Single server retrial queues with setup time

Tuan Phung-Duc

doi:10.3934/jimo.2016075

Article Contents

2017: 1329-1345. Doi: 10.3934/jimo.2016075

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Single server retrial queues with setup time

Tuan Phung-Duc

Division of Policy and Planning Sciences, Faculty of Engineering, Information and Systems, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan

Received: September 2015

Revised: May 2016

Published: September 2017

The reviewing process of the paper was handled by Wuyi Yue and Yutaka Takahashi as Guest Editors.

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Abstract

This paper considers single server retrial queues with setup time where the server is aware of the existence of retrying customers. In the basic model, the server is switched off immediately when the system becomes empty in order to save energy consumption. Arriving customers that see the server occupied join the orbit and repeat their attempt after some random time. The new feature of our models is that an arriving customer that sees the server off waits at the server and the server is turned on. The server needs some setup time to be active so as to serve the waiting customer. If the server completes a service and the orbit is not empty, it stays idle waiting for either a new or retrial customer. Under the assumption that the service time and the setup time are arbitrarily distributed, we obtain explicit expressions for the generating functions of the joint queue length. We also obtain recursive formulas for computing the moments of the queue length. We then consider an extended model where the server has a closedown time before being turned off for which an explicit solution is also obtained.

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Mathematics Subject Classification: 60K25, 68M20, 90B22.

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Figure 1. Transitions among states

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Figure 2. Transition among states

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Figure 3. Probability of OFF state against $\alpha$ ($\rho = 0.7$)

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Figure 4. Mean number of jobs in orbit against $\alpha$ ($\rho = 0.7$)

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Figure 5. Probability of OFF state against $\alpha$ ($\rho = 0.1$)

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Figure 6. Mean number of jobs in orbit against $\alpha$ ($\rho = 0.1$)

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