On the analysis of randomized load balancing schemes

Abstract

It is well known that simple randomized load balancing schemes can balance load effectively while incurring only a small overhead, making such schemes appealing for practical systems. In this paper, we provide new analyses for several such dynamic randomized load balancing schemes. Unlike previous analyses, we do not assume that in equilibrium each server is stochastically independent from other servers. Our work extends a previous analysis of the supermarket model, a model that abstracts a simple, efficient load balancing scheme in the setting where jobs arrive at a large system of parallel processors. In this model, customers arrive at a system of n servers as a Poisson stream of rate λn, λ<1, with service requirements exponentially distributed with mean 1. Each customer chooses d servers independently and uniformly at random from the n servers, and is served at the choice with the fewest customers according to the First In First Out (FIFO) protocol. We wish to understand how the system behaves in equilibrium. Here we examine several variations, including constant service times and threshold models, where a customer makes up to d successive choices until finding one below a set threshold. Our approach involves studying limiting, deterministic models representing the behavior of these systems as the number of servers n goes to infinity. Important results of our work include useful general theorems for showing that these deterministic systems are stable or converge exponentially to fixed points. We also demonstrate that allowing customers two choices instead of just one leads to exponential improvements in the expected time a customer spends in the system in several of the related models we study.