The hydrodynamic limit of a randomized load balancing network

TL;DR

This paper studies the large-scale behavior of a randomized load balancing network with general service times, introducing a novel particle representation and proving convergence to a deterministic hydrodynamic limit.

Contribution

It extends previous models by analyzing non-exponential service distributions using measure-valued processes and establishes a hydrodynamic limit and propagation of chaos.

Findings

Convergence of scaled state processes to a deterministic measure-valued solution.

Introduction of a novel particle representation for the network state.

Proof of asymptotic independence (propagation of chaos) among queues.

Abstract

Randomized load balancing networks arise in a variety of applications, and allow for efficient sharing of resources, while being relatively easy to implement. We consider a network of parallel queues in which incoming jobs with independent and identically distributed service times are assigned to the shortest queue among a randomly chosen subset of $d$ queues, and leave the network on completion of service. Prior work on dynamical properties of this model has focused on the case of exponential service distributions. In this work, we analyze the more realistic case of general service distributions. We first introduce a novel particle representation of the state of the network, and characterize the state dynamics via a sequence of interacting measure-valued stochastic processes. Under mild assumptions, we show that the sequence of scaled state processes converges, as the number of servers goes to infinity, to a hydrodynamic limit that is characterized as the unique solution to a countable system of coupled deterministic measure-valued equations. We also establish a propagation of chaos result that shows that finite collections of queues are asymptotically independent. The general framework developed here is potentially useful for analyzing a larger class of models arising in diverse fields including biology and materials science.