1 Introduction
Selfsuspension behavior has been demonstrated to appear in complex cyberphysical realtime systems, e.g., multiprocessor locking protocols, computation offloading, and multicore resource sharing, as demonstrated in (Chen et al. (2019), Section 2). Although the impact of selfsuspension behavior has been investigated since 1990, the literature of this research topic has been flawed as reported in the review by Chen et al. (2019).
Although the review by Chen et al. (2019) provides a comprehensive survey of the literature, two unresolved issues are listed in the concluding remark. One of them is regarding the “correctness of Theorem 8 in (Devi (2003), Section 4.5) \(\dots \)supported with a rigorous proof, since selfsuspension behavior has induced several nontrivial phenomena”. This paper provides a counterexample of Theorem 8 in (Devi (2003), Section 4.5) and disproves the schedulability test.
We consider a set of implicitdeadline periodic tasks \({\mathbb {T}} = \{\tau _1, \dots , \tau _n\}\), in which each task \(\tau _i\) has its period \(T_i\), worstcase selfsuspension time \(S_i\), and worstcase execution time \(C_i\). The relative deadline \(D_i\) is set to \(T_i\). There are two main models of selfsuspending tasks: the dynamic selfsuspension and segmented (or multisegment) selfsuspension models. Devi’s analysis in Devi (2003) considers the dynamic selfsuspension model. That is, a task instance (job) released by a task \(\tau _i\) can suspend arbitrarily as long as the total amount of suspension time of the job is not more than \(S_i\).
Devi’s analysis for implicitdeadline task systems is rephrased as follows:
Theorem 1
(Devi 2003) Let \({\mathbb {T}} = \left\{ {\tau _1, \tau _2, \ldots , \tau _n}\right\} \)be a system of n implicitdeadline periodic tasks, arranged in order of nondecreasing periods. The task set \({\mathbb {T}}\)is schedulable using preemptive EDF if for all k with \(1 \le k \le n\)inequality \( \frac{B_k+B_k'}{T_k} + \sum _{i=1}^{k}\frac{C_i}{T_i} \le 1 \)holds, where \( B_k = \sum _{i=1}^{k} \min \{S_i, C_i\}\)and \( B_k' = \max _{1 \le i \le k}\left( \max \{0, S_i  C_i\}\right) \).
Note that the notation follows the survey paper by Chen et al. (2019) instead of the original paper by Devi (2003). Moreover, Devi considered arbitrarydeadline task systems with asynchronous arrival times. Our counterexample is valid by considering two implicitdeadline periodic tasks released at the same time and disproves also the general case.
2 Counterexample for Devi’s analysis
The following task set \({\mathbb {T}}\) with only two tasks provides a counterexample for Devi’s analysis:

\(\tau _1: (T_1=D_1=6, C_1=5, S_1=1)\) and

\(\tau _2: (T_2=D_2=8, C_2=\epsilon , S_2=0)\), for any \(0 <\epsilon \le 1/3\).
The test of Theorem 1 is as follows:

When \(k=1\), we have \(B_1 = 1\) and \(B_1'=0\). Therefore, when \(k=1\), we obtain \(\frac{B_k+B_k'}{T_k} + \sum _{i=1}^{k}\frac{C_i}{T_i} = 1\).

When \(k=2\), we have \(B_2 = 1\) and \(B_2'=0\). Therefore, when \(k=2\), we obtain \(\frac{B_k+B_k'}{T_k} + \sum _{i=1}^{k}\frac{C_i}{T_i} = \frac{1}{8} + \frac{\epsilon }{8} + \frac{5}{6} = \frac{23+3\epsilon }{24} \le 1\), since \(\epsilon \le 1/3\).
Therefore, Devi’s schedulability test concludes that the task set is feasibly scheduled by preemptive EDF. But, a concrete schedule as demonstrated in Figure 1 shows that one of the jobs of task \(\tau _1\) misses its deadline even when both tasks release their first jobs at the same time.
The example in Fig. 1 shows that a job of task \(\tau _1\) may be blocked by a job of task \(\tau _2\), which results in a deadline miss of the job of task \(\tau _1\). The counterexample only requires task \(\tau _1\) to suspend once. It shows that applying Devi’s analysis in Devi (2003) is unsafe even for the segmented selfsuspension model under EDF scheduling. We note that the above counterexample is only for Theorem 8 in Devi (2003). We do not examine any other schedulability tests in Devi (2003).
References
Chen JJ, Nelissen G, Huang WH, Yang M, Brandenburg B, Bletsas K, Liu C, Richard P, Ridouard F, Audsley N, Rajkumar R, de Niz D, von der Brüggen G (2019) Many suspensions, many problems: a review of selfsuspending tasks in realtime systems. RealTime Systems 55(1):144–207
Devi UC (2003) An improved schedulability test for uniprocessor periodic task systems. In 15th Euromicro Conference on RealTime Systems (ECRTS), pages 23–32
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Günzel, M., Chen, JJ. Correspondence Article: Counterexample for suspensionaware schedulability analysis of EDF scheduling. RealTime Syst 56, 490–493 (2020). https://doi.org/10.1007/s11241020093530
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DOI: https://doi.org/10.1007/s11241020093530