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Some results for large closed queueing networks with and without bottleneck: Up and downcrossings approach
 Queueing Systems
, 2001
"... Abstract. The paper provides the up and downcrossing method to study the asymptotic behavior of queuelength and waiting time in closed Jacksontype queueing networks. These queueing networks consist of central node (hub) and k singleserver satellite stations. The case of infinite server hub with ..."
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Abstract. The paper provides the up and downcrossing method to study the asymptotic behavior of queuelength and waiting time in closed Jacksontype queueing networks. These queueing networks consist of central node (hub) and k singleserver satellite stations. The case of infinite server hub with exponentially distributed service times is considered in the first section to demonstrate the up and downcrossing approach to such kind of problems and help to understand the readers the main idea of the method. The main results of the paper are related to the case of singleserver hub with generally distributed service times depending on queuelength. Assuming that the first k − 1 satellite nodes operate in light usage regime, we consider three cases concerning the kth satellite node. They are the light usage regime and limiting cases for the moderate usage regime and heavy usage regime. The results related to light usage regime show that, as the number of customers in network increases to infinity, the network is decomposed to independent singleserver queueing systems. In the limiting cases of moderate usage regime, the diffusion approximations of queuelength and waiting time processes are obtained. In the case of heavy usage regime it is shown that the joint limiting nonstationary queuelengths distribution at the first k − 1 satellite nodes is represented in the product form and coincides with the product of stationary GI/M/1 queuelength distributions with parameters depending on time.
The Finite Element Method for Computing the Stationary Distribution of an SRBM in a Hypercube with Applications to Finite Buffer Queueing Networks
, 2002
"... This paper proposes an algorithm, referred to as BNA/FM (Brownian network analyzer with finite element method), for computing the stationary distribution of a semimartingale reflecting Brownian motion (SRBM) in a hypercube. The SRBM serves as an approximate model of queueing networks with finite buf ..."
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Cited by 15 (3 self)
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This paper proposes an algorithm, referred to as BNA/FM (Brownian network analyzer with finite element method), for computing the stationary distribution of a semimartingale reflecting Brownian motion (SRBM) in a hypercube. The SRBM serves as an approximate model of queueing networks with finite buffers. Our BNA/FM algorithm is based on finite element method and an extension of a generic algorithm developed by Dai and Harrison (1991). It uses piecewise polynomials to form an approximate subspace of an infinite dimensional functional space. The BNA/FM algorithm is shown to produce good estimates for stationary probabilities, in addition to stationary moments. This is in contrast to BNA/SM (Brownian network analyzer with spectral method) of Dai and Harrison (1991), where global polynomials are used to form the approximate subspace and it sometime fails to produce meaningful estimates of these stationary probabilities. Extensive computational experiences from our implementation are reported that may be useful for future numerical research on SRBMs. A threestation tandem network with finite buffers are presented to illustrate the effectiveness of the Brownian approximation model and our BNA/FM algorithm.
Diffusion Approximations for Some Multiclass Queueing Networks with FIFO Service Disciplines
 Mathematics of Operations Research
, 1997
"... The diffusion approximation is proved for a class of multiclass queueing networks under FIFO service disciplines. In addition to the usual assumptions for a heavy traffic limit theorem, a key condition that characterizes this class is that a J \Theta J matrix G, known as the workload contents matrix ..."
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Cited by 7 (2 self)
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The diffusion approximation is proved for a class of multiclass queueing networks under FIFO service disciplines. In addition to the usual assumptions for a heavy traffic limit theorem, a key condition that characterizes this class is that a J \Theta J matrix G, known as the workload contents matrix, has a spectral radius less than unity, where J represents the number of service stations. The (j; `)th component of matrix G can be interpreted as the amount of future work for station j that is embodied in per unit of immediate work at station ` at time t. This class includes RybkoStolyar network with FIFO service discipline as a special case. The result extends existing diffusion limiting theorems to nonfeedforward multiclass queueing networks. In establishing the diffusion limit theorem, a new approach is taken. The traditional approach is based on an oblique reflection mapping, but such a mapping is not welldefined for the network under consideration. Our approach takes two steps: f...
Strong Approximations for Multiclass Feedforward Queueing Networks
 Annals of Applied Probability
"... This paper derives the strong approximation for a multiclass queueing network, where jobs after service completion can only move to a downstream service station. Job classes are partitioned into groups. Within a group, jobs are served in the order of arrival, i.e., a rstinrstout (FIFO) discipline ..."
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Cited by 7 (2 self)
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This paper derives the strong approximation for a multiclass queueing network, where jobs after service completion can only move to a downstream service station. Job classes are partitioned into groups. Within a group, jobs are served in the order of arrival, i.e., a rstinrstout (FIFO) discipline is in force, and among groups, jobs are served under a preassigned preemptive priority discipline. We obtain the strong approximation for the network, through an inductive application of an inputoutput analysis for a single station queue. Specically, we show that if the input data (i.e., the arrival and the service processes) satisfy an approximation (such as the functional lawofiterated logarithm approximation or the strong approximation), then the output data (i.e., the departure processes) and the performance measures (such as the queue length, the workload and the sojourn time processes) satisfy a similar approximation. Based on the strong approximation, some procedures are propo...
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 6 (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.
Synchronous Constrained Fluid Systems
 IBM Research Division
, 1995
"... This paper introduces the framework of synchronous constrained fluid systems (SCFS) to model ..."
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This paper introduces the framework of synchronous constrained fluid systems (SCFS) to model
Control Problems in Telecommunications: The Heavy Traffic Approach
, 2000
"... The goal of this chapter is to demonstrate the usefulness of analytical and numerical methods of stochastic control theory in the design, analysis and control of telecommunication networks. The emphasis will be concentrated on the heavy traffic approach for queueing type systems in which there is li ..."
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The goal of this chapter is to demonstrate the usefulness of analytical and numerical methods of stochastic control theory in the design, analysis and control of telecommunication networks. The emphasis will be concentrated on the heavy traffic approach for queueing type systems in which there is little idle time and the queue length processes can be approximated by reected diffusion processes under suitable scaling. Three principal problems are considered: the multiplexer system, controlled admission in multiserver systems such as ISDN, and the polling or scheduling problem.