Consider an n-component system where component i, i = 1, ... , n, functions for an exponential time with rate λi and then fails; upon failure, repair begins on component i, with the repair taking an exponentially distributed time with rate μi . Once repaired, a component is as good as new. The components act independently except that when there is only one working com- ponent the system is temporarily shut down until a repair has been completed; it then starts up again with two working components.
(a) What proportion of time is the system shut down?
(b) What is the (limiting) averaging number of components that are being repaired?