Firmness Analysis of Real-time Tasks

(m,k)-firm real-time tasks require meeting the deadline of at least m jobs out of any k consecutive jobs. When compared to hard real-time tasks, (m,k)$-firm tasks open up the possibility of tighter resource-dimensioning in implementations. Firmness analysis verifies the satisfaction of (m,k)-firmness conditions. Scheduling policies under which a set of periodic tasks runs on a resource influence the number of deadline missed jobs. Therefore, the nature of the firmness analysis problem depends on scheduling policies. In this work, we present Firmness Analysis (FAn) methods for three common scheduling policies-synchronous and asynchronous Static Priority Preemptive (SPP) policies and Time Division Multiple Access (TDMA). We first introduce the Balloon and Rake problem-the problem of striking the maximum number of balloons in a balloon line with a rake. We show that the common core of firmness analysis problems can be abstracted as the Balloon and Rake problem. Next, we prove that the Finite Point method is a solution to the Balloon and Rake problem. We illustrate how existing FAn methods for the TDMA and asynchronous SPP policies can be adapted to use the same solution framework for the Balloon and Rake problem. Using the solution of the Balloon and Rake problem, we adapt the existing FAn methods to synchronous SPP scheduling policies. The scalability of the FAn methods is compared with that of a timed-automata approach, a brute-force approach, and a Mixed Integer Linear Programing method. The FAn methods scale substantially better to firmness analysis problem instances with a large k and a high number of tasks.