We present a sufficient linear-time schedulability test for preemptable, asynchronous, periodic task systems with arbitrary relative deadlines, scheduled on a uniprocessor by an optimal scheduling algorithm. We show analytically and empirically that this test is more accurate than the commonly-used density condition. We also present and discuss the results of experiments that compare the accuracy and execution time of our test with that of a pseudo-polynomial-time schedulability test presented previously for a restricted class of task systems in which utilization is strictly less than one.