We present an analysis of large-scale load balancing systems, where the processing time distribution of tasks depends on both the task and server types. Our study focuses on the asymptotic regime, where the number of servers and task types tend to infinity in proportion. In heterogeneous environments, commonly used load balancing policies such as Join Fastest Idle Queue and Join Fastest Shortest Queue exhibit poor performance and even shrink the stability region. Interestingly, prior to this work, finding a scalable policy with a provable performance guarantee in this setup remained an open question. To address this gap, we propose and analyze two asymptotically delay-optimal dynamic load balancing policies. The first policy efficiently reserves the processing capacity of each server for ``good" tasks and routes tasks using the vanilla Join Idle Queue policy. The second policy, called the speed-priority policy, significantly increases the likelihood of assigning tasks to the respective ``good" servers capable of processing them at high speeds. By leveraging a framework inspired by the graphon literature and employing the mean-field method and stochastic coupling arguments, we demonstrate that both policies achieve asymptotic zero queuing. Specifically, as the system scales, the probability of a typical task being assigned to an idle server approaches 1.

翻译：我们针对大规模负载均衡系统进行了分析，其中任务的处理时间分布取决于任务类型和服务器类型。研究聚焦于渐近状态，即服务器数量与任务类型数量按比例趋向无穷大。在异构环境中，常用的负载均衡策略（如加入最快空闲队列与加入最快最短队列）表现不佳，甚至缩小了稳定性区域。有趣的是，在本研究之前，如何在这种设置下找到一种具有可证明性能保证的可扩展策略仍是一个未解问题。为弥补这一空白，我们提出并分析了两种渐近延迟最优的动态负载均衡策略。第一种策略高效地为每台服务器保留处理能力，以处理"优质"任务，并采用基础的加入空闲队列策略进行任务路由。第二种策略称为速度优先策略，它显著提高了将任务分配给能高速处理该任务的相应"优质"服务器的概率。通过利用图论文献启发的框架，并采用平均场方法和随机耦合论证，我们证明这两种策略均能实现渐近零排队。具体而言，随着系统规模扩大，典型任务被分配给空闲服务器的概率趋近于1。