  We consider a system consisting of a server alternating between two service
points. At both service points there is an infinite queue of customers that
have to undergo a preparation phase before being served. We are interested in
the waiting time of the server. The waiting time of the server satisfies an
equation very similar to Lindley's equation for the waiting time in the GI/G/1
queue. We will analyse this Lindley-type equation under the assumptions that
the preparation phase follows a phase-type distribution while the service times
have a general distribution. If we relax the condition that the server
alternates between the service points, then the model turns out to be the
machine repair problem. Although the latter is a well-known problem, the
distribution of the waiting time of the server has not been studied yet. We
shall derive this distribution under the same setting and we shall compare the
two models numerically. As expected, the waiting time of the server is on
average smaller in the machine repair problem than in the alternating service
system, but they are not stochastically ordered.
