Admission controllers are used to prevent overload in systems with dynamically arriving tasks. Typically, these admission controllers are based on sufficient (but not necessary) capacity bounds in order to maintain a low computational complexity. In this paper we present how exact admission-control for aperiodic tasks can be efficiently obtained. Our first result is an admission controller for purely aperiodic task sets where the test has the same runtime complexity as utilization-based tests. Our second result is an extension of the previous controller for a baseload of periodic tasks. The runtime complexity of this test is lower than for any known exact admission-controller.