Consider a GI/M/1 queue with multiple vacations. As soon as the system becomes empty, the server either begins an ordinary vacation with probability q (0≤q≤1) or takes a working vacation with probability 1-q. We assume the vacation interruption is controlled by Bernoulli. If the system is non-empty at a service completion instant in a working vacation period, the server can come back to the normal busy period with probability p (0≤p≤1) or continue the vacation with probability 1 - p. Using the matrix-analytic method, we obtain the steady-state distributions for the queue length both at arrival and arbitrary epochs. The waiting time and sojourn time are also derived by different methods. Finally, some numerical examples are presented. © 2012 Elsevier Inc.
Tao, L., Wang, Z., & Liu, Z. (2013). The GI/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption. Applied Mathematical Modelling, 37(6), 3724–3735. https://doi.org/10.1016/j.apm.2012.07.045