Firmness analysis of real-time applications under static-priority preemptive scheduling

(m, k)-firm real-time tasks must meet the deadline of at least m jobs out of any k consecutive jobs to satisfy the firmness requirement. Scheduling of an (m,k)-firm task requires firmness analysis, whose results are used to provide system-level guarantees on the satisfaction of firmness conditions. We address firmness analysis of an (m, k)-firm task that is intended to be added to a set of asynchronous tasks scheduled under a Static-Priority Preemptive (SPP) policy. One of the main causes of deadline misses in periodic tasks running under an SPP policy is interference from higher priority tasks. Since the synchrony between the newly added task and higher priority tasks is unknown, the interference from the higher priority tasks is also unknown. We propose an analytic Firmness Analysis (FAn) method to obtain a synchrony that results in the maximum minimum number of deadline hit jobs in any k consecutive jobs of the task. Scalability of FAn is compared with that of existing work - a brute-force search approach - and a timed-automata model of the problem that is analysed using the reachability check of the Uppaal model checker. Our method substantially reduces the complexity of the analysis.