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：我们分析了大尺度负载均衡系统，其中任务的处理时间分布既依赖于任务类型也依赖于服务器类型。研究聚焦于渐近情形，即服务器数量和任务类型数量按比例趋于无穷大。在异构环境中，常用的负载均衡策略（如“加入最快空闲队列”和“加入最短队列”）表现不佳，甚至缩小了稳定域。有趣的是，在本工作之前，为这一设定设计具有可证明性能保证的可扩展策略仍是一个未解难题。为填补这一空白，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个服务器预留处理能力以服务“优质”任务，并通过原始的“加入空闲队列”策略路由任务。第二种策略称为“速度优先策略”，显著提高了将任务分配给能够高速处理它们的相应“优质”服务器的概率。借助图论文献启发的框架、均场方法和随机耦合论证，我们证明两种策略均实现了渐近零排队。具体而言，随着系统规模的扩展，典型任务被分配给空闲服务器的概率趋近于1。