Heavy-traffic limits for queues with periodic arrival processes

We establish conventional heavy-traffic limits for the number of customers in a Gt/GI/sGt/GI/s queue with a periodic arrival process. We assume that the arrival counting process can be represented as the composition of a cumulative stochastic process that satisfies an FCLT and a deterministic cumulative rate function that is the integral of a periodic function. We establish three different heavy-traffic limits for three different scalings of the deterministic arrival rate function. The different scalings capture the three cases in which the predictable deterministic variability (i) dominates, (ii) is of the same order as, or (iii) is dominated by the stochastic variability in the arrival and service processes.