Bounds on Herman's algorithm

TL;DR

This paper analyzes Herman's self-stabilization algorithm, providing bounds on expected stabilization time and supporting the conjecture that three equally-spaced tokens maximize this time.

Contribution

It proves exact results on a related cost function and establishes a bound on expected stabilization time close to the conjectured maximum.

Findings

Expected stabilization time is maximized with three equally-spaced tokens.

Derived bounds on expected stabilization time are close to the conjectured maximum.

Provided exact results on a related cost function.

Abstract

Herman's self-stabilisation algorithm allows a ring of $N$ processors having any odd number of tokens to reach a stable state where exactly one token remains. McIver and Morgan conjecture that the expected time taken for stabilisation is maximised when there are three equally-spaced tokens. We prove exact results on a related cost function, and obtain a bound on expected time which is very close to the conjectured bound.