Using a lag-balance property to tighten tardiness bounds for global EDF

Several tardiness bounds for global EDF and global-EDF-like schedulers have been proposed over the last decade. These bounds contain a component that is explicitly or implicitly proportional to how much the system may be cumulatively lagging behind, in serving tasks, with respect to an ideal schedule. This cumulative lag is in its turn upper-bounded by upper-bounding each per-task component in isolation, and then summing individual per-task bounds. Unfortunately, this approach leads to an over-pessimistic cumulative upper bound. In fact, it does not take into account a lag-balance property of any work-conserving scheduling algorithm. In this paper we show how to get a new tardiness bound for global EDF by integrating this property with the approach used to prove the first tardiness bounds proposed in the literature. In particular, we compute a new tardiness bound for implicit-deadline tasks, scheduled by preemptive global EDF on a symmetric multiprocessor. According to our experiments, as the number of processors increases, this new tardiness bound becomes tighter and tighter than the tightest bound available in the literature, with a maximum tightness improvement of 29 %. A negative characteristic of this new bound is that computing its value takes an exponential time with a brute-force algorithm (no faster exact or approximate algorithm is available yet). As a more general result, the property highlighted in this paper might help to improve the analysis for other scheduling algorithms, possibly on different systems and with other types of task sets. In this respect, our experimental results also point out the following negative fact: existing tardiness bounds for global EDF, including the new bound we propose, may become remarkably loose if every task has a low utilization (ratio between the execution time and the minimum inter-arrival time of the jobs of the task), or if the sum of the utilizations of the tasks is lower than the total capacity of the system.