Earliest-Deadline-First is known to be the optimal service discipline to guarantee time deadlines. We investigate here the performance of a single server queue under this discipline. The physical system is described at each time t, by an atomic measure, the atoms of which are the residual deadlines at time t. Working with measure-valued semi-martingales, we are able to derive the fluid limit of this system in critical and super-critical regimes. This yields to some approximations of the loss probability and several other performance parameters.