利用元件可靠度之beta後驗分配來建立系統可靠度分配

Establish Reliability Probability Distribution Model by Beta Posterior Distribution

董政衢(Cheng-Chu Tung)

指導教授 ： 陳慧芬

中原大學/電機資訊學院/工業與系統工程研究所/碩士(2020年)

https://doi.org/10.6840/cycu202000695

摘要

可靠度的定義為在給定的環境下，系統或元件的壽命大於特定時間的機率。本論文考慮串聯系統、並聯系統及混聯系統(包含串並聯系統、並串聯系統及某個特定的串並混合系統)，假設元件的可靠度為相互獨立且以beta分配做為先驗分配的隨機變數，並假設元件會在失效時被更換。本論文探討當給定元件壽命值是否大於特定時間的資料的條件下，研究問題為建立系統可靠度的beta近似分配。本論文與胡偉元 (2009) 不同之處為假設元件的可靠度為以beta分配做為先驗分配的隨機變數，並假設元件會在失效時被更換，在給定元件壽命值是否大於特定時間單位的資料的條件下，建立系統可靠度的beta近似分配。
本論文的研究方法為當某個子系統是由相鄰兩個元件組成時，找出與子系統可靠度真正分配最接近的beta分配，目標函數為真正分配與beta近似分配的絕對誤差。接下來將由這兩個元件組成的子系統與下一個元件組成三元件子系統，找出與子系統可靠度真正分配最接近的beta分配。以此類推可求得由多個元件組成的串聯系統、並聯系統及混聯系統可靠度的beta近似分配。
由實驗結果可以看出當串聯系統、並聯系統及混聯系統的可靠度真正分配為beta分配時，則這三個系統可靠度的beta近似分配與真正分配很接近。所以當這三個系統可靠度的真正分配不是beta分配時，只要真正分配與beta分配家族的機率分配形狀不會相差甚遠，我們猜測本論文提出的beta近似分配也和真正分配相距不遠。

關鍵字

beta近似分配 ； 可靠度

並列摘要

Reliability is defined as the probability that the lifetime of a system or component is greater than a specified time under a given environment. The research problem of this thesis is to establish the beta approximate distribution of system reliability. The research scope and assumptions of this thesis are as follows: we assume that the reliabilities of the components are independent random variables, using the beta distribution as a prior distribution, and that the components will be replaced with new components when they fail. Each component has lifetime data on whether the component's lifetime is greater or smaller than a specific time. The systems that we discuss in this thesis include the serial system, the parallel system, and the hybrid system (including the serial-parallel system, the parallel-serial system, and the specific serial-parallel hybrid system). The difference between this thesis and Hu’s (2009) is that the reliability of the component is assumed to be a random variable based on the beta distribution as a prior distribution and that the component will be replaced when it fails. Through the condition of the component’s lifetime data is greater or smaller than a specific time and establishes the beta distribution of the system reliability.
The research method of this thesis is that when a sub-system is composed of two adjacent components, we find the beta distribution that is closest to the real distribution of the sub-system’s reliability. The objective function is the absolute error between the real distribution and the approximate beta distribution. Next, the sub-system composed of these two components and the next component is composed of a three-element sub-system. We find the beta distribution closest to the real distribution of the sub-system’s reliability. By analogy, the approximate beta distribution of the reliability of the series system, the parallel system and the hybrid system composed of multiple components can be obtained.
The experimental result shows that when the reliability of the series system, the parallel system, and the hybrid system’s real distribution is the beta distribution, the real distribution of reliability of these three systems has a very small gap with the approximate beta distribution. Therefore, when the real distribution of the reliability of these three systems is not the beta distribution, as long as the real distribution and the probability distribution shape of the beta distribution family are not very different, we estimate that the approximate beta distribution proposed in this thesis is not far from the real distribution. Since the probability distribution shape of the beta distribution family has diversified characteristics, it can cover a wide range of the feasible distributions. Therefore, it is unlikely that the real distribution of the system and the shape of the beta distribution family are far apart.

並列關鍵字

beta approximate distribution ； reliability

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