Analysis of Vacation Fluid M / M /1 Queue in Multi-Phase Random Environment

Sherif I. Ammar (), Yousef F. Alharbi and Yiqiang Q. Zhao
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Sherif I. Ammar: Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia
Yousef F. Alharbi: Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia
Yiqiang Q. Zhao: School of Mathematics and Statistics, Carleton University, Ottawa, ON K1S 5B6, Canada

Mathematics, 2023, vol. 11, issue 21, 1-14

Abstract: An M / M /1 fluid queue with various vacations is studied in the context of a multi-phase random environment. When the system is in operation ( i = 1, 2, …, n ), it behaves according to the M / M /1 fluid queue model. However, in any other situation, the system is on vacation, so this leads it to transition into the vacation phase ( i = 0). This transition occurs only when there is no data in the system. If the system returns from a vacation and finds it still empty of jobs, it will initiate a new vacation and continue in this pattern until jobs become available in the system, at which point it resumes working. When the vacation phase ends, the probability of the system transitioning to the operational phase is denoted as q i ( i = 1, 2, …, n ). Subsequently, we derive the stationary probability and analyze the buffer content in relation to the modified Bessel function of the first kind. We utilize the generating function approach and the Laplace–Stieltjes transform to achieve this, enabling us to accomplish our objectives. We provide numerical results to elucidate the overall behavior of the system under consideration.

Keywords: vacation; fluid single server queue; generating functions; buffer content; random environment (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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