本文的目的是以解析解計算所謂的 「系統可靠度」,定義為某隨機系統最後的產出值會大於或等於一個預訂需求量的機率。本文研究的隨機系統是指由許多工作站連結而成的網路,其中每個工作站都有個別的隨機產能限制,且每個進入工作站的待加工品有特定的比率成為良品或不良品。上述定義的 「系統可靠度」 是隨機系統的一個重要績效指標。製造系統、網際網路系統、物流系統都是上述隨機系統的應用。
根據最新文獻 [19] , 有18篇文獻具有共同的錯誤,使得他們計算出來的系統可靠度值是錯誤的。雖然此最新文獻提出的 Song Rule 是一個正確計算可靠度的解析法,但是只限於單一生產線重工狀況,而且該方法計算相當費時。本研究首先提出改良版Song Rule 以減少計算時間,在本論文的許多例子中顯示減少的計算時間可高達99%。更進一步,本論文也將改良版的 Song Rule 延伸至兩條生產線多個重工的狀況。本論文的解析法都使用模擬法得到驗證。

The goal of this thesis is to obtain the so-called “system reliability”, defined as the probability that the production output meets a predetermined demand for a stochastic system with many workstations, each of which has random capacity following a discrete probability distribution. Moreover, with positive probability, a workstation successfully passes an input unit to the next node in the network. The “system reliability” is an important performance measure in stochastic systems. The application of stochastic systems are manufacturing system, network system and logistics system.
According to the latest literature [19], we have found that eighteen papers have actually incorrectly calculated the system reliability. Paper [19] presents a correct version of analytical approach, named Song Rule, limited to one production line with one rework system. Motivated by the problem that the Song rule is computational inefficient, this thesis proposes an extended Song rule to improve the efficiency of computation. Moreover, this thesis extends the Song rule to present a general analytical theory for networks with multiple reworks. The
proposed method is more computational efficient than the Song rule. Specifically, the proposed extended Song rule accomplishes an computation-time-reduction of above 90% for networks with more than 2 workstations. Moreover, all analytical results presented in this paper are validated via simulation approaches including C and Flexsim.

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