Overlap Times in the $GI^B/GI/\infty$ Queue

Abstract: Overlap times have been studied as a way of understanding the time of interaction between customers in a service facility. Most of the previous analysis relies on the single jump assumption for arrivals, which implies the queue increases by one for each arrival epoch. In this paper, we relax the single arrival assumption and explore the impact of having batch arrivals. Unfortunately, with batch arrivals it is not clear how one measures an overlap time between batches of customers. Thus, we develop two ways of capturing the notion of an overlap time in a batch setting and derive exact results in the infinite server queue with batch arrivals. Finally, we derive new results for analyzing overlap times of more than two batches.