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.

This paper deals with a batch arrivals queue with general retrial time, breakdowns, repairs and reserved time. Here we assume that customers arrive according to compound Poisson processes. Any arriving batch of primary customers finds the server free, one of the customers from the batch enters into the service area and the rest of them join into the orbit. The primary customers who find the server busy or failed are allowed to balk or are queued in the orbit in accordance with FCFS retrial policy. The customer whose service is interrupted can stay at the server waiting for repair or enter into service orbit. After the repair is completed, the server resumes service immediately if the customer in service has remained in the service position. This model has a potential applications in various fields, such as in the cognitive radio network and the manufacturing systems. By using supplementary variables technique, we carry out an extensive analysis of the considered model. Then, we obtain some important performance measures, stochastic decomposition property of the system size distribution and the reliability indices. Next, by setting the appropriate parameters, some special cases are discussed. Finally, some numerical examples and cost analysis are presented.

This paper deals with a finite capacity multi-server Markovian queueing model with Bernoulli feedback, synchronous multiple vacation policy and customers’ impatience (balking and reneging). By employing certain customer retention mechanism, impatient customers can be retained in the system. Applications of the suggested queueing model can be found in a wide variety of practical systems including modern information and communication technology (ICT) networks, call centers, and manufacturing systems. Using the recursive method, the steady state probabilities of the model are obtained. Various performance measures are derived. Then, some important particular cases are provided. Finally, different numerical examples are presented to demonstrate how the different parameters of the model influence the behavior of the stationary characteristics of the system.

In this paper, we analyze a multi-server queue with customers impatience and Bernoulli feedback under a variant of multiple vacations. On arrival, a customer decides whether to join or balk the system, based on the observation of the system size as well as the status of the servers. It is supposed that customer impatience can arise both during busy and vacation period because of the long wait already experienced in the system. The latter can be retained via certain mechanism used by the system. The feedback occurs as returning a part of serviced customers to get a new service. The queue under consideration can be used to model the processes of information transmission in telecommunication networks. We develop the Chapman-Kolmogorov equations for the steady-state probabilities and solve the differential equations by using the probability generating function method. In addition, we obtain explicit expressions of some important system characteristics. Different queueing indices are derived such as the probabilities when the servers are in different states, the mean number of customers served per unit of time, and the average rates of balking and reneging.