Rate monotonic scheduling of real-time control systems with the minimum number of priority levels

When applying the Rate Monotonic discipline to schedule a set of periodic preemptible real-time tasks, the scheduler may be able to distinguish only a limited number of priority levels. This is common in control applications using low cost embedded controllers. If the number of tasks to be scheduled is larger than the number of distinguishable levels, the set of tasks must be partitioned in a set of priority classes. RM can be used only to arbitrate conflicts between tasks of different classes. In this paper a method to determine the minimum number of priority levels necessary to schedule the set of tasks is formally proved and its complexity analysed. Finally, a systematic method to obtain all the possible partitions with the minimum number of classes, resembling the Quine´s method to minimize Boolean functions, is also given