Queueing models with backup servers and service breakdowns

In this paper we consider two models in which more than one server is available to service a queue, but only one server at a time is actually used. All servers are subject to breakdown, and are repaired one at a time. We consider the case where servers can only break down when they are servicing the queue, and we then consider the case where they can break down at any time. In one model there are two ranked servers, a primary server and a backup server; when operable, the primary servers always serves the queue. In the second model, there are finite number of identical unranked servers. We find methods for determining the queue length distribution by matrix-geometric methods.