Marginal queue length approximations for a two-layered network with correlated queues

J.L. Dorsman, O.J. Boxma, M. Vlasiou

Research output: Contribution to journalArticleAcademicpeer-review

5 Citations (Scopus)

Abstract

We consider an extension of the classical machine-repair model, where we assume that the machines, apart from receiving service from the repairman, also serve queues of products. The extended model can be viewed as a layered queueing network, where the first layer consists of the queues of products and the second layer is the ordinary machine-repair model. As the repair time of one machine may affect the time the other machine is not able to process products, the downtimes of the machines are correlated. This correlation leads to dependence between the queues of products in the first layer. Analysis of these queue length distributions is hard, as the exact dependence structure for the downtimes, or the queue lengths, is not known. Therefore, we obtain an approximation for the complete marginal queue length distribution of any queue in the first layer, by viewing such a queue as a single server queue with correlated server downtimes. Under an explicit assumption on the form of the downtime dependence, we obtain exact results for the queue length distribution for that single server queue. We use these exact results to approximate the machine-repair model. We do so by computing the downtime correlation for the latter model and by subsequently using this information to fine-tune the parameters we introduced to the single server queue. As a result, we immediately obtain an approximation for the queue length distributions of products in the machine-repair model, which we show to be highly accurate by extensive numerical experiments. Keywords: Layered queueing networks; Machine-repair model; Queues with server vacations
Original languageEnglish
Pages (from-to)29-63
Number of pages35
JournalQueueing Systems: Theory and Applications
Volume75
Issue number1
DOIs
Publication statusPublished - 2013

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Queue Length
Queue
Repair
Queue Length Distribution
Servers
Single Server Queue
Approximation
Queueing networks
Queueing Networks
Exact Results
Model
Server Vacations
Dependence Structure
Marginal Distribution
Immediately
Server
Numerical Experiment
Computing

Cite this

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title = "Marginal queue length approximations for a two-layered network with correlated queues",
abstract = "We consider an extension of the classical machine-repair model, where we assume that the machines, apart from receiving service from the repairman, also serve queues of products. The extended model can be viewed as a layered queueing network, where the first layer consists of the queues of products and the second layer is the ordinary machine-repair model. As the repair time of one machine may affect the time the other machine is not able to process products, the downtimes of the machines are correlated. This correlation leads to dependence between the queues of products in the first layer. Analysis of these queue length distributions is hard, as the exact dependence structure for the downtimes, or the queue lengths, is not known. Therefore, we obtain an approximation for the complete marginal queue length distribution of any queue in the first layer, by viewing such a queue as a single server queue with correlated server downtimes. Under an explicit assumption on the form of the downtime dependence, we obtain exact results for the queue length distribution for that single server queue. We use these exact results to approximate the machine-repair model. We do so by computing the downtime correlation for the latter model and by subsequently using this information to fine-tune the parameters we introduced to the single server queue. As a result, we immediately obtain an approximation for the queue length distributions of products in the machine-repair model, which we show to be highly accurate by extensive numerical experiments. Keywords: Layered queueing networks; Machine-repair model; Queues with server vacations",
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Marginal queue length approximations for a two-layered network with correlated queues. / Dorsman, J.L.; Boxma, O.J.; Vlasiou, M.

In: Queueing Systems: Theory and Applications, Vol. 75, No. 1, 2013, p. 29-63.

Research output: Contribution to journalArticleAcademicpeer-review

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