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 presents an analysis of a Markovian feedback queueing system with reneging and retention of reneged customers, multiple working vacations and Bernoulli schedule vacation interruption, where customers’ impatience is due to the servers’ vacation. The reneging times are assumed to be exponentially distributed. After the completion of service, each customer may reenter the system as a feedback customer for receiving another regular service with some probability or leave the system. A reneged customer can be retained in many cases by employing certain convincing mechanisms to stay in queue for completion of service. Thus, a reneged customer can be retained in the queueing system with some probability or he may leave the queue without receiving service. We establish the stationary analysis of the system. The probability generating functions of the stationary state probabilities is obtained, we deduce the explicit expressions of the system sizes when the server is in a normal service period and in a Bernoulli schedule vacation interruption, respectively. Various performance measures of the system are derived. Finally, we present some numerical examples to demonstrate how the various parameters of the model influence the behavior of the system.

This paper concerns the cost optimisation analysis of a discrete-time finite-capacity multiserver queueing system with Bernoulli feedback, synchronous multiple and single working vacations, balking, and reneging during both busy and...

Machining systems are essential for many industrial applications, such as manufacturing, processing, and assembly. However, these systems are often exposed to various sources of uncertainty and disruption, such as disasters and customer impatience. These factors can adversely affect the performance, reliability, and profitability of the machining systems. Consequently, modeling and analyzing machining systems under these conditions becomes crucial. In this paper, we deal with a Markovian multi-server queueing system with batch arrival, Bernoulli feedback, and customers? impatience (balking and reneging). The system undergoes disastrous interruptions that force all customers? whether waiting or currently in service?to exit, leading to server failures. Moreover, the system dynamically alternates between main servers and substitute servers based on the occurrence of disasters. These substitute servers operate at reduced rates compared to the main servers. Our contributions include deriving the stability condition for the system and obtaining the probability generating function of steady-state probabilities, enabling us to derive essential performance measures. Additionally, we develop a cost model and conduct an economic analysis for the system.