An M/G/1 Queue with Regular and Optional Phase Vacation and with State Dependent Arrival Rate

We consider an M/G/1 queue with regular and optional phase vacation and with state dependent arrival rate. The vacation policy is after completion of service if there are no customers in the system, the server takes vacation consisting of K-phases, each phase is generally distributed. Here the first phase is compulsory where as the other phases are optional. For this model the supplementary variable technique has been applied to obtain the probability generating functions of number of customers in the queue at the different server states. Some particular models are obtained and a numerical study is also carried out.