Stability of the Greedy Algorithm on the Circle

TL;DR

This paper proves that in a circular single-server system with a greedy routing mechanism, having a service rate higher than the arrival rate guarantees positive recurrence regardless of the server's speed.

Contribution

It confirms Coffman and Gilbert's 1987 conjecture that a service rate exceeding the arrival rate ensures system stability for any server speed.

Findings

The conjecture by Coffman and Gilbert is validated.

System stability is guaranteed when service rate > arrival rate.

Stability holds for all server speeds.

Abstract

We consider a single-server system with service stations in each point of the circle. Customers arrive after exponential times at uniformly-distributed locations. The server moves at finite speed and adopts a greedy routing mechanism. It was conjectured by Coffman and Gilbert in~1987 that the service rate exceeding the arrival rate is a sufficient condition for the system to be positive recurrent, for any value of the speed. In this paper we show that the conjecture holds true.