本研究旨在以連續時間馬可夫鏈建構退化系統在隨機衝擊下的失效模型，並探討其對系統可用度分析和決策支援的應用。由於決策者或操作者無法隨時觀察系統狀態，因此本研究提出以連續時間馬可夫鏈作為數學模型，用於描述系統在連續時間且受到隨機衝擊的情況下之失效過程，考慮系統的退化速率和隨機衝擊引起的轉移速率，並結合系統當前運作狀態，以快速分析系統之狀態轉移機率。
本研究建立連續時間馬可夫鏈並描述系統狀態之間的轉移機率，並運用Kolmogorov's Backward Equations計算系統從運作狀態到失效狀態的分布情況，最終得出系統的可靠度函數。Kolmogorov's Backward Equations是一組描述連續時間馬可夫鏈轉移機率的微分方程系統，能夠表示狀態機率隨時間變化的模式，並提供計算轉移機率的方法。透過可靠度函數和失效函數的分析，能夠有效預測系統的失效頻率和模式，並協助決策者制定適當的置換和維修策略，以降低故障風險並提高系統的可靠性。
系統可用性分析是評估系統在特定時間內可靠運行的能力，對於制定有效的管理策略和提供決策支援至關重要。透過連續時間馬可夫鏈建構的失效模型，透過系統每一狀態之極限機率，並計算系統平均可運作時間及不可運作時間，對其進行可用性分析，從而幫助決策者制定維修或預防等措施，以利提高系統可用度。

This study aims to construct a failure model for deteriorating systems under random shocks using continuous-time Markov chains and explore its applications in system availability analysis and decision support. Since decision-makers or operators cannot observe the system state continuously, this study proposes using continuous-time Markov chains as a mathematical model to describe the failure process of systems under continuous-time and random shocks. It considers the degradation rate of the system and the transition rate induced by random shocks, combined with the current operating state of the system, to analyze the state transition probabilities of the system rapidly.
In this study, we establish a continuous-time Markov chain to describe the transition probabilities between system states and utilize Kolmogorov's Backward Equations to calculate the distribution of transitions from the operating state to the failure state, ultimately obtaining the system's reliability function. Kolmogorov's Backward Equations are a set of differential equations that describe the transition probabilities of continuous-time Markov chains. They can represent the time-varying patterns of state probabilities and provide a method for calculating transition probabilities. Through the analysis of the reliability and failure functions, it is possible to effectively predict the failure frequency and patterns of the system and assist decision-makers in formulating appropriate replacement and maintenance strategies to reduce failure risks and maximize system reliability.
System availability analysis is crucial for evaluating the ability of a system to operate reliably within a specific time frame and is essential for formulating effective management strategies and providing decision support. By constructing a failure model using continuous-time Markov chains, and calculating the limiting probabilities of each system state, the study performs availability analysis by determining the average operational and non-operational times of the system. This helps decision-makers develop maintenance or preventive measures to enhance system availability.

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