Precautious-RM: a predictable non-preemptive scheduling algorithm for harmonic tasks

Abstract

A major requirement of many real-time embedded systems is to have time-predictable interaction with the environment. More specifically, they need fixed or small sampling and I/O delays, and they cannot cope with large delay jitters. Non-preemptive execution is a known method to reduce the latter delay; however, the corresponding scheduling problem is NP-Hard for periodic tasks. In this paper, we present Precautious-RM as a predictable linear-time online non-preemptive scheduling algorithm for harmonic tasks which can also deal with the former delay, namely sampling delay. We derive conditions of optimality of Precautious-RM and show that satisfying those conditions, tight bounds for best- and worst-case response times of the tasks can be calculated in polynomial-time. More importantly, response time jitter of the tasks is analyzed and it is proven that under specific conditions, each task has either one or two values for response time, which leads to improving the predictability of the system interaction with the environment. Simulation results demonstrate efficiency of Precautious-RM in increasing accuracy of control applications.

Appendix

Lemma 4

Suppose a non-preemptive task set $T$ consistent with Conditions (1), (2) and (3); the worst-case response time of task $T_n$ follows (4). For task $T_{n-1}$, if the processor is idle at the release of $T_{n-1}$, its response time will be $c_n+c_{n-1}$ according to Precautious-RM, and also there will be an empty interval with length $p_n-c_n$ before the third release of $T_n$. Otherwise, if at the release of task $T_{n-1}$, the processor is busy by task $T_i, 1 \le i< n-1$, the response time of $T_{n-1}$ will be bounded by $p_n+c_n+c_{n-1}$.

Proof

As shown in Line 4 of Algorithm 1, each task is scheduled only if it does not cause the next release of task $T_n$ to miss its deadline. Hence, none of the instances of $T_n$ will miss their deadline because of blocking caused by a low priority task. However, the WCRT of $T_n$ is bounded by its period since in the cases where the execution time of a task is equal to $2(p_n - c_n)$, the latest $T_n$ has to be scheduled at the end of its period.

According to the definition of harmonic tasks in Sect. 2, on release of $T_{n-1}$, all other tasks with periods smaller than $T_{n-1}$ are also released. In our case, one instance of $T_n$ is released simultaneously with $T_{n-1}$. We call it the first $T_n$. If the processor is idle at the release time of $T_{n-1}$ (as shown in Fig. 23), this task can be scheduled right after the first $T_n$ and its response time will be $c_n+c_{n-1}$ which is definitely smaller than (4). Besides, since it has been executed before the third release of task $T_n$, then an empty interval with length $p_n-c_n$ will be created after finish of the third $T_n$.

Fig. 23

Schedule of two tasks $T_n$ and $T_{n-1}$, when the processor is idle

Fig. 24

Schedule of two tasks $T_n$ and $T_{n-1}$, when the processor is busy

Moreover, if at the release time of $T_{n-1}$, the processor is busy by task $T_i$, $1 \le i < n-1$, when it becomes idle again, Precautious-RM will select the first $T_n$, since it has the highest priority. This situation has been shown in Fig. 24. In the worst-case, when the first $T_n$ is finished, $T_{n-1}$ cannot fit into the remaining interval according to Line 4 of Algorithm 1; consequently, an idle time will be inserted until release of the second $T_n$. At that time, the algorithm will select the second $T_n$ but when it is finished at time $p_n+c_n$, the execution of $T_{n-1}$ is definitely started because according to Line 4 of the algorithm, the previous executed task has been $T_n$. In this case, the response time of $T_{n-1}$ will be $p_n+c_n+c_{n-1}$ which is smaller than $3p_n-c_n$ since $p_n+c_n+c_i \le p_n+c_n+2(p_n-c_n ) \le 3p_n-c_n$. Thus, the proof is complete. $\square$