A note on GI/M/1 queues with phase-type setup times or server vacations

We study a GI/M/1 queue with phase-type (PH) setup times or vacations. In this queuing system, the service facility is shut down or is switched to do other jobs (taking vacations whenever the system becomes empty). When the next customer arrives at the system to initiate another busy period, there is a setup time for the server or there is a residual vacation time. The setup time or the vacation is a PH distributed random variable. Using the matrix analytic method, we obtain explicit expressions for the stationary distributions of queue length and waiting time and prove stochastic decomposition properties of the queue length and waiting time in the system. The model can be used to model a production and inventory control system.