A correlated M/G/1-type queue with randomized server repair and maintenance modes

We consider a single machine, subject to breakdown, that produces items to inventory, continuously and uniformly. While the machine is working (an ON period), there is a deterministic net flow into the buffer. There are also two different types of OFF periods; one is a repair operation initiated after a breakdown, and the other is a preventive maintenance operation. The length of an OFF period depends on the length of the preceding ON period. During OFF periods there is a uniform outflow from the inventory buffer. The buffer content process can then be described as a fluid model. The main tool employed in determining its steady-state law exploits an equivalence relationship between the buffer content process and the virtual waiting time process of a special single-server queue in which the interarrival times and the service requirements depend on each other. We present an appropriate cost function and provide an optimization scheme illustrated by some numerical results.