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22
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 361 (29 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...
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 ..."
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Cited by 76 (12 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 ...
Validity of heavy traffic steadystate approximations in open queueing networks
, 2006
"... We consider a single class open queueing network, also known as a generalized Jackson network (GJN). A classical result in heavytraffic theory asserts that the sequence of normalized queue length processes of the GJN converge weakly to a reflected Brownian motion (RBM) in the orthant, as the traffic ..."
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Cited by 45 (7 self)
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We consider a single class open queueing network, also known as a generalized Jackson network (GJN). A classical result in heavytraffic theory asserts that the sequence of normalized queue length processes of the GJN converge weakly to a reflected Brownian motion (RBM) in the orthant, as the traffic intensity approaches unity. However, barring simple instances, it is still not known whether the stationary distribution of RBM provides a valid approximation for the steadystate of the original network. In this paper we resolve this open problem by proving that the rescaled stationary distribution of the GJN converges to the stationary distribution of the RBM, thus validating a socalled “interchangeoflimits” for this class of networks. Our method of proof involves a combination of Lyapunov function techniques, strong approximations and tail probability bounds that yield tightness of the sequence of stationary distributions of the GJN.
Blocking a Transition in a Free Choice Net, and what it tells about its throughput
 in "Journal of Computer and System Sciences
"... In a live and bounded Free Choice Petri net, pick a nonconflicting transition. Then there exists a unique reachable marking in which no transition is enabled except the selected one. For a routed live and bounded Free Choice net, this property is true for any transition of the net. Consider now a l ..."
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Cited by 17 (5 self)
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In a live and bounded Free Choice Petri net, pick a nonconflicting transition. Then there exists a unique reachable marking in which no transition is enabled except the selected one. For a routed live and bounded Free Choice net, this property is true for any transition of the net. Consider now a live and bounded stochastic routed Free Choice net, and assume that the routings and the firing times are independent and identically distributed. Using the above results, we prove the existence of asymptotic firing throughputs for all transitions in the net. Furthermore, the vector of the throughputs at the different transitions is explicitly computable up to a multiplicative constant. 1.
Global Stability of TwoStation Queueing Networks
, 1996
"... This paper summarizes results of Dai and Vande Vate [15, 14] characterizing explicitly, in terms of the mean service times and average arrival rates, the global pathwise stability region of twostation open multiclass queueing networks with very general arrival and service processes. The conditions ..."
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Cited by 11 (3 self)
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This paper summarizes results of Dai and Vande Vate [15, 14] characterizing explicitly, in terms of the mean service times and average arrival rates, the global pathwise stability region of twostation open multiclass queueing networks with very general arrival and service processes. The conditions for pathwise global stability arise from two intuitively appealing phenomena: virtual stations and push starts. These phenomena shed light on the sources of bottlenecks in complicated queueing networks like those that arise in wafer fabrication facilities. We show that a twostation open multiclass queueing network is globally pathwise stable if and only if the corresponding fluid model is globally weakly stable. We further show that a twostation fluid model is globally (strongly) stable if and only if the average service times are in the interior of the global weak stability region. As a consequence, under stronger distributional assumptions on the arrival and service processes, the queue...
Stationary Ergodic Jackson Networks: Results and CounterExamples
, 1996
"... This paper gives a survey of recent results on generalized Jackson networks, where classical exponential or i.i.d. assumptions on services and routings are replaced by stationary and ergodic assumptions. We first show that the most basic features of the network may exhibit unexpected behavior. Sever ..."
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Cited by 8 (1 self)
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This paper gives a survey of recent results on generalized Jackson networks, where classical exponential or i.i.d. assumptions on services and routings are replaced by stationary and ergodic assumptions. We first show that the most basic features of the network may exhibit unexpected behavior. Several probabilistic properties are then discussed, including a strong law of large numbers for the number of events in the stations, the existence, uniqueness and representation of stationary regimes for queue size and workload.
The stability of jointheshortestqueue models with general input and output processes
, 2005
"... The paper establishes necessary and sufficient conditions for the stability of different jointheshortestqueue models including the loadbalanced network with general input and output processes. It is shown that the necessary and sufficient condition for the stability of the loadbalanced networ ..."
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Cited by 4 (4 self)
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The paper establishes necessary and sufficient conditions for the stability of different jointheshortestqueue models including the loadbalanced network with general input and output processes. It is shown that the necessary and sufficient condition for the stability of the loadbalanced network is related to the solution of the linear programming problem precisely formulated in the paper. It is proved that if the minimum of the objective function of that linear programming problem is less than 1, then the associated loadbalanced network is stable.
STABILITY OF JACKSONTYPE QUEUEING NETWORKS, I
, 1999
"... This paper gives a pathwise construction of Jacksontype queueing networks allowing the derivation of stability and convergence theorems under general probabilistic assumptions on the driving sequences; namely, it is only assumed that the input process, the service sequences and the routing mechanis ..."
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Cited by 4 (0 self)
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This paper gives a pathwise construction of Jacksontype queueing networks allowing the derivation of stability and convergence theorems under general probabilistic assumptions on the driving sequences; namely, it is only assumed that the input process, the service sequences and the routing mechanism are jointly stationary and ergodic in a sense that is made precise in the paper. The main tools for these results are the subadditive ergodic theorem, which is used to derive a strong law of large numbers, and basic theorems on monotone stochastic recursive sequences. The techniques which are proposed here apply to other and more general classes of discrete event systems, like Petri nets or GSMP’s. The paper also provides new results on the Jacksontype networks with i.i.d. driving sequences which were studied in the past.
Products of irreducible random matrices . . .
, 1995
"... We consider the recursive equation “x(n + 1) = A(n) ⊗ x(n) ” where x(n + 1) and x(n) are R kvalued vectors and A(n) is an irreducible random matrix of size k × k. The matrixvector multiplication in the (max,+) algebra is defined by (A(n) ⊗ x(n))i = maxj(Aij(n) + xj(n)). This type of equation ..."
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Cited by 3 (0 self)
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We consider the recursive equation “x(n + 1) = A(n) ⊗ x(n) ” where x(n + 1) and x(n) are R kvalued vectors and A(n) is an irreducible random matrix of size k × k. The matrixvector multiplication in the (max,+) algebra is defined by (A(n) ⊗ x(n))i = maxj(Aij(n) + xj(n)). This type of equation can be used to represent the evolution of Stochastic Event Graphs which include cyclic Jackson Networks, some manufacturing models and models with general blocking (such as Kanban). Let us assume that the sequence {A(n), n ∈ N} is i.i.d or more generally stationary and ergodic. The main result of the paper states that the system couples in finite time with a unique stationary regime if and only if there exists a set of matrices C such that