Distributed Proportional Fair Load Balancing in Heterogenous Systems

Abstract

We consider the problem of distributed load balancing in heterogenous parallel server systems, where the service rate achieved by a user at a server depends on both the user and the server. Such heterogeneity typically arises in wireless networks (e.g., servers may represent frequency bands, and the service rate of a user varies across bands). We assume that each server equally shares in time its capacity among users allocated to it. Users initially attach to an arbitrary server, but at random instants of time, they probe the load at a new server and migrate there if this improves their service rate. The dynamics under this distributed load balancing scheme, referred to as Random Local Search (RLS), may be interpreted as those generated by strategic players updating their strategy in a load balancing game. In closed systems, where the user population is fixed, we show that this game has pure Nash Equilibriums (NEs), and that these equilibriums get close to a Proportionally Fair (PF) allocation of users to servers when the user population grows large. We provide an anytime upper bound of the gap between the allocation under RLS and the PF allocation. In open systems, where users randomly enter the system and leave upon service completion, we establish that the RLS algorithm stabilizes the system whenever this it at all possible under centralized load balancing schemes, i.e., it is throughput-optimal. The proof of this result relies on a novel Lyapounov analysis that captures the dynamics due to both users' migration and their arrivals and departures. To our knowledge, the RLS algorithm constitutes the first fully distributed and throughput-optimal load balancing scheme in heterogenous parallel server systems. We extend our analysis to various scenarios, e.g. to cases where users can be simultaneously served by several servers. Finally we illustrate through numerical experiments the efficiency of the RLS algorithm.