The GI/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption

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)(0⩽q⩽1) or takes a working vacation with probability 1-q1-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)(0⩽p⩽1) or continue the vacation with probability 1-p1-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.