This paper is devoted to automatic competitive analysis of real-time scheduling algorithms for firm-deadline tasksets, where only completed tasks contribute some utility to the system. Given such a taskset $${\mathcal {T}}$$T, the competitive ratio of an on-line scheduling algorithm $${\mathcal {A}}$$A for $${\mathcal {T}}$$T is the worst-case utility ratio of $${\mathcal {A}}$$A over the utility achieved by a clairvoyant algorithm. We leverage the theory of quantitative graph games to address the competitive analysis and competitive synthesis problems. For the competitive analysis case, given any taskset $${\mathcal {T}}$$T and any finite-memory on-line scheduling algorithm $${\mathcal {A}}$$A, we show that the competitive ratio of $${\mathcal {A}}$$A in $${\mathcal {T}}$$T can be computed in polynomial time in the size of the state space of $${\mathcal {A}}$$A. Our approach is flexible as it also provides ways to model meaningful constraints on the released task sequences that determine the competitive ratio. We provide an experimental study of many well-known on-line scheduling algorithms, which demonstrates the feasibility of our competitive analysis approach that effectively replaces human ingenuity (required for finding worst-case scenarios) by computing power. For the competitive synthesis case, we are just given a taskset $${\mathcal {T}}$$T, and the goal is to automatically synthesize an optimal on-line scheduling algorithm $${\mathcal {A}}$$A, i.e., one that guarantees the largest competitive ratio possible for $${\mathcal {T}}$$T. We show how the competitive synthesis problem can be reduced to a two-player graph game with partial information, and establish that the computational complexity of solving this game is Np-complete. The competitive synthesis problem is hence in Np in the size of the state space of the non-deterministic labeled transition system encoding the taskset. Overall, the proposed framework assists in the selection of suitable scheduling algorithms for a given taskset, which is in fact the most common situation in real-time systems design.

中文翻译：

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使用图形游戏对实时调度进行自动竞争分析

本文致力于对固定期限任务集的实时调度算法进行自动竞争分析，其中只有已完成的任务才能为系统提供一些效用。给定这样一个任务集 $${\mathcal {T}}$$T，在线调度算法 $${\mathcal {A}}$$A 对 $${\mathcal {T}}$ 的竞争率$T 是 $${\mathcal {A}}$$A 与千里眼算法实现的效用的最坏情况效用比。我们利用定量图游戏的理论来解决竞争分析和竞争综合问题。对于竞争分析案例，给定任何任务集 $${\mathcal {T}}$T 和任何有限内存在线调度算法 $${\mathcal {A}}$$A，我们表明，$${\mathcal {A}}$$A 在 $${\mathcal {T}}$T 中的竞争比率可以在 $${\ 状态空间大小的多项式时间内计算数学{A}}$$A。我们的方法很灵活，因为它还提供了对已发布任务序列的有意义的约束进行建模的方法，这些约束决定了竞争比率。我们提供了许多著名的在线调度算法的实验研究，这证明了我们的竞争分析方法的可行性，该方法可以通过计算能力有效地取代人类的独创性（寻找最坏情况所必需的）。对于竞争性综合案例，我们只是给定了一个任务集 $${\mathcal {T}}$$T，目标是自动合成一个最优的在线调度算法 $${\mathcal {A}}$$一个，即 一种保证 $${\mathcal {T}}$$T 可能的最大竞争比率。我们展示了如何将竞争合成问题简化为具有部分信息的两人图游戏，并确定解决该游戏的计算复杂度是 Np 完全的。因此，竞争性综合问题在编码任务集的非确定性标记转换系统的状态空间大小的 Np 中。总的来说，所提出的框架有助于为给定的任务集选择合适的调度算法，这实际上是实时系统设计中最常见的情况。因此，竞争性综合问题在编码任务集的非确定性标记转换系统的状态空间大小的 Np 中。总的来说，所提出的框架有助于为给定的任务集选择合适的调度算法，这实际上是实时系统设计中最常见的情况。因此，竞争性综合问题在编码任务集的非确定性标记转换系统的状态空间大小的 Np 中。总的来说，所提出的框架有助于为给定的任务集选择合适的调度算法，这实际上是实时系统设计中最常见的情况。