Availability of a one-unit system supported by several spares and repair facilities

We consider a system consisting of one operating unit, n−1 spares and r repair facilities. As soon as the operating unit fails, one of the spares, if available, takes over the operation. The failed unit joins the repair queue and is serviced as soon as one of the repair facilities becomes free. After a perfect repair the unit becomes a viable spare and remains on cold stand-by. The system fails when the operating unit fails and there is no viable spare. We obtain the instantaneous availability function of such a system, when both life and repair times are exponentially distributed with possibly different scale parameters.