在全光封包交換網路之中，設計並解決封包爭奪相同資源的光緩衝器，是一個兼具實際與挑戰性的問題。因此在過去的二十多年中，有許多構建各種光佇列的研究。在這些種類之中，優先佇列是最常見的緩衝器類型，包含下列兩種情況：先進先出（First-In First-Out）和後進先出（Last-In First-Out）佇列。最近，透過由 (M + 2) × (M + 2)（無緩衝）光交換機和 M 條光纖延遲線（SDLs）所組成的回饋系統，可以達到 2^{\Theta(\sqrt{\mu M})} 的緩衝器大小，其中 \mu 是一個常數，取決於回饋系統中的參數。這種架構大大的改進了先前所有僅是 \Theta(M^c) 的結果，其中 c 是一個正整數。本論文將在相同結構的特定情況下驗證緩衝器大小。更準確地說，我們首先在 2s+3\le k\le 4s+4 的特定情況下推導緩衝器 B_1,B_2,...,B_k 的解析解。之後，我們估計最大緩衝器 U_k 的大小和光纖延遲線 M 的數量，而無須求助於差分方程理論。最後，我們分析 U_k 的複雜度為 2^{\Theta(\sqrt{M})}。

Designing the optical buffers that address the packets competing for the same resources is a practical and challenging problem in all-optical packet-switched networks. Therefore, there has been a great deal of work on constructing various kinds of optical queues over the past two decades. Among those categories, priority queues are the most common type of buffering schemes, including the following two cases: first in, first out (FIFO) and last in, first out (LIFO) queues. Lately, through the feedback system that consists of an (M + 2) × (M + 2) (bufferless) optical crossbar Switch and M fiber Delay Lines (SDLs), it can achieve the buffers of 2^{\Theta(\sqrt{\mu M})} size, where \mu is a constant associated with the parameters in this construction. This construction makes significant progress on all the previous results, which are only \Theta(M^c) buffers, where c is a positive integer. This thesis will verify the buffer sizes under the specific case on the same construction. To be more precisely, we first derive analytic expressions for the buffer sizes B_1,B_2,...,B_k under the certain case that 2s+3\le k\le 4s+4. After that, we estimate the maximum buffer size U_k and fiber delay lines M without resorting to the theory of difference equations. Finally, we analyze the complexity of U_k is 2^{\Theta(\sqrt{M})}.

Abstract iii
Acknowledgements v
Contents vii
List of Figures ix
1 Introduction 1
1.1 Background...............................................1
1.2 Motivation...............................................3
1.3 Contributions............................................5
1.4 Thesis Organization......................................5
2 Analytic Expressions for the Buffer Sizes 7
2.1 The constraints for the Buffer Sizes.....................7
2.2 Analytic Expressions for the Buffer Sizes................8
3 Complexity Analysis for the Maximum Buffer Size 13
4 Conclusions and Future Work 21
4.1 Conclusions.............................................21
4.2 Future Work.............................................21
Bibliography 23

[1] A. D. Sarwate and V. Anantharam, "Exact emulation of a priority queue with a switch and delay lines," Queueing Systems: Theory and Applications, vol. 53, no. 3, pp. 115-125, July 2006.

[2] J. Cheng, S.-H. Yang, C.-Y. Wang, H.-H. Tang, and B. Tang, "On efficient constructions of optical priority queues," IEEE Transactions on Communications (TCOM), vol. 70, no. 3, pp. 1861-1874, 2022.

[3] H.-C. Chiu, C.-S. Chang, J. Cheng, and D.-S. Lee, “Using a single switch with O(M inputs/outputs for the construction of an optical priority queue with O(M^3) buffer,” in Proceedings 26th IEEE International Conference on Computer Communications (INFOCOM’07 Minisymposium), Anchorage, AK, USA, May 2007, pp. 2501–2505.

[4] ——, "A simple proof for the constructions of optical priority queues," Queueing Systems: Theory and Applications, vol. 56, no. 2, pp. 73–77, June 2007.

