M (X)/G/1 WITH TWO PHASE OF HETEROGENEOUS SERVICE UNDER DIFFERENT VACATION POLICY , RESTRICTED ADMISSIBILITY AND SET UP

Abstract

In this paper, we consider a batch arrival queueing system with two stage of heterogeneous service with different vacation policy subject to Restricted admissibility  and set up time is considered. Customers arrive in batches according to compound Poisson process with rate  and are served one by one in FIFO basis. After first-stage service the server must provide the second stage service. The service times of two phase of heterogenous services follow arbitrary (general) distribution with different vacation policies. Before providing service to a new customer or a batch of customers that joins the system in the renewed busy period, the server enters into a setup time process such that setup time follows exponential distribution. In addition we assume restricted admissibility of arriving batches in which not all batches are allowed to join the system at all times . The probability generating function for the number of customers in the queue is found using the supplementary variable technique. The mean number of customers in the queue and the system are also found.