Simple Load Balancing

TL;DR

This paper analyzes a simple load balancing process on a complete graph, showing it rapidly achieves near-perfect balance within a bound depending on the number of nodes and initial load disparity.

Contribution

It provides a straightforward analysis demonstrating that the process converges quickly to balanced loads within a specific number of steps.

Findings

Reaches near-perfect balance within O(n log n + n log Δ) steps

Convergence time depends on initial load disparity Δ

Process is simple and effective for load balancing

Abstract

We consider the following load balancing process for $m$ tokens distributed arbitrarily among $n$ nodes connected by a complete graph: In each time step a pair of nodes is selected uniformly at random. Let $\ell_1$ and $\ell_2$ be their respective number of tokens. The two nodes exchange tokens such that they have $\lceil(\ell_1 + \ell_2)/2\rceil$ and $\lfloor(\ell_1 + \ell_2)/2\rfloor$ tokens, respectively. We provide a simple analysis showing that this process reaches almost perfect balance within $O(n\log{n} + n \log{\Delta})$ steps, where $\Delta$ is the maximal initial load difference between any two nodes.