We study large-scale systems operating under the JSQ$(d)$ policy in the presence of stringent task-server compatibility constraints. Consider a system with $N$ identical single-server queues and $M(N)$ task types, where each server is able to process only a small subset of possible task types. Each arriving task selects $d\geq 2$ random servers compatible to its type, and joins the shortest queue among them. The compatibility constraint is naturally captured by a fixed bipartite graph $G_N$ between the servers and the task types. When $G_N$ is complete bipartite, the meanfield approximation is proven to be accurate. However, such dense compatibility graphs are infeasible due to their overwhelming implementation cost and prohibitive storage capacity requirement at the servers. Our goal in this paper is to characterize the class of sparse compatibility graphs for which the meanfield approximation remains valid. To achieve this, first, we introduce a novel graph expansion-based notion, called proportional sparsity, and establish that systems with proportionally sparse compatibility graphs match the performance of a fully flexible system, asymptotically in the large-system limit. Furthermore, for any $c(N)$ satisfying $$\frac{Nc(N)}{M(N)\ln(N)}\to \infty\quad \text{and}\quad c(N)\to \infty,$$ as $N\to\infty$, we show that proportionally sparse random compatibility graphs can be designed, so that the degree of each server is at most $c(N)$. This reduces the server-degree almost by a factor $N/\ln(N)$, compared to the complete bipartite compatibility graph, while maintaining the same asymptotic performance. Extensive simulation experiments are conducted to corroborate the theoretical results.

中文翻译：

---

严格兼容约束下的负载均衡

我们研究了在存在严格的任务服务器兼容性约束的情况下在 JSQ$(d)$ 策略下运行的大型系统。考虑一个具有 $N$ 相同单服务器队列和 $M(N)$ 任务类型的系统，其中每个服务器只能处理可能任务类型的一小部分。每个到达的任务选择 $d\geq 2$ 个与其类型兼容的随机服务器，并加入其中最短的队列。兼容性约束自然由服务器和任务类型之间的固定二部图 $G_N$ 捕获。当 $G_N$ 是完全二分时，均值场近似被证明是准确的。然而，这种密集的兼容性图是不可行的，因为它们压倒性的实施成本和服务器上过高的存储容量要求。我们在本文中的目标是表征均值场近似仍然有效的稀疏兼容性图类。为了实现这一点，首先，我们引入了一种新的基于图扩展的概念，称为比例稀疏性，并建立具有比例稀疏兼容性图的系统与完全灵活系统的性能相匹配，渐近地在大系统限制中。此外，对于任何满足 $$\frac{Nc(N)}{M(N)\ln(N)}\to \infty\quad \text{and}\quad c(N)\到 \infty,$$ 为 $N\to\infty$，我们表明可以设计成比例稀疏的随机兼容性图，使得每个服务器的度数最多为 $c(N)$。与完整的二部兼容性图相比，这将服务器度几乎减少了一个因子 $N/\ln(N)$，同时保持相同的渐近性能。