A multi-station unreliable machine model with working vacation policy and customers’ impatience

ABSTRACT

Working vacation queues with breakdowns and customers impatience have many applications in different real-life situations. The development of these models to determine their performance is extremely important. In this paper, we deal with a finite population Markovian multi-server machine system with breakdowns, repairs, Bernoulli feedback, balking, reneging, and retention of reneged customers, under multiple synchronous working vacations. The investigated model has a potential application in the real-world machine systems, such as the automated manufacturing systems with limited space. For the analysis purpose, we employ Q-matrix method in order to obtain the steady-state probabilities as well as closed-form expressions for several system characteristics. Particular cases of the current study are presented. We construct the expected cost function and develop an optimization problem to determine the optimum cost. The direct search method and the Quasi–Newton method are applied to find the optimum system capacity, the minimum number of servers, and the optimum service rates during both working vacation and regular busy periods at minimum cost. Lastly, a sensitivity analysis for the numerical simulation is carried out.

KEYWORDS:

• System reliability
• customers’ impatience
• working vacation
• Q-matrix method
• optimization

MATHEMATICSSUBJECT CLASSIFICATION (2000):

• 60K25
• 68M20
• 90B22

Acknowledgments

The authors are pleased to thank the anonymous referees and Editor for their valuable comments and suggestions, which improved the content and the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Amina Angelika Bouchentouf

Amina Angelika Bouchentouf is a Full Professor of Mathematics at the Djillali Liabes University of Sidi Bel Abbes, Algeria. She received her Doctorate and University Habilitation from the same university. She is a Permanent Researcher at the Laboratory of Mathematics of Sidi Bel Abbes. Her research interests include queueing theory, performance evaluation, and statistic.

Mohamed Boualem

Mohamed Boualem is a Full Professor of Mathematics at the Department of Automation, Telecommunications and Electronics, University of Bejaia, Algeria. He received his Master’s degree in Stochastic Methods of Operational Research from the University of Sciences and Technology Houari Boumediene, USTHB in 2003. He received a Doctorate in Applied Mathematics from University of Béjaia in 2009, and from the same university in 2012 his University Habilitation (HDR) in Mathematics. He is a permanent researcher at the Research Unit LaMOS (Modeling and Optimization of Systems). His main current research interests include Queueing Theory, Retrial Queues, Performance Evaluation, Reliability Theory, Stochastic Orders, Monotonicity, Ageing Distributions and Statistics. He has already published several scientific papers in reputed journals and conference proceedings.

Lahcene Yahiaoui

Lahcene Yahiaoui is an Associate Professor at the Department of Mathematics, University of Saida, Algeria. He is a Permanent Researcher at Laboratory of Stochastic Models, Statistic and Applications. His research interests include queueing theory, stochastic modelling, and statistics.

Hijaz Ahmad

Hijaz Ahmad is a researcher at University of Engineering and Technology, Peshawar, Pakistan. His main research interests include numerical analysis, numerical and computational mathematics, numerical methods, and partial differential equations.