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 the servers are exchangeable. However, modern data-centers have data locality constraints, where 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 in the two partitions represent the task types and servers, respectively, and an edge represents the server's ability to process the corresponding task type. Due to the lack of exchangeability in this model, the mean-field techniques fundamentally break down. Recent progress has been made by considering graphs with strong edge-expansion properties, i.e., where any two large subsets of vertices are well-connected. However, data locality often leads to spatial constraints, where edges are local. As a result, these bipartite graphs do not have strong expansion properties. In this paper, we consider the power-of-$d$ choices algorithm and develop a novel coupling-based approach to establish mean-field approximation for a large class of graphs that includes spatial graphs. As a corollary, we also show that, as the system size becomes large, the steady-state occupancy process for arbitrary regular bipartite graphs with diverging degrees, is indistinguishable from a fully flexible system on a complete bipartite graph. 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 $N$. Such a large-scale mixing-time result might be of independent interest. Numerical experiments are conducted, which positively support the theoretical results.