A Note on Approximating Peak Congestion in Mt/G/∞ Queues with Sinusoidal Arrivals
研究顾客到达率呈正弦波动的M t/G/∞排队系统，发现峰值拥塞时的期望顾客数可近似为(λ+A)/μ，时间滞后则取决于服务时间分布，近似精度随服务率μ增大而提高。
We study the M t /G/∞ queue where customers arrive according to a sinusoidal function λ t = λ + A sin(2πt/T) and the service rate is μ. We show that the expected number of customers in the system during peak congestion can be closely approximated by (λ + A)/μ for service distributions with coefficient of variation between 0 and 1. Motivated by a result derived by Eick, Massey, and Whitt that the time lag of the peak congestion from the peak of the customer arrivals is 1/2μ for models with deterministic service times, we show that the time lag for exponential service times is closely approximated by 1/μ. Based on a cycle length of 24 hours and regardless of the values of other system parameters, these approximations are of the order of 1% accuracy for μ = 1, and the accuracy increases rapidly with increasing μ.