Errata: Timing Analysis of Fixed Priority Self-Suspending Sporadic Tasks

In the paper "Timing Analysis of Fixed Priority Self-Suspending Sporadic Tasks" published in ECRTS 2015, a MILP formulation is provided to compute an upper-bound on the worst-case response time (WCRT) of one selfsuspending task running concurrently with a set of higher priority non-self-suspending tasks. Section VI of that paper extends the MILP formulation to the case where the higher priority tasks are also self-suspending. This generalisation is incorrect. We present the problem and its solution in this technical report. Errata: Timing Analysis of Fixed Priority Self-Suspending Sporadic Tasks Geoffrey Nelissen∗, José Fonseca∗, Gurulingesh Raravi∗ and Vincent Nélis∗ CISTER/INESC-TEC, ISEP, Polytechnic Institute of Porto, Portugal Email: {grrpn, jcnfo, gurhi, nelis}@isep.ipp.pt I. INCORRECT STATEMENT In [1], a MILP formulation is provided to compute an upper bound on the worst-case response time (WCRT) of one self-suspending task running concurently with a set of higher priority non-self-suspending tasks. Section VI of [1] extends the MILP formulation to the case where the higher priority tasks are also self-suspending. It is stated that: Claim 1 (in [1]). “[...] each higher priority self-suspending task τk can safely be replaced by a non-self-suspending task τ ′ k def = 〈(Ck), Dk, Tk, Jk〉 in the response time analysis. The new parameter Jk is the jitter and is given by Jk def = WCRTk −Ck. The worst-case execution time Ck of the equivalent task τ ′ k is defined as the sum of the worstcase execution times of all τk’s execution regions, that is, Ck def = ∑mk j=1 Ck,j .” This claim is supported by Theorem 2 repeated below. Theorem 2 (in [1]). The interference caused by τk ∈ hp(τi) on a self-suspending task τi is upper bounded by the interference caused by the transformed task τ ′ k def = 〈(Ck), Dk, Tk, Jk〉. Although Theorem 2 is correct, Claim 1 is not. It is demonstrated with a counter-example below. Counter-Example 1. Assume the task set composed of three tasks τ1 = 〈(1), 4, 4, 0〉, τ2 = 〈(1, 9, 1), 29, 29, 0〉 and τ3 = 〈(3, 5, 3), 100, 100, 0〉. τ1 has the highest priority and τ3 the lowest. We are interested in computing the WCRT of τ3. Since τ1 does not self-suspend we get τ ′ 1 = τ1 and using the definition provided in Claim 1, we get τ ′ 2 = 〈(2), 29, 29, J2〉 where J2 = WCRT2 −C2 = WCRT2 −2. Since the minimum inter-arrival time of τ1 is smaller than the suspension time of τ2, task τ1 generates the worst-case interference when it is released synchronously with each execution region of τ2 (see Figure 1(b)). In which case, we get WCRT2 = 13 and thus J2 = 13− 2 = 11. Figure 1(a) depicts one of the release patterns that generates the WCRT of τ3 when executed concurrently with the modified tasks τ ′ 1 and τ ′ 2. In that execution scenario, the WCRT of τ3 is 16. Indeed, due to its large inter-arrival time, task τ ′ 2 can interfere at most once with τ3 since, even considering its release jitter, the earliest possible release for its second job is at time T2 − J2 = 18 (see Figure 1(a)). Figure 1(b) shows the WCRT of τ3 when it executes concurrently with the actual tasks τ1 and τ2. As it can be seen, 0 t τ1 τ2 ' τ3 4 8 12 τ1