Abstract
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. Arriving tasks select d≥2 random compatible servers, and join the shortest queue among them. The compatibility constraints are captured by a fixed bipartite graph GN between the servers and the task types. When GN is complete bipartite, the meanfield 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 meanfield 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 to the complete bipartite compatibility graph.
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