During the past decades batch-service queueing models have been studied extensively, especially with regard to the system content. Some researchers have studied the distribution of the customer delay, but not in the case of batch arrivals, which is a non-trivial extension. In this paper, we compute the probability generating function of the delay in a discrete-time batch-service queueing model with batch arrivals and single-slot service times. We make extensive use of residue theory. It is further shown that moments of the delay can be derived from the obtained probability generating function.