Résumé en anglais

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 is proved by compactness-uniqueness methods, and the deterministic fluid limit is the solution of an integrated equation in the space $\S^{\prime}$ of tempered distributions. We then establish the corresponding central limit theorem, i.e. the approximation of the normalized error process by a $\S^{\prime}$-valued diffusion.