Delay analysis of a two-server discrete-time queue where one server is only intermittently available

In this work we look at the delay analysis of a customer in a discrete-time queueing system with one permanent server and one occasional extra server. The arrival process is assumed to be general independent, the buffer size infinite and the service times deterministically equal to one slot. The system resides in one of two different states defined by the number of available servers. In the UP-state 2 servers are available and in the DOWN-state 1 server is available. State changes can only occur at slot boundaries. When the extra server becomes available, an UP-period starts (DOWN-period ends) and when the extra server becomes unavailable a DOWN-period starts (UP-period ends). The lengths of these periods, expressed in their number of slots, are assumed to follow a geometric distribution, with different parameter for UP-periods and DOWN-periods. Also, the extension is made to DOWN-periods according to a mixture of M geometric distributions. Using the technique of the dominant singularity, we provide a method to evaluate the tail characteristics of the delay of an arbitrary customer. The method is illustrated with a numerical example