Transient and steady state solution of a single server queue with modified Bernoulli schedule vacations based on exhaustive service and a single vacation policy

We analyze a single server queue with modified Bernoulli server vacations based on exhaustive service. Unlike other vacation policies, we assume that only at the completion of service of the last customer in the system, the server has the option to take a vacation or to remain idle in the system waiting for the next customer to arrive. The service times of the customers have been assumed to be exponential and vacations are phase type exponential. We obtain explicit steady state results for the probability generating functions of the queue length, the expected number of customers in the queue and the expected waiting time of the customer. Some earlier known results of the single server have been derived as a particular case.