Finite-capacity $M/M/2$ machine repair model with impatient customers, triadic discipline, and two working vacation policies

Document Type : Research Article

Authors

1 University of Bejaia, Department of Mathematics, Laboratory of Applied Mathematics, 06000 Bejaia, Algeria

2 University of Bejaia, Faculty of Exact Sciences, Research Unit LaMOS (Modeling and Optimization of Systems), 06000 Bejaia, Algeria

3 Laboratory of Mathematics, Djillali Liabes University of Sidi Bel Abbes, 22000 Sidi Bel Abbes, Algeria

4 University of Bejaia, Faculty of Technology, Research Unit LaMOS (Modeling and Optimization of Systems), 06000 Bejaia, Algeria

Abstract

In this paper, we model and analyze a machine repair system characterized as an $M/M/2$ queue with finite source $L$, operating under the triadic policy $(0,Q,N,M)$, considering impatience, and both multiple and single working vacations. The two servers can be active, on working vacation, or dormant depending on the number of failed machines in the system, following the triadic policy $(0,Q,N,M)$. We analyze the system's steady-state using the matrix-geometric method. Various performance measures are numerically presented and accurately interpreted. Finally, the Quadratic Fit Search method is employed to determine the optimal service rate $\mu_{v}^{*}$ and the optimal expected cost. Additionally, the effect of system parameters on the cost function is investigated. This study offers a comprehensive analytical framework for complex queueing environments, informing decision-making and operational efficiency across various industrial sectors.

Keywords

Main Subjects

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Journal of Mathematical Modeling
Volume 13, Issue 1
March 2025
Pages 183-200

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