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 $$M_t/GI/\infty $$ M t / G I / ∞ 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 $$M_t/GI/\infty $$ M t / G I / ∞ model that should be seen with data if the $$M_t/GI/\infty $$ M t / G I / ∞ 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 $$M_t/GI/\infty $$ M t / G I / ∞ model is appropriate. We also establish many-server heavy-traffic fluid limits for the steady-state distribution in the periodic $$M_t/GI/\infty $$ M t / G I / ∞ model. For the special case of sinusoidal arrival rates, the limiting steady-state distribution has an arcsine law.