Time-reversibility of schedulability tests

For timing guarantees of a set of real-time tasks under a target scheduling algorithm, a number of schedulability tests have been studied. However, there still exist many task sets that are potentially schedulable by a target scheduling algorithm, but proven schedulable by none of existing schedulability tests, especially on a multiprocessor platform. In this paper, we propose a new notion of time-reversibility of schedulability tests, which yields tighter schedulability guarantees by viewing real-time scheduling under a change in the sign of time. To this end, we first define the notion of a time-reversed scheduling algorithm against a target scheduling algorithm, for example, the time-reversed scheduling algorithm against EDF (Earliest Deadline First) is LCFS (Last-Come, First-Served), and the converse also holds. Then, a schedulability test for a scheduling algorithm is said to be time-reversible with respect to schedulability, if all task sets deemed schedulable by the test are also schedulable by its time-reversed scheduling algorithm. To exploit the notion of time-reversibility for tighter schedulability guarantees, we not only prove time-reversibility of an existing schedulability test, but also develop a new time-reversible schedulability test, both of which cover additional schedulable task sets. Next, we generalize the time-reversibility theory towards partial execution. Utilizing the notion, we can assure the schedulability of a task under a target scheduling algorithm in a divide-and-conquer manner: (i) the first some units of execution guaranteed by a schedulability test for the scheduling algorithm, and (ii) the remaining execution guaranteed by a time-reversible (with respect to partial execution) schedulability test for its time-reversed scheduling algorithm. Such a divide-and-conquer approach has not been directly applied to existing schedulability tests in that they cannot address (ii) effectively. As a case study, this paper develops RTA (Response-Time Analysis) for LCFS, proves its time-reversibility, and applies the divide-and-conquer approach to the test along with an existing EDF schedulability test. Our simulation results show that the time-reversibility theory helps to find up to 13.1% additional EDF-schedulable task sets on a multiprocessor platform.