Analysis of a two-node closed queueing cyclic network with one non-exponential node

Maddah, Bacel; El-Taha, Muhammad

doi:10.1016/j.cie.2017.06.007

Cited by 2 publications

(11 citation statements)

References 22 publications

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“…3 Journal of Applied Mathematics which is consistent with the approach given by Maddah and El-Taha [1].…”

Section: Birth Processsupporting

confidence: 85%

“…In our model, this is equivalent to the special case of M = N. The fact that M can be larger than N complicates the development of the imbedded MC in our model since the transition rate to the second node is not as smooth as in the case of M/G/1 and M/G/1//N. Except for Maddah and El-Taha [1], no other article utilizes an imbedded Markov chain approach for the closed cyclic two-node network model as the one we consider.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, Oz et al [3] utilized conditional residual service times and a recursive scheme to compute the stationary distribution function. Maddah and El-Taha [1] use the imbedded Markov chain approach representing node 2 system size probabilities at departure epochs. They compute the transition probabilities using an exponential algorithm.…”

Section: Introductionmentioning

confidence: 99%

“…They compute the transition probabilities using an exponential algorithm. This paper's contribution is a significant improvement over the results of Maddah and El-Taha [1] by giving a more efficient polynomial algorithm to compute transition probabilities. The imbedded Markov chain technique has been successfully applied to solve the related machine repairman model with no spares, M/G/1//N (see, for example, Van Hoorn [4], Saaty [5], and Takagi [6]).…”

Section: Introductionmentioning

confidence: 99%

“…Boxma and Daduna [9] discuss sojourn time distributions and approximate methods for two-node queueing networks. A more detailed literature review can be found in Maddah and El-Taha [1].…”

Section: Introductionmentioning

confidence: 99%

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An Efficient Convolution Algorithm for the Non-Markovian Two-Node Cyclic Network

El-Taha

Michaud

2021

Journal of Applied Mathematics

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Consider a closed cyclic queueing model that consists of two nodes and a total of M customers. Each node buffer can accommodate all M customers. Node 1 has N ≤ M servers, each having an exponential service time with rate λ . The second node consists of a single server with a general service time distribution function B . . The well-known machine repair model with spares, where a set of identical machines, N , is served by a single repair person, is a key application of this model. This model has several other applications in performance evaluation, manufacturing, computer networks, and in reliability studies as it can be easily used to compute system availability. In this article, we give an efficient algorithm to derive an exact solution for the steady state system size probabilities. Our approach is based on developing an efficient polynomial convolution method to compute the transition probabilities of a birth process over node 2 service times and solving an imbedded Markov chain at node 2 service completion epochs. This is a significant improvement over the exponential algorithm given in an earlier paper. Numerical examples are given to demonstrate the performance of our method.

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“…3 Journal of Applied Mathematics which is consistent with the approach given by Maddah and El-Taha [1].…”

Section: Birth Processsupporting

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Boxma and Daduna [9] discuss sojourn time distributions and approximate methods for two-node queueing networks. A more detailed literature review can be found in Maddah and El-Taha [1].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An Efficient Convolution Algorithm for the Non-Markovian Two-Node Cyclic Network

El-Taha

Michaud

2021

Journal of Applied Mathematics

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On the Application of Mean Value Analysis Without the Normalization Constant in the Closed Queuing Networks

O¹

2021

NJPAS

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An example of closed queue network could be view when patients arrive at a doctor’s ofﬁce to update their medical records, then it’s off to the nurse’s station for various measurements like weight, blood pressure, and so on. The next stop is generally to queue (i.e., wait patiently) for one of the doctors to arrive and begin the consultation and examination. Perhaps it may be necessary to have some X-rays taken, an ultrasound may be called for, and so on. After these procedures have been completed, it may be necessary to talk with the doctor once again. The ﬁnal center through which the patient must pass is always the billing ofﬁce. In this work, multiple-node” system in which a customer requires service at more than one node, which may be viewed as a network of nodes, and each node is a service center having storage room for queues to form and perhaps with multiple servers to handle customer requests is investigated in order to provide some insight into the performance measure analysis. Our quest is to exempt the normalization constant in the computation of performance measure in the closed queueing network. The arrival properties and Little’s law are use with the help of some existing equations and formulas in queueing network. Performance measures, such as Mean number of customers, response time, throughput, and marginal probability distribution are obtained for central server and load dependent server closed queuing networks for nodes 4 and 5, and also for k = 3 and k = 10.

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Analysis of a two-node closed queueing cyclic network with one non-exponential node

Cited by 2 publications

References 22 publications

An Efficient Convolution Algorithm for the Non-Markovian Two-Node Cyclic Network

An Efficient Convolution Algorithm for the Non-Markovian Two-Node Cyclic Network

On the Application of Mean Value Analysis Without the Normalization Constant in the Closed Queuing Networks

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