A better polynomial-time schedulability test for real-time fixed-priority scheduling algorithms

The problem of scheduling real-time periodic task has been studied extensively since its first introduction by C.L. Liu and J.W. Layland in their classic paper (1973). Due to several merits of the fixed-priority scheduling scheme, a lot of research work has focused on the analysis of fixed-priority scheduling algorithms. For the case that the deadlines of the executions of all the tasks coincide with the ends of their corresponding periods. Liu and Layland derived a worst-case utilization bound for a task set to be schedulable by the rate-monotonic (RM) algorithm. A. Burchard et al. (1995) presented another schedulability condition for RM, which has a higher utilization bound under a certain task condition. Although their closed-form utilization bounds provide a convenient way for testing the schedulability of a task set under the RM algorithm, the schedulability test using their bounds is too pessimistic since a lot of task sets with total utilizations larger than their bounds (and less than or equal to 1) are still schedulable by RM. In this paper, we propose a polynomial-time schedulability test and prove that it is better than Liu and Layland's and Burchard's utilization bounds in the sense that as long as the total utilization of a task set is less than or equal to their bounds, our schedulability test will always answer positively for the schedulability of the task set under RM and even if a feasible task set has a total utilization larger than their bounds, our schedulability test will still answer positively with a high probability. We also show how to generalize our polynomial-time schedulability test to handle general task sets scheduled by arbitrary fixed-priority scheduling algorithms.