Abstract
In this paper, we develop an M/M/c queueing system in a Markovian environment with waiting servers, balking and reneging, under both synchronous single and multiple working vacation policies. When the system is in operative phase j, \(j=\overline{1,K},\) customers are served one by one. Once the system is empty, the servers have to wait a random period of time before leaving, causing the system to move to vacation phase 0 at which new arrivals can be served at lower rate. Using the method of the probability generating functions, we establish the steady-state analysis of the system. Special cases of the queueing model are presented. Then, explicit expressions of the useful system characteristics are derived. In addition, a cost model is constructed to define the optimal values of service rates, simultaneously, to minimize the total expected cost per unit time via a quadratic fit search method. Numerical examples are provided to display the impact of different system characteristics.
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Houalef, M., Bouchentouf, A.A. & Yahiaoui, L. A Multi-server Queue in a Multi-phase Random Environment with Waiting Servers and Customers’ Impatience Under Synchronous Working Vacation Policy. J. Oper. Res. Soc. China 11, 459–487 (2023). https://doi.org/10.1007/s40305-021-00384-3
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DOI: https://doi.org/10.1007/s40305-021-00384-3
Keywords
- Queueing models
- Random environment
- Working vacation policy
- Impatient customers
- Probability generating function
- Optimization