A simple approach for analyzing feedback vacation queues with levy input process

We introduce a simple approach for modeling and analyzing a SII/G/I queue where the server may take repeated vacations. Whenever a busy period ends the server takes a vacation of random duration. At the end of each vacation the server may either take a new vacation or resume service; if the queue is found empty the server always takes a new vacation. Furthermore, the queuing system allows Bernoulli feedback of customers. Three classes of service disciplines, random gated, 1-limited and exhaustive, are considered. The random gated service discipline generalizes several known service disciplines. The customers arrival process is assumed to be a Levy process (i.e., satisfies the stationary and independent increments (SII property). We obtain explicit expressions for several performance measures of the system. These performance measures include the mean and second moment of the cycle time, the mean queue length at the beginning of a cycle of service and the expected delay observed by a customer. Furthermore, our analysis provides a uniform method to get several results previously obtained by Baba, Chiarawongse and Sriniwasan, and Takine, Takagi and Hasegawa.