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An admission control test is responsible for deciding whether a new task may be accepted by a set of running tasks, such that the already admitted and the new task are all schedulable. Admission control decisions have to betaken on-line and, hence, there is a strong interest in developing efficient algorithms for different setups. In this paper, we propose a novel constant-time admission control test for tasks scheduled on identical processors under partitioned Earliest Deadline First (EDF), i.e., once tasks have been assigned to a processor they remain on that processor. In particular, to model demanding real-time systems, we consider the case where relative deadlines may be less than the minimum separation between two consecutive task activations or jobs. The main advantage of the proposed test is that the time it takes is independent of the number of tasks currently admitted in the system. While it is possible to adapt polynomial-time schedulability tests from the literature to design a linear or even constant-time admission control for this setup, the test we propose provides a better accuracy/complexity ratio. We evaluate this test through a set of detailed experiments based on synthetic tasks and a realistic case study consisting of a real-time multimedia server.