Self-suspension behavior has been demonstrated to appear in complex cyber-physical real-time systems, e.g., multiprocessor locking protocols, computation offloading, and multicore resource sharing, as demonstrated in [3, Section 2]. Although the impact of self-suspension behavior has been investigated since 1990, the literature of this research topic has been flawed as reported in the review by Chen et al. .
Although the review by Chen et al.  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 [4, Section 4.5] supported with a rigorous proof, since self-suspension behavior has induced several non-trivial phenomena”. This paper provides a counterexample of Theorem 8 in [4, Section 4.5] and disproves the schedulability test.
We consider a set of implicit-deadline periodic tasks, in which each task has its period , worst-case self-suspension time , and worst-case execution time . The relative deadline is set to . There are two main models of self-suspending tasks: the dynamic self-suspension and segmented (or multi-segment) self-suspension models. Devi’s analysis in  considers the dynamic self-suspension model. That is, a task instance (job) released by a task can suspend arbitrarily as long as the total amount of suspension time of the job is not more than .
The analysis by Devi in Theorem 8 in [4, Section 4.5] extended the analysis proposed by Jane W.S. Liu in her book [6, Page 164-165] for uniprocessor preemptive fixed-priority scheduling to uniprocessor preemptive EDF scheduling. Under preemptive EDF scheduling, the job that has the earliest absolute deadline has the highest priority. Despite the non-optimality of EDF for scheduling self-suspending task systems as shown in [7, 1], EDF remains one of the most adopted scheduling strategies.
Devi’s analysis quantifies the additional interference due to self-suspensions from the higher-priority jobs by setting up the blocking time induced by self-suspensions. The correctness of the analysis by Liu in [6, Page 164-165] has been proved by Chen et al.  in 2016 for fixed-priority scheduling. The authors in  noted that “Even though the authors in this paper are able to provide a proof to support the correctness, the authors are not able to provide any rationale behind this method which treats suspension time as blocking time.”
Devi’s analysis for implicit-deadline task systems is rephrased as follows:
Theorem 1.1 (Devi )
Let be a system of implicit-deadline periodic tasks, arranged in order of non-decreasing periods. The task set T is schedulable using preemptive EDF if
Note that the notation follows the survey paper by Chen et al.  instead of the original paper by Devi . Moreover, Devi considered arbitrary-deadline task systems with asynchronous arrival times. Our counterexample is valid by considering two implicit-deadline periodic tasks released at the same time.
2 Counterexample for Devi’s Analysis
The following task set with two tasks provides a counterexample for Devi’s analysis:
, for any .
The test of Theorem 1.1 is as follows:
When , we have and . Therefore, when , .
When , we have and . Therefore, when , , since .
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 misses its deadline even when both tasks release their first jobs at the same time.
The example in Figure 1 shows that a job of task may be blocked by a job of task , which results in a deadline miss of the job of task . However, in Devi’s schedulability analysis, such blocking is never considered since and do not have any term related to .
3 Conclusion and Discussions
The counterexample in Section 2 only requires task to suspend once. The counterexample shows that applying Devi’s analysis in  is unsafe for the segmented self-suspension model under EDF scheduling.
Although there have been many different analyses for preemptive fixed-priority scheduling, Devi’s analysis was the only existing suspension-aware analysis for hard real-time task systems under preemptive EDF scheduling for long time until 2016, where Dong and Liu  developed a utilization-based schedulability test for global EDF in multiprocessor systems. The special case when there is only one processor, i.e., in , can be applied for testing the schedulability of preemptive EDF in uniprocessor systems. We note that the analysis in  is limited to implicit-deadline task systems. For task systems that are not with implicit deadlines, the invalidation of Devi’s analysis implies that there is no suspension-aware schedulability analysis for preemptive EDF. The only safe schedulability test is the trivial suspension-oblivious analysis, which considers suspension time of the self-suspending tasks as if they are usual execution time. (Detailed discussions can be found in [3, Section 4].
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