Overlap Times in the Infinite Server Queue

Abstract: Imagine, you enter a grocery store to buy food. How many peopledo you overlap with in this store? How much time do you overlap witheach person in the store? In this paper, we answer these questions bystudying the overlap times between customers in the infinite serverqueue. We compute in closed form the steady state distribution ofthe overlap time between a pair of customers and the distribution ofthe number of customers that an arriving customer will overlap with.Finally, we define a residual process that counts the number of over-lapping customers that overlap in the queue for at least{\delta}time unitsand compute its mean, variance, and distribution in the exponentialservice setting