M[x]/G/1 Queue with Two Phase of Service and Optional Server Vacation

In this paper we analyze a single server queue with batch arrival Poisson input, two heterogeneous service with different general (arbitrary) service time distributions and two phase (compulsory and optional) server vacations with general (arbitrary) vacation period. The first phase of service is essential for all customers, as soon as the first service of a customer is completed, then with probability θ, he may opt for the second service or else with probability (1θ), he leaves the system. After completion of each service, the server will take compulsory vacation. The vacation period of the server has two heterogeneous phases. However, after returning from first compulsory vacation the server may take one more optional vacation with probability p or return back to the system with probability (1p). No server can take more than two vacations at a time. The probability generating function for the number of customers in the queue is found using the supplementary variable technique. The mean number of customers and the mean waiting time in the queue also found. Some particular cases are discussed. Numerical results are also obtained. AMS Subject Classification: 60K25, 60K30