Load Balancing Under Strict Compatibility Constraints

Consider a system with N identical single-server queues and a number of task types, where each server is able to process only a small subset of possible task types. Arriving tasks select $d\ge 2$ random compatible servers and join the shortest queue among them. The compatibility constraints are captured by a fixed bipartite graph between the servers and the task types. When the graph is complete bipartite, the mean-field approximation is accurate. However, such dense compatibility graphs are infeasible for large-scale implementation. We characterize a class of sparse compatibility graphs for which the mean-field approximation remains valid. For this, we introduce a novel notion, called proportional sparsity, and establish that systems with proportionally sparse compatibility graphs asymptotically match the performance of a fully flexible system. Furthermore, we show that proportionally sparse random compatibility graphs can be constructed, which reduce the server degree almost by a factor $N/\ln (N)$ compared with the complete bipartite compatibility graph.

Funding: This work was supported by the National Science Foundation [Grant CCF-2113027].