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120
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 352 (28 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 Of Queueing Networks And Scheduling Policies
 IEEE Transactions on Automatic Control
, 1995
"... Usually, the stability of queueing networks is established by explicitly determining the invariant distribution. However, outside of the narrow class of queueing networks possessing a product form solution, such explicit solutions are rare, and consequently little is known concerning stability too. ..."
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Cited by 130 (16 self)
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Usually, the stability of queueing networks is established by explicitly determining the invariant distribution. However, outside of the narrow class of queueing networks possessing a product form solution, such explicit solutions are rare, and consequently little is known concerning stability too. We develop here a programmatic procedure for establishing the stability of queueing networks and scheduling policies. The method uses linear or nonlinear programming to determine what is an appropriate quadratic functional to use as a Lyapunov function. If the underlying system is Markovian, our method establishes not only positive recurrence and the existence of a steadystate probability distribution, but also the geometric convergence of an exponential moment. We illustrate this method on several example problems. For an example of an open reentrant line, we show that all stationary nonidling policies are stable for all load factors less than one. This includes the well known First Com...
Contact centers with a callback option and realtime delay information
 Operations Research
, 2004
"... doi 10.1287/opre.1030.0088 ..."
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 66 (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.
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.
Departures from Many Queues in Series
, 1990
"... We consider a series of n singleserver queues, each with unlimited waiting space and the firstin firstout service discipline. Initially, the system is empty; then k customers are placed in the first queue. The service times of all the customers at all the queues are i.i.d. with a general distribu ..."
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Cited by 57 (5 self)
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We consider a series of n singleserver queues, each with unlimited waiting space and the firstin firstout service discipline. Initially, the system is empty; then k customers are placed in the first queue. The service times of all the customers at all the queues are i.i.d. with a general distribution. We are interested in the time D(k, n) required for all k customers to complete service from all n queues. In particular, we investigate the limiting behavior of D(k, n) as n and/or k . There is a duality implying that D(k, n) is distributed the same as D(n , k) so that results for large n are equivalent to results for large k. A previous heavytraffic limit theorem implies that D(k, n) satisfies an invariance principle as n , converging after normalization to a functional of kdimensional Brownian motion. We use the subadditive ergodic theorem and a strong approximation to describe the limiting behavior of D(k n , n) where k n as n . The case of k n = xn corresponds to a hydrodyna...
Sequencing and routing in multiclass queueing networks part I: Feedback regulation
 SIAM J. Control Optim
"... Abstract. Part II continues the development of policy synthesis techniques for multiclass queueing networks based upon a linear fluid model. The following are shown: (i) A relaxation of the fluid model based on workload leads to an optimization problem of lower dimension. An analogous workloadrelax ..."
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Cited by 55 (12 self)
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Abstract. Part II continues the development of policy synthesis techniques for multiclass queueing networks based upon a linear fluid model. The following are shown: (i) A relaxation of the fluid model based on workload leads to an optimization problem of lower dimension. An analogous workloadrelaxation is introduced for the stochastic model. These relaxed control problems admit pointwise optimal solutions in many instances. (ii) A translation to the original fluid model is almost optimal, with vanishing relative error as the networkload ρ approaches one. It is pointwise optimal after a short transient period, provided a pointwise optimal solution exists for the relaxed control problem. (iii) A translation of the optimal policy for the fluid model provides a policy for the stochastic networkmodel that is almost optimal in heavy traffic, over all solutions to the relaxed stochastic model, again with vanishing relative error. The regret is of order  log(1 − ρ).
Fractal traffic: measurements, modelling and performance evaluation
 in Proc. IEEE INFOCOM '95
, 1995
"... Observations of both Ethernet traffic and variable bit rate (VBR) video traffic have demonstrated that these traffics exhibit “selfsimilarity ” and/or infinit l e asymptotic index of dispersion for counts (IDC). We report here on measurements of traffic an a commercialpublic broadband network where ..."
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Cited by 54 (9 self)
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Observations of both Ethernet traffic and variable bit rate (VBR) video traffic have demonstrated that these traffics exhibit “selfsimilarity ” and/or infinit l e asymptotic index of dispersion for counts (IDC). We report here on measurements of traffic an a commercialpublic broadband network where similar characteristics have been observed. For the purpose of analysis and dimensioning of the central links of an ATM network we analyse in this paper the performance of a single server queue fed by Gaussian trafic with infjlnite IDC. The analysis leads to an approximation fo,r the performance of a queue in which the arriving trafic is “fractal ” Gaussian and consequently where there does not exist a dominant negativeexponential tail. The term ufractal ” is used here in the sense thtzt the autocovariance of the traffic exhibits selfsimilarity, that is to say, where the autocovariance of an aggregate of the trafic is the same, or asymptotically the same for large time lags, as the original traffic. We are not concerned with proving or exploiting this selfsimilarity property as such, but only with performancle analysis techniques which are effective for such processes. In order to be able to test the performance analysis formulae, we show that trafic with the same arutocovariawe as measured an a real network over a wide range of lags (sufficiently wide a range for the traffic to be equivalent from the point of view of queueing performance) can be generated as a mixture of two Gaussian AR(1) processes. In this way we demonstrate that the analytic performance formulae are accurate. 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...
Reflected Brownian Motion in an Orthant: Numerical Methods for SteadyState Analysis
 Annals of Applied Probability
, 1992
"... This paper is concerned with a class of multidimensional diffusion processes, variously known as reflected Brownian motions, regulated Brownian motions, or just RBM's, that arise as approximate models of queueing networks. We develop an algorithm for numerical analysis of a semimartingale RBM w ..."
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Cited by 49 (13 self)
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This paper is concerned with a class of multidimensional diffusion processes, variously known as reflected Brownian motions, regulated Brownian motions, or just RBM's, that arise as approximate models of queueing networks. We develop an algorithm for numerical analysis of a semimartingale RBM with state space S = R d + (the nonnegative orthant of ddimensional Euclidean space). This algorithm lies at the heart of the QNET method [13] for approximate twomoment analysis of open queueing networks. KEY WORDS: Brownian system model, reflected Brownian motion, stationary distribution, numerical analysis, open queueing networks, performance analysis Contents 1. Introduction 1 2. Definitions and Preliminaries 3 3. The Basic Adjoint Relationship 6 4. An Algorithm 9 5. Choosing a Reference Density and fHng 12 6. Numerical Comparisons 16 7. Analysis of an Illustrative Queueing Network 22 AMS 1980 Subject Classification: primary 60J70, 60K30, 65U05; secondary 65P05, 68M20. Abbreviated titl...