A Queueing System with First Essential Phase of Service Followed by Optional Second Phase of Service and Server’s Choice for a Working Vacation

Abstract

We investigate the steady state behavior of a batch arrival single server queue in which the first phase of service with general service time is essential and the second phase of service with heterogeneous general service time is optional. We term such a two-phase service as generalized Coxian-2 service, which means that every customer may take only the first essential phase of service or the first essential phase of service followed by second optional phase of service.  It is further assumed that after completion of service(s) selected by a customer,the server, with a certain probability, may opt to take a working vacation with exponentially distributed length of time. As soon as the server’s working vacation time completes, it instantly stops the undergoing service and the customer with incomplete service joins the head of the queue. This customer whose service is interrupted, will be taken up by the server for the regular first essential phase of the Coxian-2 service.   We obtain steady state probability generating functions for the queue size at a random epoch of time. Some particular cases of interest have been discussed.