A Non-Markovian Batch Arrival Queue with Service Interruption & Extended Server Vacation

A single server provides service to all arriving customers with service time following general distribution. After every service completion the server has the option to leave for phase one vacation of random length with probability p or continue to stay in the system with probability 1-p. As soon as the completion of phase one vacation, the server may take phase two vacation with probability q or to remain in the system with probability 1 - q, after phase two vacation again the server has the option to take phase three vacation with probability r or to remain in the system with probability 1-r. The vacation times are assumed to be general. The server is interrupted at random and the duration of attending interruption follows exponential distribution. Also we assume, the customer whose service is interrupted goes back to the head of the queue where the arrivals are Poisson. The time dependent probability generating functions have been obtained in terms of their Laplace transforms and the corresponding steady state results have been obtained explicitly. Also the mean number of customers in the queue and system and the waiting time in the queue and system are also derived. Particular cases and numerical results are discussed.