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 [3, 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. [3].
Although the review by Chen et al. [3] 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 selfsuspension behavior has induced several nontrivial phenomena”. This paper provides a counterexample of Theorem 8 in [4, Section 4.5] and disproves the schedulability test.
We consider a set of implicitdeadline periodic tasks, in which each task has its period , worstcase selfsuspension time , and worstcase execution time . The relative deadline is set to . There are two main models of selfsuspending tasks: the dynamic selfsuspension and segmented (or multisegment) selfsuspension models. Devi’s analysis in [4] considers the dynamic selfsuspension 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 164165] for uniprocessor preemptive fixedpriority scheduling to uniprocessor preemptive EDF scheduling. Under preemptive EDF scheduling, the job that has the earliest absolute deadline has the highest priority. Despite the nonoptimality of EDF for scheduling selfsuspending 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 selfsuspensions from the higherpriority jobs by setting up the blocking time induced by selfsuspensions. The correctness of the analysis by Liu in [6, Page 164165] has been proved by Chen et al. [2] in 2016 for fixedpriority scheduling. The authors in [2] 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 implicitdeadline task systems is rephrased as follows:
Theorem 1.1 (Devi [4])
Let be a system of implicitdeadline periodic tasks, arranged in order of nondecreasing periods. The task set T is schedulable using preemptive EDF if
where
(1) 
(2) 
Note that the notation follows the survey paper by Chen et al. [3] instead of the original paper by Devi [4]. 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.
2 Counterexample for Devi’s Analysis
The following task set with two tasks provides a counterexample for Devi’s analysis:

and

, 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 [4] is unsafe for the segmented selfsuspension model under EDF scheduling.
Although there have been many different analyses for preemptive fixedpriority scheduling, Devi’s analysis was the only existing suspensionaware analysis for hard realtime task systems under preemptive EDF scheduling for long time until 2016, where Dong and Liu [5] developed a utilizationbased schedulability test for global EDF in multiprocessor systems. The special case when there is only one processor, i.e., in [5], can be applied for testing the schedulability of preemptive EDF in uniprocessor systems. We note that the analysis in [5] is limited to implicitdeadline task systems. For task systems that are not with implicit deadlines, the invalidation of Devi’s analysis implies that there is no suspensionaware schedulability analysis for preemptive EDF. The only safe schedulability test is the trivial suspensionoblivious analysis, which considers suspension time of the selfsuspending tasks as if they are usual execution time. (Detailed discussions can be found in [3, Section 4].
References
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