Mean-field analysis for load balancing on spatial graphs

The analysis of large-scale, parallel-server load balancing systems has relied heavily on mean-field analysis. A pivotal assumption for this framework is that servers are exchangeable. However, modern data-centers have data locality constraints, such that tasks of a particular type can only be routed to a small subset of servers. An emerging line of research, therefore, considers load balancing algorithms on bipartite graphs where vertices represent task types and servers, respectively. Due to the lack of exchangeability in this model, mean-field techniques fundamentally break down. Recent progress has been made on graphs with strong edge-expansion properties, that is, where any two large subsets of vertices are well-connected. However, data locality often leads to spatial graphs that do not have strong expansion properties. In this paper, we develop a novel coupling-based approach to establish mean-field approximation for a large class of graphs which includes spatial graphs. The method extends the scope of mean-field analysis far beyond the classical full-flexibility setup. En route, we prove that, starting from suitable states, the occupancy process becomes close to its steady state in a time that is independent of system size, which might be of independent interest. Numerical experiments are conducted, which positively support the theoretical results.