The Multi-class Fifo M/g/1 Queue with Exponential Working Vacations

We consider a stationary multi-class FIFO M/G/1 queue with exponential working vacations, where a server works at two different processing rates. There are K classes of customers, and the arrival rates and the distributions of the amount of service requirements of arriving customers depend on both their customer classes and the server state. When the system becomes empty, the server takes a working vacation, during which customers are served at processing rate γ (γ > 0). If the system is empty at the end of the working vacation, the server takes another working vacation. On the other hand, if a customer is being served at the end of the working vacation, the server switches its processing rate to one and continues to serve customers in a preemptive-resume manner, until the system becomes empty. For this queue, we derive various quantities of interest, including the Laplace-Stieltjes transforms of the actual waiting time and sojourn time distributions, and the joint transform of the numbers of customers and the amounts of unfinished work in respective classes. As by-products, we also obtain various results of a stationary multi-class FIFO M/G/1 queue with Poisson disasters.