A Single Server Non-Markovian queue with K Phase of Vacation, Two Types of Services and an Optional Service

A single server queue with two types of services and with vacation has been considered. The type 1 service is a phase type service with two service phases. Both the service time distributions are generally distributed. The type 2 service has only one phase of service. In addition the server also provides an optional service. This service time distribution is also general. At each time the system becomes empty, the server takes K phases of vacation and the vacation time distribution of each phase is general. For this model the probability generating function for the number of customers in the queue at different server's state are obtained using the supplementary variable technique. Some performance measures and particular model are calculated and numerical results are presented.