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27
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 34 (10 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).
Stability of Multiclass Queueing Networks Under Priority Service Disciplines
 Mathematics of Operations Research
, 1996
"... In this paper, we establish a sufficient condition for the stability of a multiclass fluid network and queueing network under priority service disciplines. The sufficient condition is based on the existence of a linear Lyapunov function, and it is stated in terms of the feasibility of a set of inequ ..."
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Cited by 31 (9 self)
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In this paper, we establish a sufficient condition for the stability of a multiclass fluid network and queueing network under priority service disciplines. The sufficient condition is based on the existence of a linear Lyapunov function, and it is stated in terms of the feasibility of a set of inequalities that are defined by network parameters. In all the networks we have tested, this sufficient condition actually gives a necessary and sufficient condition for their stability.
Stability of Adversarial Queues via Fluid Models
 In Proceedings of the 39th Annual Symposium on Foundations of Computer Science
, 1998
"... The subject of this paper is stability properties of adversarial queueing networks. Such queueing systems are used to model packet switch communication networks, in which packets are generated and routed dynamically, and have become a subject of research focus recently. Adversarial queueing networks ..."
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Cited by 30 (3 self)
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The subject of this paper is stability properties of adversarial queueing networks. Such queueing systems are used to model packet switch communication networks, in which packets are generated and routed dynamically, and have become a subject of research focus recently. Adversarial queueing networks are defined to be stable, if the number of packets stays bounded over time. A central question is determining which adversarial queueing networks are stable, when an arbitrary greedy packet routing policy is implemented. In this paper we show how stability of a queueing network can be determined by considering an associated fluid models. Our main result is that the stability of the fluid model implies the stability of an underlying adversarial queueing network. This opens an opportunity for analyzing stability of adversarial networks, using established stability methods from continuous time processes, for example, the method of Lyapunov function or trajectory decomposition. We demonstrate t...
Simple Necessary and Sufficient Conditions for the Stability of Constrained Processes
, 2000
"... In recent years a new approach has emerged for analyzing the stability properties of constrained stochastic processes. In this approach, one associates with the stochastic model a deterministic model (or a family of deterministic models), and, under appropriate conditions, stability of the stochasti ..."
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Cited by 20 (11 self)
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In recent years a new approach has emerged for analyzing the stability properties of constrained stochastic processes. In this approach, one associates with the stochastic model a deterministic model (or a family of deterministic models), and, under appropriate conditions, stability of the stochastic model follows if all solutions of the deterministic model are attracted to the origin. In the present work we show that a rather sharp characterization for the stability of the deterministic model is possible when it can be represented in terms of what we call a "regular" Skorokhod Map. Let G be a convex polyhedral cone with vertex at the origin given as the intersection of half spaces {G i , i = 1, ..., N}, where n i and d i denote the inward normal and direction of constraint associated with G i . Suppose that the Skorokhod Problem defined by the data {(n i , d i ), i = 1, ..., N} is regular. Under these conditions, the deterministic model mentioned above will correspond to a law of...
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 18 (1 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.
Necessary Conditions for Global Stability of Multiclass Queueing Networks
 Operations Research Letters
, 1996
"... In this paper, we obtain necessary conditions for the global stability of a dstation multiclass queueing network. The conditions are given explicitly in terms of the average service and arrival rates of the network. Although these conditions are in general not sufficient for d ? 2, they may still h ..."
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Cited by 15 (3 self)
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In this paper, we obtain necessary conditions for the global stability of a dstation multiclass queueing network. The conditions are given explicitly in terms of the average service and arrival rates of the network. Although these conditions are in general not sufficient for d ? 2, they may still highlight hidden bottlenecks in complex manufacturing systems such as wafer fabrication processes. Keywords: multiclass queueing networks; stability; virtual station 1 Introduction Recently, there has been renewed interest in the stability of multiclass queueing networks. This is due primarily to two factors: a series of counterexamples demonstrating that the station traffic intensities may not be sufficient to determine the stability region, and insight into the close relationship between discrete queueing networks and their associated fluid models. In the first area, Kumar and Seidman [15], Lu and Kumar [17], and Rybko and Stolyar [19] gave examples of queueing networks which are unstable...
On Positive Recurrence of Constrained Diffusion Processes
"... this paper we consider the stability properties of constrained di#usion processes when both the drift and the di#usion coe#cients may be state dependent. Let G ..."
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Cited by 15 (9 self)
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this paper we consider the stability properties of constrained di#usion processes when both the drift and the di#usion coe#cients may be state dependent. Let G
The Stability of TwoStation MultiType Fluid Networks
 Operations Research
, 1997
"... This paper studies the uid models of twostation multiclass queueing networks with deterministic routing. A uid model is globally stable if the uid network eventually empties under each nonidling dispatch policy. We explicitly characterize the global stability region in terms of the arrival and serv ..."
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Cited by 12 (5 self)
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This paper studies the uid models of twostation multiclass queueing networks with deterministic routing. A uid model is globally stable if the uid network eventually empties under each nonidling dispatch policy. We explicitly characterize the global stability region in terms of the arrival and service rates. We show that the global stability region is dened by the nominal workload conditions and the \virtual workload conditions" and we introduce two intuitively appealing phenomena: virtual stations and push starts, that explain the virtual workload conditions. When any of the workload conditions is violated, we construct a uid solution that cycles to innity, showing that the uid network is unstable. When all of the workload conditions are satised, we solve a network ow problem to nd the coecients of a piecewise linear Lyapunov function. The Lyapunov function decreases to zero proving that the uid level eventually reaches zero under any nonidling dispatch policy. Under certain assumptions on the interarrival and service time distributions, a queueing network is stable or positive Harris recurrent if the corresponding uid network is stable. Thus, the workload conditions are sucient to ensure the global stability of twostation multiclass queueing networks with deterministic routing. To appear in Operations Research
Using Fluid Models to Prove Stability of Adversarial Queueing Networks
 IEEE Transactions on Automatic Control
, 1999
"... A digital communication network can be modeled as an adversarial queueing network. An adversarial queueing network is defined to be stable if the number of packets stays bounded over time. A central question is to determine which adversarial queueing networks are stable under every workconserving pa ..."
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Cited by 10 (2 self)
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A digital communication network can be modeled as an adversarial queueing network. An adversarial queueing network is defined to be stable if the number of packets stays bounded over time. A central question is to determine which adversarial queueing networks are stable under every workconserving packet routing policy. Our main result is that stability of an adversarial queueing network is implied by stability of an associated fluid queueing network.