為了實現一個全光的封包交換網路，我們需要光記憶體來解決封包間相互競爭有限資源所產生的衝突。近年來，透過光交換機以及光纖延遲線來建造光記憶體的方法引起很多研究學者的興趣。而從實際層面觀點來看，容錯能力是在設計光記憶體時的一個很重要考量。

在這篇論文中，我們著重在具有容錯能力的二對一先進先出光學多工器，我們考慮由一個(M+2)×(M+2)的光交換機，與M條具有延遲為d1,d2,…,dM的光纖延遲線組成的回授系統，其中M條光纖延遲線從光交換機的M個輸出連接到M個輸入，剩下兩個輸入當作二對一多工器的輸入，另兩個輸出則是二對一多工器的輸出。在文獻[20]中，作者提供一種選擇延遲的方法，使得上述回授系統在最多F條光纖斷掉時，仍可正常操作成二對一先進先出光學多工器，而建造效率也在此篇文獻中被用來比較不同延遲選擇下所對應的系統性能。作者最後提出一個猜測，關於在最佳的延遲選擇下，漸近建造效率的下界恰好就是漸近建造效率。

在這篇論文中，我們想要證明文獻[20]中的猜測為真。但我們並沒有解決這個問題，我們只提供一些新的結果和觀察來幫助我們解決上述的猜測。首先，我們推導出針對某些F，在最佳的延遲選擇下，延遲為 的光纖個數的數學表示法。其次，假設當系統容錯能力為F時，在最佳的延遲選擇下，我們得到的延遲數列為di*，而容錯能力為F+1時相對應的延遲數列為hi*，則我們可證明di*≧hi*。最後我們提出一些猜測可用來獲得漸近建造效率的數學表示式。

To resolve packet conflicts due to limited resources, optical buffers are necessary in all-optical packet-switched networks. Recently, constructing optical buffers directly via optical Switches and fiber Delay Lines (SDL) has attracted a lot of attention. Fault tolerant capability is an important design issue in the constructions of optical buffers from a practical perspective, and such an issue has seldom been theoretically addressed before. In this thesis, we focus on fault tolerant 2-to-1 FIFO multiplexers. We consider a feedback system consisting of an
(M +2)×(M +2) optical crossbar switch and M fiber delay lines with delays d1, d2, . . . , dM. In [20], a class of choices of the fiber delays d1, d2, . . . , dM such that the feedback system can still be operated as a 2-to-1 FIFO multiplexer even after up to F of the fibers are broken was provided. To compare various choices of the fiber delays,
the construction efficiency was used in [20] as a performance measure for a choice of the fiber delays. In this thesis, we give some new results and observations that could be helpful in solving the conjecture in [20] regarding the closed-form expression of the asymptotic
construction efficiency for the optimal choice (in the sense of maximizing the buffer size) of the fiber delays. First, we derive a closed-form expression for n_(ℓ+2)*, the number of fibers with delay 2^(ℓ+1) when the choice of fiber delays is optimal, for some values of F. Furthermore, we show that di* ≧ hi* for all i = 1, 2, . . . ,M, where di* (resp. hi* ) denote the sequence of fiber delays given by the optimal choice such that the feedback system can tolerate up to F (resp. F +1) failures of the fiber delay lines. Finally, we give some conjectures which (if true) could be used to obtain the closed-form expression for the asymptotic construction efficiency.

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