System delay versus system content for discrete-time queueing systems subject to server interruptions

This paper concerns discrete-time queueing systems operating under a first-come-first-served queueing discipline, with deterministic service times of one slot and subject to independent server interruptions. For such systems, we derive a relationship between the probability generating functions of the system content during an arbitrary slot and of the system delay of an arbitrary customer. This relationship is valid regardless of the nature of the arrival process. From this relationship we derive a relationship between the first- and second-order moments of the distributions involved. It is shown that the relationship also applies to subsystems of the queueing system being discussed, and to the waiting time and queue content of a multi-server queueing system with geometric service times and uninterrupted servers.