The discrete-time GI/Geo/1 queue with working vacations and vacation interruption

In this paper, we consider a GI/Geo/1 queue with working vacations and vacation interruption. The server takes the original work at the lower rate rather than completely stopping during the vacation period. Meanwhile, we introduce vacation interruption policy: the server can come back to the normal working level once there are customers after a service completion during the vacation period, thus the server may not accomplish a complete vacation. Using matrix-geometric solution method, we obtain the steady-state distributions for the number of customers in the system at arrival epochs, and waiting time for an arbitrary customer. Meanwhile, we explain the stochastic decomposition properties of queue length and waiting time.