In this paper we study the semantics of the $\lambda$-calculus induced by Milner's encoding into the $\pi$-calculus. We show that the resulting may testing preorder on $\lambda$-terms coincides with the inclusion of Lévy-Longo trees. To establish this result, we use a refinement of the $\lambda$-calculus where the argument of a function may be of limited availability. In our $\lambda$-calculus with multiplicities, evaluation is deterministic, but it may deadlock, due to the lack of resources. We show that this feature is enough to make the $\lambda$-calculus as discriminating as the $\pi$-calculus.