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Performance of multiclass Markovian queueing networks via piecewise linear Lyapunov functions
 Annals of Applied Probability
, 2001
"... We study the distribution of steadystate queue lengths in multiclass queueing networks under a stable policy. We propose a general methodology based on Lyapunov functions, for the performance analysis of infinite state Markov chains and apply it specifically to Markovian multiclass queueing network ..."
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Cited by 22 (3 self)
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We study the distribution of steadystate queue lengths in multiclass queueing networks under a stable policy. We propose a general methodology based on Lyapunov functions, for the performance analysis of infinite state Markov chains and apply it specifically to Markovian multiclass queueing networks. We establish a deeper connection between stability and performance of such networks by showing that if there exist linear and piecewise linear Lyapunov functions that show stability, then these Lyapunov functions can be used to establish geometric type lower and upper bounds on the tail probabilities, and thus bounds on the expectation of the queue lengths. As an example of our results, for a reentrant line queueing network with two processing stations operating under a workconserving policy we showthat E[L] =O 1 (1; ) 2 � where L is the total number ofcustomers in the system, and is the maximal actual or virtual traffic intensity inthenetwork. In a Markovian setting, this extends a recent result by Daiand Vande Vate, which states that a reentrant line queueing network with two stations is globally stable if < 1: We also present several results on the
Stabilizing Queueing Networks with Setups
, 2002
"... For multiclass queueing networks, dispatch policies govern the assignment of servers to the jobs they process. Production policies perform the analogous task for queueing networks whose servers are subject to switchover delays or setups, a model we refer to as setup networks. ..."
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Cited by 5 (0 self)
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For multiclass queueing networks, dispatch policies govern the assignment of servers to the jobs they process. Production policies perform the analogous task for queueing networks whose servers are subject to switchover delays or setups, a model we refer to as setup networks.
Existence condition for the diffusion approximations of multiclass priority queueing networks
 Faculty of Commerce and Business Administration, UBC
, 2001
"... In this paper, we extend the work of Chen and Zhang (2000b) and establish a new sufficient condition for the existence of the (conventional) diffusion approximation for multiclass queueing networks under priority service disciplines. This sufficient condition relates to the weak stability of the flu ..."
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Cited by 5 (1 self)
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In this paper, we extend the work of Chen and Zhang (2000b) and establish a new sufficient condition for the existence of the (conventional) diffusion approximation for multiclass queueing networks under priority service disciplines. This sufficient condition relates to the weak stability of the fluid networks and the stability of the high priority classes of the fluid networks that correspond to the queueing networks under consideration. Using this sufficient condition, we prove the existence of the diffusion approximation for the lastbufferfirstserved reentrant lines. We also study a threestation network example, and observe that the diffusion approximation may not exist, even if the “proposed” limiting semimartingale reflected Brownian motion (SRBM) exists.
Lyapunov Method for the Stability of Fluid Networks
 Operations Research Letters
, 2000
"... One of the primary tools in establishing the stability of a fluid network is to construct a Lyapunov function. In this paper, we establish the sufficiency in the use of a Lyapunov function. Specifically, we show that a necessary and sufficient condition for the stability of a generic fluid network ( ..."
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Cited by 3 (3 self)
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One of the primary tools in establishing the stability of a fluid network is to construct a Lyapunov function. In this paper, we establish the sufficiency in the use of a Lyapunov function. Specifically, we show that a necessary and sufficient condition for the stability of a generic fluid network (GFN) is the existence of a Lyapunov function for its fluid level process. Then by applying this result to various specific fluid networks, including a fluid network under all workconserving service disciplines, a fluid network under a priority service discipline, and a fluid network under a FIFO service discipline, we establish the existence of a Lyapunov function for their fluid level processes is a necessary and sufficient condition for their stabilities. The result is also applied to various fluid limit models and a linear Skorohod problem.
Stability of fluid networks with proportional routing. Queueing Systems: Theory and Applications
 in Russian); Ann. Phys. (N.Y
, 2001
"... In this paper we investigate the stability of a class of twostation multiclass fluid networks with proportional routing. We obtain explicit necessary and sufficient conditions for the global stability of such networks. By virtue of a stability theorem of Dai [14], these results also give sufficient ..."
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Cited by 2 (1 self)
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In this paper we investigate the stability of a class of twostation multiclass fluid networks with proportional routing. We obtain explicit necessary and sufficient conditions for the global stability of such networks. By virtue of a stability theorem of Dai [14], these results also give sufficient conditions for the stability of a class of related multiclass queueing networks. Our study extends the results of Dai and VandeVate [19], who provided a similar analysis for fluid models without proportional routing, which arise from queueing networks with deterministic routing. The models we investigate include fluid models which arise from a large class of twostation queueing networks with probabilistic routing. The stability conditions derived turn out to have an appealing intuitive interpretation in terms of virtual stations and pushstarts which were introduced in earlier work on multiclass networks.
Printed in U.S.A. ON DECIDING STABILITYOF CONSTRAINED HOMOGENEOUS RANDOM WALKS AND QUEUEING SYSTEMS
"... We investigate stability of scheduling policies in queueing systems. To this day no algorithmic characterization exists for checking stability of a given policy in a given queueing system. In this paper we introduce a certain generalized priority policy and prove that the stability of this policy is ..."
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We investigate stability of scheduling policies in queueing systems. To this day no algorithmic characterization exists for checking stability of a given policy in a given queueing system. In this paper we introduce a certain generalized priority policy and prove that the stability of this policy is algorithmically undecidable. We also prove that stability of a homogeneous random walk in � d is undecidable. Finally, we show that the problem of computing a fluid limit of a queueing system or of a constrained homogeneous random walk is undecidable. To the best of our knowledge these are the first undecidability results in the area of stability of queueing systems and random walks in �d +. We conjecture that stability of common policies like FirstInFirstOut and priority policy is also an undecidable problem. 1. Introduction. We