MAP/Ek/1 Queue with Working Vacation Providing Main Service Only in Normal Mode of Service

Abstract

In this paper, we consider a MAP/Ek/1 queue with working vacation. Customers
arrive according to a Markovian Arrival Process and service time follows generalized
Erlang distribution of order n. Service in the first k stages is called the preliminary service and service in the remaining n − k stages is called the main service. When the system becomes empty at the time of completion of a service, the server goes on working vacation. During working vacation server provides only the preliminary service. After availing of the preliminary service, the customers leave the system with probability p. Those who require the main service, join a buffer of finite capacity N with complementary probability 1 − p. The server switches to normal mode when the vacation expires, or N customers accumulate in the buffer during working vacation, whichever occurs first. The customer in service at the working vacation expiration epoch, continues to get his service in normal mode. Steady state analysis of this system is performed. Several performance characteristics of interest are computed. A cost function is constructed and the optimal values of N for the positive, zero and negative correlation values of the Markovian arrival process are obtained.