Global Deadline-Monotonic Scheduling of Arbitrary-Deadline Sporadic Task Systems

This paper presents an error in the schedulability test for global deadlinemonotonic scheduling of arbitrary-deadline sporadic task systems in identical multiprocessor systems proposed by Baruah and Fisher in OPODIS 2007. This erratum provides a simple fix. Fortunately, the speedup bound 2+ √ 3 claimed in their paper remains valid with this simple fix. In the sporadic task model, a task τi is characterized by its relative deadline Di, its minimum inter-arrival time (period) Ti, and its worst-case execution time Ci. An arbitrary-deadline sporadic task set does not assume any relation between the relative deadlines and the periods of the tasks. Baruah and Fisher [1] considered an arbitrarydeadline sporadic task set executed on m ≥ 2 identical processors based on global deadline-monotonic (DM) scheduling, in which D1 ≤ D2 ≤ . . . ,≤ Dn. They proposed a schedulability test that is the state of the art of this problem with respect to the resource augmentation (speedup) bound. We here recall their notation as follows: – density δi of task τi: Ci/min(Di, Ti); – maximum density δmax(k) among the first k tasks: maxi=1(δi); – demand bound function of task τi: DBF(τi, t) = max ( 0, (⌊ t−Di Ti ⌋ + 1 ) Ci ) ; – load LOAD(k) of the first k tasks: LOAD(k) = maxt>0 ∑k i=1 DBF(τi,t) t . The schedulability test of task τk under global DM by Baruah and Fisher [1] is as follows: ( 1 + Dk ∆ ) LOAD(k) + (dμke − 1)δmax(k) ≤ μk (1) since Dk≤∆ ⇐ 2LOAD(k) + (dμke − 1)δmax(k) ≤ μk (2) where μk is defined as m− (m− 1)δk. The schedulability test in Eqs. (1) and (2) is based on an incorrect Lemma 3 from the original analysis [1], stated as follows: “The total remaining execution requirement of all the carry-in jobs of each task τi (that has carry-in jobs at time-instant t0) is < ∆× δmax(k).” There was one unsound step in the second part of the equation set (5) in the original proof [1]. They stated that the conditionmφi−(m−1)yi < (m−(m−1)δk)φi implies ? This paper has been supported by DFG, as part of the Collaborative Research Center SFB876 (http://sfb876.tu-dortmund.de/). that yi > φiδmax(k). The fact is that mφi − (m − 1)yi < (m − (m − 1)δk)φi only implies yi > φiδk. The original implication holds when δk ≥ δi, i.e., δk is equal to δmax(k). Without such an implication, the remaining execution requirement of task τi can only be safely stated as < ∆δi + φi(δi − δk) in their proof. However, this correct inequality introduces an unknown variable φi. One simple solution to fix their analysis is to define μk asm−(m−1)δmax(k). With this solution, all the analysis steps are valid and their Lemma 3 is correct. Therefore, the schedulability test in Eq. (1) and Eq. (2) both remain valid if μk is defined as m− (m− 1)δmax(k). Based on the above discussion, the (sufficient) schedulability test in Corollary 1 by Baruah and Fisher [1] should be restated as LOAD(k) ≤ 1 2 (m− (m− 1)δmax(k)) (1− δmax(k)). (3) Fortunately, the above schedulability test in Eq. (3) still leads to the speedup bound 2+ √ 3, as the procedure in the proof of Lemma 5 in the original analysis remains valid by using only the condition δmax(k) ≤ x.