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

Consider a GI/Geo/1 queue with multiple vacation policies described as follows: when 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  . During a working vacation period, customers can be served at a lower rate. If there are customers in the system at a service completion instant, the vacation can be interrupted and the server will come back to the normal busy period with probability p(0≤p≤1) or continue the working 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 sojourn time is also derived. Finally, we present some numerical examples and use the parabolic method to search the optimum value of service rate in working vacation period.