Analysis of renewal input batch service queue with impatient customers and multiple working vacations

This paper presents a renewal input single server batch service queue with impatient customers and multiple working vacations. An arriving customer may join the system with probability d or balk with probability 1 - d. The service is carried out in batches with a maximum threshold limit b. The server takes a working vacation when there are no customers in the system, and continues taking multiple working vacations as long as there are no customers at working vacation completion epochs. Applying the embedded Markov chain approach and the displacement operator method, we find the probability distribution of the queue-length at pre-arrival epoch. With the help of the Markov renewal theory and semi-Markov processes, we compute the probability distribution of the queue-length at an arbitrary epoch. We also obtain the system-length distributions at both pre-arrival and arbitrary epochs and present a potential application of the model in cloud systems. Some corresponding results underneath special cases are found by putting suitable values. We present various performance measures and waiting-time distributions in the queue. Numerical results are demonstrated to emphasize the qualitative features of the model.