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34
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 225 (17 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...
Performance and Stability of Communication Networks via Robust Exponential Bounds
 IEEE/ACM Transactions on Networking
, 1993
"... We propose a new way for evaluating the performance of packet switching communication networks under a fixed (session based) routing strategy. Our approach is based on properly bounding the probability distribution functions of the system input processes. The bounds we suggest, which are decaying ex ..."
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Cited by 124 (3 self)
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We propose a new way for evaluating the performance of packet switching communication networks under a fixed (session based) routing strategy. Our approach is based on properly bounding the probability distribution functions of the system input processes. The bounds we suggest, which are decaying exponentials, possess three convenient properties: When the inputs to an isolated network element are all bounded, they result in bounded outputs, and assure that the delays and queues in this element have exponentially decaying distributions; In some network settings bounded inputs result in bounded outputs; Natural traffic processes can be shown to satisfy such bounds. Consequently, our method enables the analysis of various previously intractable setups. We provide sufficient conditions for the stability of such networks, and derive upper bounds for the interesting parameters of network performance. 1 Introduction In this paper we consider data communication networks, and the problem of ev...
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 110 (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...
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 78 (31 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...
Adversarial Queuing Theory
, 2001
"... We consider packet routing when packets are injected continuously into a network. We develop an adversarial theory of queuing aimed at addressing some of the restrictions inherent in probabilistic analysis and queuing theory based on timeinvariant stochastic generation. We examine the stability of ..."
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Cited by 75 (0 self)
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We consider packet routing when packets are injected continuously into a network. We develop an adversarial theory of queuing aimed at addressing some of the restrictions inherent in probabilistic analysis and queuing theory based on timeinvariant stochastic generation. We examine the stability of queuing networks and policies when the arrival process is adversarial, and provide some preliminary results in this direction. Our approach sheds light on various queuing policies in simple networks, and paves the way for a systematic study of queuing with few or no probabilistic assumptions.
Performance Bounds for Queueing Networks and Scheduling Policies
, 1994
"... Except for the class of queueing networks and scheduling policies admitting a product form solution for the steadystate distribution, little is known about the performance of such systems. For example, if the priority of a part depends on its class (e.g., the buffer that the part is located in), t ..."
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Cited by 61 (14 self)
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Except for the class of queueing networks and scheduling policies admitting a product form solution for the steadystate distribution, little is known about the performance of such systems. For example, if the priority of a part depends on its class (e.g., the buffer that the part is located in), then there are no existing results on performance, or even stability. However, in most applications such as manufacturing systems, one has to choose a control or scheduling policy, i.e., a priority discipline, that optimizes a performance objective. In this paper we introduce a new technique for obtaining upper and lower bounds on the performance of Markovian queueing networks and scheduling policies. Assuming stability, and examining the consequence of a steadystate for general quadratic forms, we obtain a set of linear equality constraints on the mean values of certain random variables that determine the performance of the system. Further, the conservation of time and material gives an au...
Exponential and Uniform Ergodicity of Markov Processes
 Ann. Probab
, 1995
"... Geometric convergence of Markov chains in discrete time on a general state has been studied in detail in [15]. Here we develop a similar theory for 'irreducible continuous time processes, and consider the following types of criteria for geometric convergence: (a) the existence of exponentially boun ..."
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Cited by 46 (13 self)
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Geometric convergence of Markov chains in discrete time on a general state has been studied in detail in [15]. Here we develop a similar theory for 'irreducible continuous time processes, and consider the following types of criteria for geometric convergence: (a) the existence of exponentially bounded hitting times on one and then all suitably "small" sets; (b) the existence of "FosterLyapunov" or "drift" conditions for any one and then all skeleton and resolvent chains; (c) the existence of drift conditions on the extended generator e A of the process. We use the identity e AR fi = fi(R fi \Gamma I) connecting the extended generator and the resolvent kernels R fi , to show that, under a suitable aperiodicity assumption, exponential convergence is completely equivalent to any of (a)(c). These conditions yield criteria for exponential convergence of unbounded as well as bounded functions of the chain. They enable us to identify the dependence of the convergence on the initial state ...
A Lyapunov Bound for Solutions of Poisson's Equation
 Ann. Probab
, 1996
"... In this paper we consider /irreducible Markov processes evolving in discrete or continuous time, on a general state space. We develop a Lyapunov function criterion that permits one to obtain explicit bounds on the solution to Poisson's equation and, in particular, obtain conditions under which the ..."
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Cited by 43 (25 self)
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In this paper we consider /irreducible Markov processes evolving in discrete or continuous time, on a general state space. We develop a Lyapunov function criterion that permits one to obtain explicit bounds on the solution to Poisson's equation and, in particular, obtain conditions under which the solution is square integrable. These results are applied to obtain sufficient conditions that guarantee the validity of a functional central limit theorem for the Markov process. As a second consequence of the bounds obtained, a perturbation theory for Markov processes is developed which gives conditions under which both the solution to Poisson's equation and the invariant probability for the process are continuous functions of its transition kernel. The techniques are illustrated with applications to queueing theory and autoregressive processes. AMS subject classifications: 68M20, 60J10 Running head: Poisson's Equation Keywords: Markov chain, Markov process, Poisson's equation, Lyapunov f...
Stability and Instability of Fluid Models for ReEntrant Lines
, 1996
"... Reentrant lines can be used to model complex manufacturing systems such as wafer fabrication facilities. As the first step to the optimal or nearoptimal scheduling of such lines, we investigate their stability. In light of a recent theorem of Dai (1995) which states that a scheduling policy is sta ..."
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Cited by 38 (11 self)
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Reentrant lines can be used to model complex manufacturing systems such as wafer fabrication facilities. As the first step to the optimal or nearoptimal scheduling of such lines, we investigate their stability. In light of a recent theorem of Dai (1995) which states that a scheduling policy is stable if the corresponding fluid model is stable, we study the stability and instability of fluid models. To do this we utilize piecewise linear Lyapunov functions. We establish stability of FirstBufferFirstServed (FBFS) and LastBufferFirstServed (LBFS) disciplines in all reentrant lines, and of all workconserving disciplines in any three buffer reentrant lines. For the four buffer network of Lu and Kumar we characterize the stability region of the Lu and Kumar policy, and show that it is also the global stability region for this network. We also study stability and instability of Kellytype networks. In particular, we show that not all workconserving policies are stable for such netw...
Piecewise Linear Test Functions for Stability and Instability of Queueing Networks
 Queueing Systems
"... We develop the use of piecewise linear test functions for the analysis of stability of multiclass queueing networks and their associated fluid limit models. It is found that if an associated LP admits a positive solution, then a Lyapunov function exists. This implies that the fluid limit model is ..."
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Cited by 36 (3 self)
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We develop the use of piecewise linear test functions for the analysis of stability of multiclass queueing networks and their associated fluid limit models. It is found that if an associated LP admits a positive solution, then a Lyapunov function exists. This implies that the fluid limit model is stable and hence that the network model is positive Harris recurrent with a finite polynomial moment. Also, it is found that if a particular LP admits a solution, then the network model is transient. Running head : Stability and Instability of Queueing Networks Keywords : Multiclass queueing networks, ergodicity, stability, performance analysis. 1 Introduction It has generally been taken for granted in queueing theory that stability of a network is guaranteed so long as the overall traffic intensity is less than unity and in recent years there has been much analysis which supports this belief for special classes of systems, such as single class queueing networks (see Borovkov [2], Sig...