An Average-Case Analysis for Rate-Monotonic Multiprocessor Real-Time Scheduling

We introduce the First Fit Matching Periods algorithm for rate-monotonic multiprocessor scheduling of periodic tasks with implicit deadlines and show that it yields asymptotically optimal processor assignments if utilization values are chosen uniformly at random. More precisely we prove that the expected waste is upper bounded by O(n(3/4) (log n)(3/S)). Here the waste denotes the ratio of idle times, cumulated over all processors and n gives the number of tasks. The algorithm can be implemented to run in time O(n log n) and even in the worst case, an asymptotic approximation ratio of 2 is guaranteed. Experiments yield an average waste proportional to n(0.70), indicating that the above upper bound on the expected waste is almost tight. While such average-case analyses are a classical topic of Bin Packing, to the best of our knowledge, this is the first result dealing with a theoretical average-case analysis for this scheduling problem, which was described by Liu and Lay-land more than 35 years ago and has received a lot of attention, especially in the real-time and embedded-systems community.