  In this paper, we present a functional fluid limit theorem and a functional
central limit theorem for a queue with an infinity of servers M/GI/$\infty$.
The system is represented by a point-measure valued process keeping track of
the remaining processing times of the customers in service. The convergence in
law of a sequence of such processes after rescaling is proved by
compactness-uniqueness methods, and the deterministic fluid limit is the
solution of an integrated equation in the space $\mathcal{S}^{\prime}$ of
tempered distributions. We then establish the corresponding central limit
theorem, that is, the approximation of the normalized error process by a
$\mathcal{S}^{\prime}$-valued diffusion. We apply these results to provide
fluid limits and diffusion approximations for some performance processes.
