University of Guilan Journal of Mathematical Modeling 2345-394X 13 1 2025 03 01 Finite-capacity $M/M/2$ machine repair model with impatient customers, triadic discipline, and two working vacation policies 183 200 8159 10.22124/jmm.2024.27095.2384 EN Kadi Abir University of Bejaia, Department of Mathematics, Laboratory of Applied Mathematics, 06000 Bejaia, Algeria Touche Nassim University of Bejaia, Faculty of Exact Sciences, Research Unit LaMOS (Modeling and Optimization of Systems), 06000 Bejaia, Algeria Amina Angelika Bouchentouf Laboratory of Mathematics, Djillali Liabes University of Sidi Bel Abbes, 22000 Sidi Bel Abbes, Algeria Boualem Mohamed University of Bejaia, Faculty of Technology, Research Unit LaMOS (Modeling and Optimization of Systems), 06000 Bejaia, Algeria Journal Article 2024 03 26 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. https://jmm.guilan.ac.ir/article_8159_ebbd4fcefad3e5bff26aed48f8ed9d8c.pdf