The GI/M/1 queue with start-up period and single working vacation and Bernoulli vacation interruption

Consider a GI/M/1 queue with start‐up period and single working vacation. When the system is in a closed state, an arriving customer leading to a start‐up period, after the start‐up period, the system becomes a normal service state. And during the working vacation period, if there are customers at a service completion instant, the vacation can be interrupted and the server will come back to the normal working level with probability p (0?p?1) or continue the vacation with probability 1‐ p. Meanwhile, if there is no customer when a vacation ends, the system is closed. Using the matrix‐analytic method, we obtain the steady‐state distributions for the queue length at both arrival epochs and arbitrary epochs, the waiting time and sojourn time.