Many greedy cleaners in a Poisson environment

TL;DR

This paper models multiple greedy cleaners operating on a Poisson-distributed dust environment on a star-like space, analyzing their asymptotic escape behavior as the number of cleaners grows large.

Contribution

It introduces a novel greedy cleaning model on a star-like space with Poisson dust, analyzing the asymptotic escape behavior based on the parameter .

Findings

All cleaners eventually escape to infinity.

The escape dynamics depend on the value of .

As N increases, the behavior of the system becomes predictable.

Abstract

We introduce a new ``greedy cleaning'' model where a star-like state space (containing N half-lines connected by the origin) is covered by a homogeneous Poisson process of ``dust particles'', and N^{\alpha} cleaners/workers proceed with cleaning in a ``greedy'' manner: each worker chooses the closest particle next. Assuming \alpha \in (0,1), we analyse the asymptotic behaviour of the workers, as N\to\infty. We show that eventually all of them escape to infinity and that the way how do they do it depends on the value of \alpha.