Heavy-traffic fluid limits for periodic infinite-server queues

To better understand what stochastic model might be appropriate in applications with system data, we study the consequences of fitting a stationary birth‐and‐death (BD) process to the sample path of a periodic $M_t/GI/\infty$ model. The fitted BD process will necessarily have the correct steady‐state distribution (appropriately defined), but will not have the correct transient behavior. Nevertheless, the fitted birth‐rate and death‐rate functions have structure determined by the $M_t/GI/\infty$ model that should be seen with data if the $M_t/GI/\infty$ model is appropriate. In this paper, we establish heavy‐traffic fluid limits that yield explicit approximation formulas for the fitted birth‐rate and death‐rate functions that can help evaluate whether a periodic $M_t/GI/\infty$ model is appropriate. We also establish many‐server heavy‐traffic fluid limits for the steady‐state distribution in the periodic $M_t/GI/\infty$ model. For the special case of sinusoidal arrival rates, the limiting steady‐state distribution has an arcsine law.