Analysis of non-Markovian batch arrival queueing model with multi stages of service of restricted admissibility, feedback service and three optional vacations in production and manufacturing

In this paper, a multi stage of heterogeneous service in a batch arrival queue is investigated. Each customer undergoes multi stages of heterogeneous service. As soon as the completion of nth stage of service, if the customer is dissatisfied with his service, he can immediately join the tail of the original queue to get a feedback service. According to vacation policy, it has three heterogeneous phases. After service completion of a customer the server may take a phase one Bernoulli vacation. Further, after completion of phase one Bernoulli vacation the server may take phase two and phase three optional vacations. The vacation times are assumed to be general. Also we assume restricted admissibility of arriving batches. By employing generating functions and supplementary variable techniques, the transient solution and the corresponding steady state results have been obtained explicitly. Also the mean number of customers in the queue and the system are also derived. Practical rationalisation of the model is described. Special cases are discussed. In order to validate the model, numerical illustration is specified.