论文标题
繁重流量的单人队列和转换方法
Heavy-traffic single-server queues and the transform method
论文作者
论文摘要
繁重的人流限制理论涉及接近批判性并面临严重排队时间的队列。令$ w $表示$ {\ rm gi}/{\ rm g}/1 $ queue中的稳态等待时间。金曼（Kingman，1961）表明，当系统的负载接近1时，$ w $在适当的缩放时会收敛到指数随机变量。1。此著名结果的原始证明使用转换方法。从$ W $（Pollaczek的轮廓积分表示）的PDF的拉普拉斯变换开始，金曼表现出变换的融合，因此涉及的随机变量的融合较弱。我们应用并扩展了此转换方法，以通过错误评估获得矩的收敛。我们还展示了如何将转换方法应用于金曼型和高斯繁重交通状态中所谓的几乎确定性队列。我们从数值上演示了各种重型交通近似值的准确性。
Heavy-traffic limit theory deals with queues that operate close to criticality and face severe queueing times. Let $W$ denote the steady-state waiting time in the ${\rm GI}/{\rm G}/1$ queue. Kingman (1961) showed that $W$, when appropriately scaled, converges in distribution to an exponential random variable as the system's load approaches 1. The original proof of this famous result uses the transform method. Starting from the Laplace transform of the pdf of $W$ (Pollaczek's contour integral representation), Kingman showed convergence of transforms and hence weak convergence of the involved random variables. We apply and extend this transform method to obtain convergence of moments with error assessment. We also demonstrate how the transform method can be applied to so-called nearly deterministic queues in a Kingman-type and a Gaussian heavy-traffic regime. We demonstrate numerically the accuracy of the various heavy-traffic approximations.
