Utilization bounds for Multiprocessor Rate-Monotonic Scheduling

In this paper we extend Liu & Layland's utilization bound for fixed priority scheduling on uniprocessors to homogeneous multiprocessor systems under a partitioning strategy. Assuming that tasks are pre-emptively scheduled on each processor according to fixed priorities assigned by the Rate-Monotonic policy, and allocated to processors by the First Fit algorithm, we prove that the utilization bound is (n − 1)(2 1/2 − 1) + (m − n + 1)(2 1/(m n+1) − 1), where m and n are the number of tasks and processors respectively. This bound is valid for arbitrary utilization factors. Moreover, if all the tasks have utilization factors under a value α, the previous bound is raised and the new utilization bound considering α is calculated. Finally, simulation provides the average-case behaviour.