In this paper we develop linear time, on-line feasibility testing algorithms for the guaranteed scheduling of firm aperiodic tasks in fixed-priority real-time systems. Firm aperiodic tasks, unlike critical periodic tasks which are guaranteed off-line, can be guaranteed dynamically only if they pass on-line feasibility tests. In this paper we derive feasibility tests drawing upon the Workload-Matrix method, an innovative technique which determines the idle capacity exactly within arbitrary intervals of periodic schedules, in time (n), where n is the number of periodic tasks. To the best of our knowledge, this is the first algorithm reported in the literature, which can accomplish this task in linear time. Previous state-of-the-art methods based on the static or dynamic slack stealing, or the EDL method by Silly and Chetto, require pseudo-polynomial time for the determination of spare capacity to guarantee one single task and they incur slack variable maintenance overhead continuously
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