Exploiting Harmonic Periods to Improve Linearly Approximated Response-Time Upper Bounds Chidiebere G. U. Okwudire, Martijn M. H. P. van den Heuvel, Reinder J. Bril and Johan J. Lukkien Exact schedulability tests for fixed-priority preemptively scheduled systems are pseudo-polynomial in complexity. A linear-time sufficient test [1,2] has therefore been developed to estimate response-time upper bounds. In line with utilization-based sufficient tests, we propose to improve this test for task sets with harmonically related task periods. Moreover, we make it possible to reuse this test in the context of hierarchically scheduled resources. In such systems several applications are given a virtual share (budget) of the processor. By modeling the unavailability of processor resources to an application as two fictive tasks, we can also use a budget’s period to improve response-time bounds. Affiliation Department of Mathematics and Computer Science Eindhoven University of Technology (TU/e) Den Dolech 2, 5612 AZ Eindhoven, The Netherlands Email: m.m.h.p.v.d.heuvel@tue.nl References [1] R. I. Davis and A. Burns, “Response time upper bounds for fixed priority real-time systems,” in Proc. RTSS, Dec. 2008, pp. 407–418. [2] E. Bini and S. Baruah, “Efficient computation of response time bounds under fixed-priority scheduling,” in Proc. RTNS, March 2007, pp. 95–104. [3] A. Easwaran, M. Anand, and I. Lee, “Compositional analysis framework using EDP resource models,” in Proc. RTSS, Dec. 2007, pp. 129–138. [4] C.G.U. Okwudire, M.M.H.P. van den Heuvel, R.J. Bril and J.J. Lukkien, “Exploiting Harmonic Periods to Improve Linearly Approximated Response- Time Upper Bounds”, in Proc. ETFA (WiP), Sept. 2010 Abstract 1. Existing Response-time Upper Bounds Figure 1: Davis and Burns’ [1] approach. Figure 2: Tangent-of-the-combined-task approach. 3. Tangent of the combined task 975 1195 800 100 975 τ 3 155 155 150 75 200 τ 2 5 5 5 5 10 τ 1 R UB (new) R UB [1] R (exact) Computation time Period ( = deadline) Task We can construct an artificial task which represents the combined workload of a set of higher-priority, harmonic tasks. Its period equals the hyper-period of all harmonic tasks, and its computation time is the sum of all job executions that can occur in such an interval. Figure 2 illustrates this approach on the example task set of Table I. Given two harmonic tasks with T 2 =k*T 1 and k  + , our approach • dominates the approach in [1] if C 2 > ½(k-1)(T 1 – C 1 ) • gives the tightest possible linear upper bound if (k-1)C 1 + C 2 ≥ (k-1)T 1 Both properties apply to the two fictive tasks that model the unavailability of an EDP resource. As a result, when the period of a task is harmonic with its budget, our unavailability approach together with the tangent-of- combined-tasks approach may lead to improved response-time upper bounds compared to a the straightforward method using lsbf  (t). Future Work We showed in [4] that we can reuse the method in [1, 2] on a shared EDP resource [3], , by substituting the linear processor supply y=t with a (tight) lower bound the EDP supply bound function lsbf  (t). We can alternatively model the unavailability of a partitioned EDP resource by two fictive tasks at the highest priority. These two tasks have the same period and execute in distinct periods, i.e. they do not interfere with each other. 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 resources (y) time (t) I UB τ 1 = 0.5t + 5(1-0.5) I UB τ 2 = 0.375t + 75(1-0.375) y=100 + I UB τ 1 + I UB τ 2 y = t 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 resources (y) time (t) I UB τ (1+2) = 0.875t + 175(1-0.875) y=100 + I UB τ (1+2) y = t Table I: Characteristics of an example task set. We consider task set mapped on a single, shared, fixed priority preemptively scheduled resource. The approach presented in [1, 2] assumes that the entire processor is available to a task set. The response-time upper bounds, R, derived in [1,2] work as follows: 1. linear approximation of the interference, I UB , of each higher priority task, τ i , is derived; 2. these linear approximations are summed up and the computation time of the task is added; and 3. the intersection of the resulting equation with the processor supply is calculated. The interference of a single task over a time interval of length t is defined: I j UB (t) =U j *t +C j (1-U j ). The linear processor supply over a time interval t is given by y = t. An upper bound on the worst-case response time, R i UB , of task τ i is [1]: R i UB = ( C i +∑ 1≤j<i C j (1-U j ) ) / (1- ∑ 1≤j<i U j ) In our paper we do not consider activation jitter. The above derivation is shown in Figure 1 for an example task set, see Table I. 2. EDP resource unavailability • Extend our analysis for tasks with activation jitter; • Compare our approach with utilization-based schedulability tests; • Exploit harmonic periods to efficiently calculate EDP budget parameters.