近幾年來，即時系統已大量應用於交通工具、醫院、電廠或工廠作業
及金融機構等，使得即時系統設計與研究成為非常重要且實用價值高的熱門學問。這些即時系統應用的特色，除了程式的執行必須保證正確性以外，還要考慮系統能否於特定的時間內完成指定的工作或運算，如果系統無法逹成每個工作或運算對於時間性的要求，則系統會造成嚴重的災難；最明顯的例子是例如控制系統上發生了問題，有可能會危及人員的性命安全或造成經濟上的巨大損失。

每個即時性工作都具有不同的工作完成期限、最晚開始執行時間，以
及所需的執行時間，而即時系統必須保證所有的即時性工作都能夠在規定的完成期限內完成。因此在即時系統執行工作之前，需要快速且即時地判斷即時工作的可排程性。即時工作的可排程性判斷主要可分為「精準性測試」或「非精準性測試」兩種類型。精準性測試為可完全正確地判斷某即時工作在特定的排程法則中能順利地執行成功，此類測試需花費較多時間；非精準性測試為不能完全確認某即時工作在特定的排程法則中是否能執行成功，此類測試需要較少時間，適用於online 測試環境中。因此，我們針對在Rate Monotonic 排程法則中執行的即時工作分別提出有效的精準性測試和非精準性測試方法。

在週期性的即時工作方面，我們首先對於精準性測試提出兩個有效率
的測試方法。藉由觀察即時工作在排程上的特性，對於測試時的初值設定及之後的演算方法皆提出改進。此外，對於非精準性測試，我們也提出考量工作週期特性的兩個方法。在多框型態的週期性即時工作方面，我們針對精準性測試和非精確性測試也分別提出考量工作週期特性的有效測試方法。

Real-time scheduling for task sets has been studied, and the corresponding schedulability analysis has been developed. Real-time systems using rate monotonic (RM) fixed priority scheduling can be checked for schedulability either by sufficient but pessimistic schedulability conditions or by exact test. In this thesis, a series of period-dependent schedulability test methods for real-time tasks scheduled by the RM policy on uniprocessor systems are proposed.

In the first portion, we find the efficient schedulability test methods for periodic tasks. Both exact schedulability test methods and sufficient but pessimistic schedulability test methods are investigated. Exact testing provides a more precise result but may not be performed in polynomial time. Audsley et al. proposed one of the earliest methods by iteratively deriving the response times of jobs. Other researchers have improved the exact test method by using different initial values for testing. In this thesis, we first improve the iterative formula used to calculate the response times of jobs. Then, we propose a new initial value for Audsley’s test method to improve the efficiency. We show that the new initial values can significantly improve the efficiency of exact testing. In addition, a better RM schedulability bound for performing the sufficient but pessimistic schedulability is also proposed. The period ratios are considered in the schedulability bound. When more period ratios are considered, the schedulability bound can be higher.

In the second portion, we find the efficient schedulability test methods for multiframe tasks. We explore both exact schedulability test methods and sufficient but pessimistic schedulability test methods. For the exact schedulability test, the period-dependent initial value is used for the test and effectively reduce the number of iterations for testing. For the sufficient but pessimistic schedulability test, we use the information of the relative period ratios between tasks in a system to derive a new schedulability condition. By considering the smallest and the largest period values in a system, we can show that the RM schedulability bound can be improved significantly. This method also can be applied to other test methods studied earlier to improve the schedulability of real-time multiframe systems.

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