Load Balancing Under Strict Compatibility Constraints

TL;DR

This paper investigates how to maintain efficient load balancing in large-scale systems with strict task-server compatibility constraints by identifying sparse graph structures that preserve performance.

Contribution

The paper introduces the concept of proportional sparsity in compatibility graphs and proves it preserves meanfield performance, reducing server degrees significantly.

Findings

Proportional sparsity maintains performance similar to fully flexible systems.

Sparse random compatibility graphs can be designed with low server degrees.

Performance remains optimal under certain growth conditions of c(N).

Abstract

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.