(m,k)-firm constraints and DBP scheduling: impact of the initial k-sequence and exact schedulability test

In this paper we study the scheduling of (m,k)-firm synchronous periodic task systems using the Distance Based Priority (DBP) scheduler. We first show three phenomena: (i) choosing, for each task, the initial k-sequence 1^k is not optimal, (ii) we can even start the scheduling from a (fictive) error state (in regard to the initial k-sequence) and (iii) the period of feasible DBP-schedules is not necessarily the task hyper-period. We then show that any feasible DBP-schedule is periodic and we upper-bound the length of that period. Lastly, based on our periodicity result we provide an exact schedulability test.