連續k系統之最優配置

Abstract

連續k系統乃由n個元件排成一線所組成。在F系統中，若n個元件中有連續k個元件不能運作，則整個系統亦不能運作；而在G系統中，則是若n個元件中有連續k個元件能運作，則整個系統即能運作。連續k系統有許多的實際應用，諸如：電信通訊系統、太空轉接站、監控系統等等，相當值得廣泛地研究。 最佳配置問題於1982由Derman, Lieberman and Ross所提出，其研究如何安排n個元件使得系統具有最佳可靠度。已知G系統僅當k£n£2k時才具有最佳不變性配置，在第二章我們探討當ki£ni£2ki時，串聯G系統的最佳不變性配置是否存在。 由於許多連續k系統並無最佳不變性配置，於是我們研究重要性指標，以提供某些位置較重要的資訊，使得我們能將較可靠的元件優先配置其上，以得最佳系統可靠度。在第三章我們研究F系統上，組合型(p=1/2)以及一致型(0<p<1)的重要性指標，並提出新的半線型(p31/2)重要性指標，除了展現新結果及訂定某些猜測外、並利用程式結果對前人所立定的某些猜測提出反例。 在第四章我們研究偶發事件型(pR0)重要性指標，在第五章我們更進一步研究通路型重要性指標，以提供更多的配置決策資訊。

A consecutive-k system consists of n components arranged in a line. A consecutive-k-out-of-n:F (con(k/n:F)) system fails if and only if some consecutive k components are all failed. A consecutive-k-out-of-n:G (con(k/n:G)) system works if and only if some consecutive k components are all working. Consecutive-k systems are used in several applications, such as telecommunication, space relay stations, monitoring systems, and so on. That is why they have attracted many researchers. In 1982, Derman, Lieberman and Ross proposed the optimal assignment problem which is to assign the n functionally exchangeable components to the n positions in a line to maximize the system reliability. The dissertation first addresses the existence of invariant series systems. It is known that a consecutive-k-out-of-n:G system has an invariant optimal assignment if and only if k£n£2k. In Chapter 2, we discuss the consecutive-ki-out-of-ni:G series system with ki £ ni £ 2ki and completely characterize the existence of invariant optimal assignments. Many consecutive-k systems, however, do not have invariant optimal assignments. At the present, we consider heuristic algorithms for optimal assignments. In Chapter 3, we summarize our knowledge of the combinatorial case (p=1/2) and the uniform case (0<p<1), propose the new half-line case (p31/2), present some new results and conjectures, and use our extensive computer-generated data to give counterexamples to some plausible conjectures for con(k/n:F) systems. After we have studied the structural Birnbaum importance on the cases p=1/2 and p31/2, we look into the rare-event case (pR0) in Chapter 4 to complete the spectrum of examining uniform Birnbaum importance over the whole range of p. Moreover, we propose the path importance to strengthen the rare-event case in Chapter 5.