The GI/M/1 Queue in a Multi-phase Service Environment with Working Vacations and Bernoulli Vacation Interruption

In this paper, we consider a GI/M/1 queue operating in a multi-phase service environment withworking vacations and Bernoulli vacation interruption. Whenever the queue becomes empty, the server begins a working vacation of random length, causing the system to move to vacation phase 0. During phase 0, the server takes service for the customers at a lower rate rather than stopping completely. When a vacation ends, if the queue is non-empty, the system switches from the phase 0 to some normal service phase i with probability q(i), i = 1, 2, center dot center dot center dot, N. Moreover, we assume Bernoulli vacation interruption can happen. At a service completion instant, if there are customers in a working vacation period, vacation interruption happens with probability p, then the system switches from the phase 0 to some normal service phase i with probability qi, i = 1, 2, center dot center dot center dot, N, or the server continues the vacation with probability 1 - p. Using the matrix geometric solution method, we obtain the stationary distributions for queue length at both arrival epochs and arbitrary epochs. The waiting time of an arbitrary customer is also derived. Finally, several numerical examples are presented.