We study a class of load-balancing algorithms for many-server systems (N servers). Each server has a buffer of size $b-1$ with $b=O(\sqrt{\log N})$, i.e. a server can have at most one job in service and $b-1$ jobs queued. We focus on the steady-state performance of load-balancing algorithms in the heavy traffic regime such that the load of the system is $\lambda = 1 - \gamma N^{-\alpha}$ for $0<\alpha<0.5$ and $\gamma > 0,$ which we call the sub-Halfin–Whitt regime ($\alpha=0.5$ is the so-called Halfin–Whitt regime). We establish a sufficient condition under which the probability that an incoming job is routed to an idle server is 1 asymptotically (as $N \to \infty$) at steady state. The class of load-balancing algorithms that satisfy the condition includes join-the-shortest-queue, idle-one-first, join-the-idle-queue, and power-of-d-choices with $d\geq \frac{r}{\gamma}N^\alpha\log N$ (r a positive integer). The proof of the main result is based on the framework of Stein’s method. A key contribution is to use a simple generator approximation based on state space collapse.

中文翻译：

---

sub-Half-Whitt 状态下负载平衡算法的稳态分析

我们研究了一类多服务器系统的负载均衡算法（ñ服务器）。每个服务器都有一个大小为$b-1$和$b=O(\sqrt{\log N})$，即服务器最多可以有一个服务工作，并且$b-1$作业排队。我们关注负载均衡算法在大流量情况下的稳态性能，使得系统的负载为$\lambda = 1 - \gamma N^{-\alpha}$为了$0<\alpha<0.5$和$\gamma > 0,$我们称之为 sub-Half-Whitt 政权（$\alpha=0.5$是所谓的Halfin-Whitt政权）。我们建立了一个充分条件，在该条件下，传入作业被路由到空闲服务器的概率渐近地为 1（如$N \to \infty$) 处于稳定状态。满足条件的负载均衡算法包括join-the-shortest-queue、idle-one-first、join-the-idle-queue和power-of-d- 选择与$d\geq \frac{r}{\gamma}N^\alpha\log N$(r一个正整数）。主要结果的证明是基于 Stein 方法的框架。一个关键贡献是使用基于状态空间崩溃的简单生成器近似。