We provide a constant time schedulability test for an on-line multiprocessor server handling aperiodic tasks. Dhall's effect is avoided by dividing the tasks in two priority classes based on task utilization: heavy and light. We prove that if the load on the multiprocessor server stays below U threshold = 3 - root 7 approximately equals 35.425%, the server can accept an incoming aperiodic task and guarantee that the deadlines of all accepted tasks will be met. The same number 35.425% is also a threshold for a task to be characterized as heavy. The bound U threshold = 3 - root 7 approximately equals 35.425% is easy-to-use, but not sharp if we know the number of processors in the multiprocessor system. Assuming the server to be equipped with m processors, we calculate a formula for the sharp bound U threshold (m), which converges to U threshold from above as m -> infinity . The results are based on a utilization function u(x) = 2(1 - x)/(2 + root 2+2x). By using this function, the performance of the multiprocessor server can in some cases be improved beyond U threshold(m) by paying the extra overhead of monitoring the individual utilization of the current tasks.