Best-Case Response Times of Real-Time Tasks Reinder J. Bril Liesbeth Steffens Wim F.J. Verhaegh Philips Research Laboratories Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands reinder.bril,liesbeth.steffens,wim.verhaegh @philips.com Abstract In this paper, we present a simple recursive equation to determine the best-case response times of periodic tasks under fixed-priority preemptive scheduling and arbitrary phasings. The approach is of a similar nature to the one used to determine worst-case response times, in the sense that, where a critical instant is considered to determine the latter, we base our analysis on an optimal instant, in which all higher priority tasks have a simultaneous release that coincides with the best- case completion of the task under consideration. The resulting recursive equation closely resembles the one for worst-case response times, apart from a term 1 difference, and the fact that the best- case response times are approached from above. The resulting iterative procedure is illustrated by means of a small example. Finally, we discuss the effect of the best-case response times on completion jitter, as well as the effect of release jitter on the best-case response times. Key words: response time, best case, periodic tasks, preemptive scheduling, fixed priority, crit- ical instant, optimal instant, jitter, real-time systems. 1 Introduction One of the main performance issues in real-time computing systems is to determine whether or not a set of periodic tasks can be processed on a resource without exceeding their deadlines, using fixed-priority preemptive scheduling, and under arbitrary phasings of the tasks [2, 6]. To answer this question on an abstract level, Liu & Layland [7] have derived constraints for workloads. An- other method is to have an exact worst-case analysis of response times of all tasks under arbitrary phasings [3] and compare them to their deadlines. This worst-case analysis is as follows. Given are a set of n tasks τ 1 τ 2 τ n , and for each task τ j a period T j of release and a worst-case computation time WC j . The worst-case response time WR i of a task τ i is defined as the maximum time between any release of task τ i and its corresponding completion, where the maximum is taken over all executions of τ i and all possible phasings of the tasks with respect to each other. Liu & Layland [7] have shown that to determine this maximum, we can restrict ourselves to so-called critical instants, which are given by situations that all tasks have a simultaneous release. From this notion of critical instants, Joseph & Pandya [3] have derived that the worst-case response time WR i of a task τ i is given by the smallest (positive) value that satisfies the following recursive equation. WR i WC i ∑ j hp i WR i T j WC j Here, hp i denotes the set of tasks with a higher priority than τ i . We can use the following iterative In: Proc. 2nd Philips Workshop on Scheduling and Resource Management (SCHARM’01), Eindhoven, The Netherlands, June 28-29, pp. 19-27, 2001, PRLE document NL-R 20.914  Koninklijke Philips Electronics N.V. 2001 19