We consider a real-time system where a single processor with variable speed executes an infinite sequence of sporadic and independent jobs. We assume that job sizes and relative deadlines are bounded by C and $$\varDelta $$ Δ respectively. Furthermore, $$S_{\max }$$ S max denotes the maximal speed of the processor. In such a real-time system, a speed selection policy dynamically chooses ( i.e. , on-line) the speed of the processor to execute the current, not yet finished, jobs. We say that an on-line speed policy is feasible if it is able to execute any sequence of jobs while meeting two constraints: the processor speed is always below $$S_{\max }$$ S max and no job misses its deadline. In this paper, we compare the feasibility region of four on-line speed selection policies in single-processor real-time systems, namely Optimal Available $${\text{(OA)}}$$ (OA) (Yao et al. in IEEE annual foundations of computer science, 1995), Average Rate $${\text{(AVR)}}$$ (AVR) (Yao et al. 1995), $${\text{(BKP)}}$$ (BKP) (Bansal in J ACM 54:1, 2007), and a Markovian Policy based on dynamic programming $${\text{(MP)}}$$ (MP) (Gaujal in Technical Report hal-01615835, Inria, 2017). We prove the following results: $$ {\text{(OA)}}$$ (OA) is feasible if and only if $$S_{\max } \ge C (h_{\varDelta -1}+1)$$ S max ≥ C ( h Δ - 1 + 1 ) , where $$h_n$$ h n is the n -th harmonic number ( $$h_n = \sum _{i=1}^n 1/i \approx \log n$$ h n = ∑ i = 1 n 1 / i ≈ log n ). $${\text{(AVR)}}$$ (AVR) is feasible if and only if $$S_{\max } \ge C h_\varDelta $$ S max ≥ C h Δ . $${\text{(BKP)}}$$ (BKP) is feasible if and only if $$S_{\max } \ge e C$$ S max ≥ e C (where $$e = \exp (1)$$ e = exp ( 1 ) ). $${\text{(MP)}}$$ (MP) is feasible if and only if $$S_{\max } \ge C$$ S max ≥ C . This is an optimal feasibility condition because when $$S_{\max } < C$$ S max < C no policy can be feasible. This reinforces the interest of $${\text{(MP)}}$$ (MP) that is not only optimal for energy consumption (on average) but is also optimal regarding feasibility.

中文翻译：

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实时系统中在线速度策略的可行性

我们考虑一个实时系统，其中具有可变速度的单个处理器执行无限序列的零星和独立作业。我们假设作业大小和相对截止日期分别受 C 和 $$\varDelta $$ Δ 的限制。此外，$$S_{\max }$$ S max 表示处理器的最大速度。在这样的实时系统中，速度选择策略动态地选择（即在线）处理器的速度来执行当前的、尚未完成的作业。我们说在线速度策略是可行的，如果它能够在满足两个约束的同时执行任何作业序列：处理器速度始终低于 $$S_{\max }$$ S max 并且没有作业错过其最后期限。在本文中，我们比较了单处理器实时系统中四种在线速度选择策略的可行性区域，即最优可用 $${\text{(OA)}}$$ (OA)（Yao 等人在 IEEE 计算机科学年度基础，1995 年），平均速率 $${\text{(AVR)}}$$ (AVR) (Yao et al. 1995), $${\text{(BKP)}}$$ (BKP) (Bansal in J ACM 54:1, 2007)，以及基于动态规划的马尔可夫策略 $${ \text{(MP)}}$$ (MP)（Gaujal 在技术报告 hal-01615835，Inria，2017 年）。我们证明以下结果： $$ {\text{(OA)}}$$ (OA) 是可行的当且仅当 $$S_{\max } \ge C (h_{\varDelta -1}+1)$ $S max ≥ C ( h Δ - 1 + 1 ) ，其中 $$h_n$$ hn 是第 n 个谐波数 ( $$h_n = \sum _{i=1}^n 1/i \approx \log n$$ hn = ∑ i = 1 n 1 / i ≈ log n )。$${\text{(AVR)}}$$ (AVR) 是可行的，当且仅当 $$S_{\max } \ge C h_\varDelta $$ S max ≥ C h Δ 。$${\text{(BKP)}}$$ (BKP) 是可行的当且仅当 $$S_{\max } \ge e C$$ S max ≥ e C （其中 $$e = \exp (1 )$$ e = exp ( 1 ) )。$${\text{(MP)}}$$ (MP) 是可行的当且仅当 $$S_{\max } \ge C$$ S max ≥ C 。这是最佳可行性条件，因为当 $$S_{\max } < C$$ S max < C 时，没有策略可行。这增强了 $${\text{(MP)}}$$ (MP) 的兴趣，它不仅对能源消耗（平均）是最佳的，而且在可行性方面也是最佳的。