论文标题
关于许多服务器重型交通状态的负载平衡的注释
A Note on Load Balancing in Many-Server Heavy-Traffic Regime
论文作者
论文摘要
在本说明中，我们采用Stein的方法来分析多服务器重型交通状态中通用负载平衡方案的性能。特别是，考虑一个$ n $服务器的负载平衡系统，到达容量区域的到达率的距离由$ n^{1-α} $给出，$α> 1 $。我们对性能感兴趣，因为$ n $在大量政策下转到了无限。我们在不同的尺度和条件下建立了不同的渐近造剂。具体而言，（i）如果第二矩与$ n $线性增加，则具有系数$σ_a^2 $和$ν_s^2 $，那么对于任何$α> 4 $，则分配$ n^{ - α} $的总和排序长度分布到均值随机变量，均为均值$ \ frac $ \ frac} $ \ frac {2^2^2^2^2^2 + n。 （3）如果第二次次次倍增$ n $，则具有系数$ \tildeσ_a^2 $和$ \tildeν_s^2 $，那么对于任何$α> 3 $ \tildeν_s^2} {2} $。这两个结果都是我们先前开发的Stein方法的简单应用，用于在\ cite {zhou2020note}中进行重型交通分析。
In this note, we apply Stein's method to analyze the performance of general load balancing schemes in the many-server heavy-traffic regime. In particular, consider a load balancing system of $N$ servers and the distance of arrival rate to the capacity region is given by $N^{1-α}$ with $α> 1$. We are interested in the performance as $N$ goes to infinity under a large class of policies. We establish different asymptotics under different scalings and conditions. Specifically, (i) If the second moments linearly increase with $N$ with coefficients $σ_a^2$ and $ν_s^2$, then for any $α> 4$, the distribution of the sum queue length scaled by $N^{-α}$ converges to an exponential random variable with mean $\frac{σ_a^2 + ν_s^2}{2}$. (3) If the second moments quadratically increase with $N$ with coefficients $\tildeσ_a^2$ and $\tildeν_s^2$, then for any $α> 3$, the distribution of the sum queue length scaled by $N^{-α-1}$ converges to an exponential random variable with mean $\frac{\tildeσ_a^2 + \tildeν_s^2}{2}$. Both results are simple applications of our previously developed framework of Stein's method for heavy-traffic analysis in \cite{zhou2020note}.
