Received: 31 December 2017 DOI: 10.1002/mma.5119 SPECIAL ISSUE PAPER Q-matrix method for the analysis and performance evaluation of unreliable M∕M∕1∕N queueing model Faïrouz Afroun 1,2 Djamil Aïssani 1 Djamel Hamadouche 2 Mohamed Boualem 1 1 Research Unit LaMOS (Modeling and Optimization of Systems), University of Bejaia, 06000 Bejaia, Algeria 2 Laboratory of Pure and Applied Mathematics, University of Tizi-Ouzou, 15000 Tizi Ouzou, Algeria Correspondence Mohamed Boualem, Research Unit LaMOS, Faculty of Technology, University of Bejaia, 06000 Bejaia, Algeria. Email: robertt15dz@yahoo.fr Communicated by: A. Debbouche MSC Classification: 60K25; 90B22; 68M20 In this work, we study an M∕M∕1∕N queuing system with multiple vacations, Bernoulli feedback, balking, reneging and retention of the impatient customers, and the possibility of a server breakdown and repair. First, by using the Q-matrix (infinitesimal generator matrix) method, we obtain the steady-state probabilities of the system. Then, some useful performance measures are derived. Finally, the influence of the reliability parameters on the performance measures of the system has been examined numerically. KEYWORDS infinitesimal generator, Markov process, performance measures, queueing models, reliability, simulation 1 INTRODUCTION In the queueing theory, a classical queue may be described as a system, where customers arrive according to an arrival process to be served by a service facility according to a service process. But in practice, various behaviors of the server(s) and customers can be identified. 1-4 Mathematical modeling of queueing systems with server vacations is much more flexible and useful while dealing with real-time congestion problems. In vacation queueing systems, the server directs to do few supplementary works at the period of its idle time, which may improve the server performance. The application of queueing systems with vacations can be established from manufacturing industries, production line systems, inventory management, and communication networks. 5-7 With the advancement of technology, performance modeling is one of the important issues that affect the design, development, configuration, and modification of any real-time system. Unreliable server makes negative impact on per- formance of any system; therefore, some measures are to be taken to maintain a desired grade of service. In many practical applications, the server may fail and requires repair (renewal). From a practical point of view, several authors have studied different models that describe the failure of a server and its renewal (repairs) as well as the rules of servicing a customer who finds a server in a broken down condition. In many waiting line systems, the role of servers is performed by mechan- ical/electronic devices, such as computers, pallets, ATM, and traffic lights; all of these are subject to accidental/random failures until the failed server is repaired; it may cause a halt in service. 8-10 Queueing models with an unreliable server under multiple vacation policy are more realistic representation of the systems. The service of the components may be interrupted when the operator encounters unpredicted breakdowns, and it is to be immediately recovered with a random time. When the repair is completed, the server immediately returns for service. 9 On the other hand, a customer can leave the system definitively without being served for various reasons. In a balk- ing scenario, customers are refusing to enter the queue given that it has reached a certain length. Other than the balking Math Meth Appl Sci. 2018;1–12. wileyonlinelibrary.com/journal/mma © 2018 John Wiley & Sons, Ltd. 1