We study a load balancing system in the many-server heavy-traffic regime. We consider a system with N servers, where jobs arrive to the system according to a Poisson process and have an exponentially distributed size with mean 1. We parametrize the arrival rate so that the arrival rate per server is \(1-N^{-\alpha }\), where \(\alpha >0\) is a parameter that represents how fast the load grows with respect to the number of servers. The many-server heavy-traffic regime corresponds to the limit as \(N\rightarrow \infty \), and subsumes several regimes, such as the Halfin–Whitt regime (\(\alpha =1/2\)), the NDS regime (\(\alpha =1\)), as \(\alpha \downarrow 0\) it approximates mean field and as \(\alpha \rightarrow \infty \) it approximates the classical heavy-traffic regime. Most of the prior work focuses on regimes with \(\alpha \in [0,1]\). In this paper, we focus on the case when \(\alpha >1\) and the routing algorithm is power-of-d choices with \(d=\lceil cN^\beta \rceil \) for some constants \(c>0\) and \(\beta \ge 0\). We prove that \(\alpha +\beta >3\) is sufficient to observe that the average queue length scaled by \(N^{1-\alpha }\) converges to an exponential random variable. In other words, if \(\alpha +\beta >3\), the scaled average queue length behaves similarly to the classical heavy-traffic regime. In particular, this result implies that if d is constant, we require \(\alpha >3\) and if routing occurs according to JSQ we require \(\alpha >2\). We provide two proofs to our result: one based on the Transform method introduced in Hurtado-Lange and Maguluri (Stoch Syst 10(4):275–309, 2020) and one based on Stein’s method. In the second proof, we also compute the rate of convergence in Wasserstein’s distance. In both cases, we additionally compute the rate of convergence in expected value. All of our proofs are powered by state space collapse.

我们研究了多服务器大流量状态下的负载平衡系统。我们考虑一个有N个服务器的系统，其中作业根据泊松过程到达系统，并且具有均值为 1 的指数分布大小。我们对到达率进行参数化，使得每个服务器的到达率是\(1-N^{- \alpha }\)，其中\(\alpha >0\)是一个参数，表示负载相对于服务器数量的增长速度。多服务器大流量机制对应的限制为\(N\rightarrow \infty \)，并包含多个机制，例如 Halfin-Whitt 机制 ( \(\alpha =1/2\) )、NDS制度（\(\alpha =1\)），如\(\alpha \downarrow 0\)它近似于平均场，而\(\alpha \rightarrow \infty \)它近似于经典的重交通状态。大多数先前的工作都集中在具有\(\alpha \in [0,1]\) 的制度上。在本文中，我们关注当\(\alpha >1\)和路由算法是幂的d选择的情况，其中\(d=\lceil cN^\beta \rceil \)对于一些常数\(c >0\)和\(\beta \ge 0\)。我们证明\(\alpha +\beta >3\)足以观察到由\(N^{1-\alpha }\)缩放的平均队列长度收敛到指数随机变量。换句话说，如果\(\alpha +\beta >3\)，按比例缩放的平均队列长度的行为类似于经典的大流量制度。特别是，这个结果意味着如果d是常数，我们需要\(\alpha >3\)，如果根据 JSQ 进行路由，我们需要\(\alpha >2\)。我们为我们的结果提供了两种证明：一种基于 Hurtado-Lange 和 Maguluri (Stoch Syst 10(4):275–309, 2020) 中引入的变换方法，另一种基于 Stein 的方法。在第二个证明中，我们还计算了 Wasserstein 距离的收敛速度。在这两种情况下，我们额外计算期望值的收敛速度。我们所有的证明都是由状态空间崩溃驱动的。