The steady-state distribution of the Mt/M/∞ queue with a sinusoidal arrival rate function

The number of customers in a stable Mt/GI/n queue with a periodic arrival rate function and n servers has a proper steady-state limiting distribution if the initial place within the cycle is chosen uniformly at random. Insight is gained by examining the special case with infinitely many servers, exponential service times and a sinusoidal arrival rate function. Heavy-traffic limits help explain an unexpected bimodal form. The peakedness (ratio of the variance to the mean) can be used for approximations with finitely many servers.