the functional strong lawoflargenumbers) and the stability (positive Harris recurrent) for a multiclass queueing network. Both of these are related to the stabilities of a linear fluid model, constructed from the firstorder parameters (i.e., longrun average arrivals, services and routings) of the queueing network. It is proved that the fluid approximation for the queueing network exists if the corresponding linear fluid model is weakly stable, and that the queueing network is stable if the corresponding linear fluid model is (strongly) stable. Sufficient conditions are found for the stabilities of a linear fluid model. Keywords and phrases: Multiclass queueing networks, fluid models, fluid approximations, stability, positive Harris recurrent, and workconserving service disciplines. Preliminary Versions: September 1993 Revisions: June 1994; September 1994; January 1995 To appear in Annals of Applied Probability AMS 1980 su...
Brownian models of open processing networks: Canonical representation of workload
 Ann. Appl. Probab
, 2003
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Optimization of multiclass queueing networks: Polyhedral and nonlinear characterizations of achievable performance
 ANNALS OF APPLIED PROBABILITY
, 1994
"... We consider open and closed multiclass queueing networks with Poisson arrivals (in open networks), exponentially distributed class dependent service times, and with class dependent deterministic or probabilistic routing. For open networks, the performance objective is to minimize, over all sequen ..."
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Cited by 73 (9 self)
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We consider open and closed multiclass queueing networks with Poisson arrivals (in open networks), exponentially distributed class dependent service times, and with class dependent deterministic or probabilistic routing. For open networks, the performance objective is to minimize, over all sequencing and routing policies, a weighted sum of the expected response times of different classes. Using a powerful technique involving quadratic or higher order potential functions, we propose variants of a method to derive polyhedral and nonlinear spaces which contain the entire set of achievable response times under stable and preemptive scheduling policies. By optimizing over these spaces, we obtain lower bounds on achievable performance. In particular, we obtain a sequence of progressively more complicated nonlinear approximations (relaxations) which are progressively closer to the exact achievable space. In the special case of single station networks (multiclass queues and Klimov's model) and homogenous multiclass networks, our characterization gives exactly the achievable region. Consequently, the proposed method can be viewed as the natural
Maximum pressure policies in stochastic processing networks
, 2005
"... Complex systems like semiconductor wafer fabrication facilities (fabs), networks of data switches, and largescale call centers all demand efficient resource allocation. Deterministic models like linear programs (LP) have been used for capacity planning at both the design and expansion stages of s ..."
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Cited by 69 (6 self)
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Complex systems like semiconductor wafer fabrication facilities (fabs), networks of data switches, and largescale call centers all demand efficient resource allocation. Deterministic models like linear programs (LP) have been used for capacity planning at both the design and expansion stages of such systems. LPbased planning is critical in setting a medium range or longterm goal for many systems, but it does not translate into a daytoday operational policy that must deal with discreteness of jobs and the randomness of the processing environment. A stochastic processing network, advanced by J. Michael Harrison (2000, 2002, 2003), is a system that takes inputs of materials of various kinds and uses various processing resources to produce outputs of materials of various kinds. Such a network provides a powerful abstraction of a wide range of realworld systems. It provides highfidelity stochastic models in diverse economic sectors including manufacturing, service, and information technology. We propose a family of maximum pressure service policies for dynamically allocating service capacities in a stochastic processing network. Under a mild assumption on network structure, we prove that a network operating under a maximum pressure policy achieves maximum throughput predicted by LPs. These policies are semilocal in the sense that each
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 66 (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
Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedrons
 Theory of Probability and Its Applications
, 1995
"... We consider the problem of existence and uniqueness of semimartingale reflecting Brownian motions (SRBM's) in convex polyhedrons. Loosely speaking, such a process has a semimartingale decomposition such that in the interior of the polyhedron the process behaves like a Brownian motion with a con ..."
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Cited by 65 (15 self)
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We consider the problem of existence and uniqueness of semimartingale reflecting Brownian motions (SRBM's) in convex polyhedrons. Loosely speaking, such a process has a semimartingale decomposition such that in the interior of the polyhedron the process behaves like a Brownian motion with a constant drift and covariance matrix, and at each of the (d \Gamma 1)dimensional faces that form the boundary of the polyhedron, the bounded variation part of the process increases in a given direction (constant for any particular face), so as to confine the process to the polyhedron. For historical reasons, this &quot;pushing &quot; at the boundary is called instantaneous reflection. For simple convex polyhedrons, we give a necessary and sufficient condition on the geometric data for the existence and uniqueness of an SRBM. For nonsimple convex polyhedrons, our condition is shown to be sufficient. It is an open question as to whether our condition is also necessary in the nonsimple case. From the uniqueness, it follows that an SRBM defines a strong Markov process. Our results have application to the study of diffusions arising as heavy traffic limits of multiclass queueing networks and in particular, the nonsimple case has application to multiclass fork and join networks. Our proof of weak existence uses a patchwork martingale problem introduced by T. G. Kurtz, whereas uniqueness hinges on an ergodic argument similar to that used by L. M. Taylor and R. J. Williams to prove uniqueness for SRBM's in an orthant.