Greedy walk on the real line

TL;DR

This paper analyzes a self-interacting greedy routing process on the real line, showing that the server's position diverges logarithmically over time, with implications for understanding such systems' long-term behavior.

Contribution

It introduces a model of a greedy server on the real line with Poisson arrivals and characterizes its asymptotic divergence, a novel analysis of self-interacting stochastic processes.

Findings

Server's position diverges logarithmically over time

The process is driven by Poisson customer arrivals

Asymptotic behavior of the server's trajectory is characterized

Abstract

We consider a self-interacting process described in terms of a single-server system with service stations at each point of the real line. The customer arrivals are given by a Poisson point processes on the space-time half plane. The server adopts a greedy routing mechanism, traveling toward the nearest customer, and ignoring new arrivals while in transit. We study the trajectories of the server and show that its asymptotic position diverges logarithmically in time.