Results 1  10
of
29
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 ..."
Abstract

Cited by 361 (29 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 113 (37 self)
 Add to MetaCart
(Show Context)
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...
Fluid Approximations And Stability Of Multiclass Queueing Networks: WorkConserving Disciplines
, 1995
"... This paper studies the fluid approximation (also known as the functional strong lawoflargenumbers) and the stability (positive Harris recurrent) for a multiclass queueing network. Both of these are related to the stabilities of a linear fluid model, constructed from the firstorder parameters (i. ..."
Abstract

Cited by 86 (9 self)
 Add to MetaCart
This paper studies the fluid approximation (also known as the functional strong lawoflargenumbers) and the stability (positive Harris recurrent) for a multiclass queueing network. Both of these are related to the stabilities of a linear fluid model, constructed from the firstorder parameters (i.e., longrun average arrivals, services and routings) of the queueing network. It is proved that the fluid approximation for the queueing network exists if the corresponding linear fluid model is weakly stable, and that the queueing network is stable if the corresponding linear fluid model is (strongly) stable. Sufficient conditions are found for the stabilities of a linear fluid model. Keywords and phrases: Multiclass queueing networks, fluid models, fluid approximations, stability, positive Harris recurrent, and workconserving service disciplines. Preliminary Versions: September 1993 Revisions: June 1994; September 1994; January 1995 To appear in Annals of Applied Probability AMS 1980 su...
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 ..."
Abstract

Cited by 63 (18 self)
 Add to MetaCart
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...
Stability of queueing networks
 Lecture Notes in Mathematics
, 2006
"... Milan Paris Tokyo2 This manuscript will appear as Springer Lecture Notes in Mathematics 1950, École d ’ Été de Probabilités de SaintFlour XXXVI2006. ..."
Abstract

Cited by 45 (1 self)
 Add to MetaCart
Milan Paris Tokyo2 This manuscript will appear as Springer Lecture Notes in Mathematics 1950, École d ’ Été de Probabilités de SaintFlour XXXVI2006.
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 ..."
Abstract

Cited by 45 (7 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 42 (3 self)
 Add to MetaCart
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...
Stability conditions for multiclass fluid queueing networks
 IEEE Transactions on Automatic Control
, 2002
"... We find necessary and sufficient conditions for the stability of all workconserving policies for multiclass fluid queueing networks with two stations. Furthermore, we find new sufficient conditions for the stability of multiclass queueing networks involving any number of stations and conjecture tha ..."
Abstract

Cited by 23 (11 self)
 Add to MetaCart
(Show Context)
We find necessary and sufficient conditions for the stability of all workconserving policies for multiclass fluid queueing networks with two stations. Furthermore, we find new sufficient conditions for the stability of multiclass queueing networks involving any number of stations and conjecture that these conditions are also necessary. Previous research had identified sufficient conditions through the use of a particular class (monotone piecewise linear convex) potential functions. We show that for twostation systems it is not possible for this class of potential function to give the new (sharp) conditions. 1
Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic
 Mathematics of Operations Research
"... In a recent paper [5] it was shown that under suitable conditions stationary distributions of the (scaled) queue lengths process for a generalized Jackson network converge to the stationary distribution of the associated reflected Brownian motion in the heavy traffic limit. The proof relied on certa ..."
Abstract

Cited by 18 (3 self)
 Add to MetaCart
(Show Context)
In a recent paper [5] it was shown that under suitable conditions stationary distributions of the (scaled) queue lengths process for a generalized Jackson network converge to the stationary distribution of the associated reflected Brownian motion in the heavy traffic limit. The proof relied on certain exponential integrability assumptions on the primitives of the network. In this note we show that the above result holds under much weaker integrability conditions. We provide an alternative proof of this result making (in addition to natural heavy traffic and stability assumptions) only standard independence and square integrability assumptions on the network primitives that are commonly used in heavy traffic analysis. Furthermore, under additional integrability conditions we establish convergence of moments of stationary distributions.
On the stability of open networks: an unified approach by stochastic dominance
 QUEUEING SYSTEMS
, 1994
"... Using stochastic dominance, in this paper we provide a new characterization of point processes. This characterization leads to a unified proof for various stability results of open Jackson networks where service times are i.i.d. with a general distribution, external interarrival times are i.i.d. wit ..."
Abstract

Cited by 17 (4 self)
 Add to MetaCart
Using stochastic dominance, in this paper we provide a new characterization of point processes. This characterization leads to a unified proof for various stability results of open Jackson networks where service times are i.i.d. with a general distribution, external interarrival times are i.i.d. with a general distribution and the routing is Bernoulli. We show that if the traffic condition is satisfied, i.e., the input rate is smaller than the service rate at each queue, then the queue length process (the number of customers at each queue) is tight. Under the traffic condition, the p th moment of the queue length process is bounded for all t if the p+1 th moment of the service times at all queues are nite. If, furthermore, the moment generating functions of the service times at all queues exist, then all the moments of the queue length process are bounded for all t. When the interarrival times are unbounded and nonlattice (resp. spreadout), the queue lengths and the remaining service times converge in distribution (resp. in total variation) to a steady state. Also, the moments converge if the corresponding moment conditions are satisfied.