Many greedy cleaners in a Poisson environment

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.