Delay analysis of a discrete-time single-server queue with an occasional extra server

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 is assumed to be in one of two different states; during the UP-state 2 servers are available and during the DOWN-state 1 server is available. State changes can only occur at slot boundaries and mark the beginnings and ends of UP-periods and DOWN-periods. The lengths of the UP-periods, expressed in their number of slots, are assumed to follow a geometric distribution, while the lengths of the DOWN-periods follow a general distribution with rational probability generating function. We provide a method to compute the tail characteristics of the delay of an arbitrary customer based on the theory of the dominant singularity. We illustrate the developed method with several numerical examples