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

Consider a GI/Geo/1 queue with start-up period and single working vacation. When the system is in a closed state, an arriving customer leads to a start-up period. After the start-up period, the system becomes a normal service state. 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 both at arrival and arbitrary epochs. The sojourn time is also obtained. Finally, some numerical examples are presented.