Critical Instant for Probabilistic Timing Guarantees: Refuted and Revisited

In soft real-time systems, tasks may occasionally miss their deadlines. This possibility has triggered research on probabilistic timing analysis for the execution time of a single program and probabilistic response time analysis of concurrently executed tasks. Under fixed-priority preemptive uniprocessor scheduling, it was shown that the classical critical instant theorem (for deriving the worst-case schedulability or response time) by Liu and Layland (in JACM 1973) can be applied to analyze the worst-case deadline failure probability (WCDFP) and the worst-case response time exceedance probability (WCRTEP). In this work, we present a counterexample for this result, showing that the WCDFP and WCRTEP derived by the classical critical instant theorem is unsound. We further provide two sound methods: one is to account for one additional carry-in job of a higher-priority task and another is to sample and inflate the execution time of certain jobs without adding one additional carry-in job. We show that these two methods do not dominate each other and, in the evaluation, apply them to two well-known approaches based on direct convolution and Chernoff bounds.