Unreliable M/M/1/1 retrial queues with set-up time

We consider a single-server retrial queue with set-up time and server breakdown. There is no waiting space in front of the server, and customers who cannot occupy the server upon arrival will go into a retrial orbit and they retry independently of each other after an exponentially distributed time. To save power, the server is turned off immediately after a service if there is no customer in the orbit. If the orbit is not empty, the server will stay idle to wait a customer coming from the external or from the orbit. A newly coming customer can reactivate the off server, and the server needs some set-up time to work. The server may fail to restart. Once failed, it will be repaired immediately and the repair time is exponentially distributed. Two models are considered according to whether the server can be perfectly repaired or not. The first model is concerned with imperfect repair and the server may still fail to activate after repair. In the second model, the server is perfectly repaired. In both models, we get the explicit expressions of stationary distribution of queue length in the system. Cost minimization is studied numerically and several numerical examples are presented.