A study of a queueing system with three-phase service

This paper considers a single server that serves customers, each of which requires three phases of service. The server alternates between two queues, a batch queue and an individual queue, giving priority to customers in the latter queue. Customers in the batch queue are served in batches whereas customers in the individual queue are served individually. The batches undergo two service phases. During the first phase, the non-gated phase, any customer entering the batch queue is allowed to joint the batch in service. However, once the batch enters the second phase, the gated phase, no other customer is allowed to enter service. At the completion of the gated phase, the customers then enter the individual queue where they acquire their third phase of service. Customers are individually served in the individual queue after which they depart the system. The paper derives an expression for the mean customer sojourn time under the assumption of Poisson arrivals, and general service times for each of the three phases.