Mean-field Analysis for Load Balancing on Spatial Graphs

Abstract

A pivotal methodological tool behind the analysis of large-scale load balancing systems is mean-field analysis. The high-level idea is to represent the system state by aggregate quantities and characterize their rate of change as the system size grows large. An assumption for the above scheme to work is that the aggregate quantity is Markovian such that its rate of change can be expressed as a function of its current state. If the aggregate quantity is not Markovian, not only does this technique break down, the mean-field approximation may even turn out to be highly inaccurate. In load balancing systems, if servers are exchangeable, then the aggregate quantity is indeed Markovian. However, the growing heterogeneity in the types of tasks processed by modern data centers has recently motivated the research community to consider systems beyond the exchangeability assumption. The main reason stems from data locality, i.e., the fact that servers need to store resources to process tasks of a particular type locally and have only limited storage space. An emerging line of work thus considers a bipartite graph between task types and servers [2, 3, 5 -7]. In this compatibility graph, an edge between a server and a task type represents the server's ability to process these tasks. In practice, storage capacity or geographical constraints force a server to process only a small subset of all task types, leading to sparse network topologies. This motivates the study of load balancing in systems with suitably sparse bipartite compatibility graphs.