Abstract: This paper considers the single working vacation and vacation interruption G-queue with negative customers and setup times. When a service is completed during a regular busy period, if there is no customer in the system, the server begins a working vacation of random length. There are two ways that the system transfers to the regular service period, one is that there are customers in the queue after the completion of a service during the working vacation. Another is that there are customers in the queue after the completion of a vacation, but the service time before is invalid for the unfinished service. If there is no customer in the system after the completion of a service during the working vacation, the system turns off. The service of the first positive customer must take a setup time from the close-down server. Using the quasi birth death process and matrix-geometric-solution method, the steady-state distribution for the steady probabilities of the system and the queue length distribution are obtained. The mean waiting time of arriving customers in the regular busy period is derived by solving the distribution function. Additionally, we provide the numerical examples to illustrate the effect of the parameters on several performance characteristics mentioned above.

Key words: single working vacation, vacation interruption, QBD process, matrix-geometric-solution