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Analyzing a single hyper-exponential working vacation queue from its governing difference equation

Published online by Cambridge University Press:  10 November 2022

Miaomiao Yu Open the ORCID record for Miaomiao Yu [Opens in a new window]  and
Yinghui Tang
Miaomiao Yu
Affiliation:
School of Mathematical Science, Sichuan Normal University, Chengdu, Sichuan 610066, P. R. China. E-mail: mmyu75@163.com; tangyh@sicnu.edu.cn
Yinghui Tang
Affiliation:
School of Mathematical Science, Sichuan Normal University, Chengdu, Sichuan 610066, P. R. China. E-mail: mmyu75@163.com; tangyh@sicnu.edu.cn
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Abstract

As the queue becomes exhausted, different maintenance tasks can be performed according to the fatigue load and wear degree of the service equipment. At the same time, considering the customer's sensitivity to time delay, the service facility will not completely remain inactive during the maintenance period. To describe this objectively existing phenomenon arising in the waiting line system, we consider a hyper-exponential working vacation queue with a batch renewal arrival process. Through the calculation of the well-structured roots of the associated characteristic equation, the shift operator method in the theory of difference equations and the supplementary variable technique for stochastic modeling plays a central role in the queue-length distribution analysis. Comparison with other ways to analyze queueing models, the advantage of our approach is that we can avoid deriving the complex transition probability matrix of the queue-length process embedded at input points. The feasibility of this approach is verified by extensive numerical examples.

Keywords

Difference equationHyper-exponential working vacationQueueing systemRouché's theoremShift operator
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 37 , Issue 4 , October 2023 , pp. 997 - 1019
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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