[5] H. Kogan and I. Keslassy, "Optimal-complexity optical router," in Proceedings IEEE International Conference on Computer Communications (INFOCOM’07), Anchorage, AK, USA, May 2007, pp. 706–714.

[6] A. Datta, "Construction of polynomial-size optical priority queues using linear switches and fiber delay lines," IEEE/ACM Transactions on Networking (TON), vol. 25, no. 2, pp. 974–987, April 2017.

[7] B. Tang, X. Wang, C.-T. Nguyen, B. Ye, and S. Lu, "Construction of subexponential-size optical priority queues with switches and fiber delay lines," IEEE/ACM Transactions on Networking (TON), vol. 28, no. 1, pp. 336–346, February 2020.

[8] C.-S. Chang, D.-S. Lee, and C.-K. Tu, "Recursive construction of FIFO optical multiplexers with switched delay lines," IEEE Transactions on Information Theory (TIT), vol. 50, no. 12, pp. 3221–3233, December 2004.

[9] C.-C. Chou, C.-S. Chang, D.-S. Lee, and J. Cheng, "A necessary and sufficient condition for the construction of 2-to-1 optical FIFO multiplexers by a single crossbar switch and fiber delay lines," IEEE Transactions on Information Theory (TIT), vol. 52, no. 10, pp. 4519–4531, October 2006.

[10] C.-S. Chang, Y.-T. Chen, and D.-S. Lee, "Constructions of optical FIFO queues," IEEE Transactions on Information Theory (TIT), vol. 52, no. 6, pp. 2838–2843, June 2006.

[11] P.-K. Huang, C.-S. Chang, J. Cheng, and D.-S. Lee, "Recursive constructions of parallel FIFO and LIFO queues with switched delay lines," IEEE Transactions on Information Theory (TIT), vol. 53, no. 5, pp. 1778–1798, May 2007.

[12] J. Cheng, "Constructions of fault-tolerant optical 2-to-1 FIFO multiplexers," IEEE Transactions on Information Theory (TIT), vol. 53, no. 11, pp. 4092–4105, 2007.

[13] ——, "Constructions of optical 2-to-1 FIFO multiplexers with a limited number of recirculations," IEEE Transactions on Information Theory (TIT), vol. 54, no. 9, pp. 4040–4052, September 2008.

[14] J. Cheng, H.-C. Chiu, C.-S. Chang, and D.-S. Lee, "Constructions of optical priority queues with multiple inputs and multiple outputs," IEEE Transactions on Information Theory (TIT), vol. 57, no. 7, pp. 4274–4301, July 2011.

[15] J. Cheng, H.-H. Chou, and C.-H. Cheng, "A necessary and sufficient condition for SDL constructions of optical FIFO queues," in Proceedings IEEE Global Communications Conference (GLOBECOM’13), Atlanta, GA, USA, December 2013, pp. 2339–2345.

[16] B. Tang, X. Wang, C.-T. Nguyen, and S. Lu, "Constructing sub-exponentially large optical priority queues with switches and fiber delay lines," in Proceedings IEEE International Symposium on Information Theory (ISIT'16), Barcelona, Spain, July 2016, pp. 1441-1445.

[17] J. Cheng, C.-S. Chang, S.-H. Yang, T.-H. Chao, D.-S. Lee and C.-M. Lien, "Greedy constructions of optical queues with a limited number of recirculations," IEEE Transactions on Information Theory (TIT), vol. 63, no. 8, pp. 5314-5326, August 2017.

[18] J. Cheng, C.-Y. Wang and B. Tang, "Construction efficiency for constructions of optical priority queues under priority-based routing," in Proceedings International Conference on Systems and Informatics (ICSAI'19), Shanghai, China, November 2019, pp. 1029-1034.

[19] ——, "Constructions of optical priority queues under a priority-based routing policy," in Proceedings IEEE International Conference on Computer and Communications (ICCC'19), Chengdu, China, December 2019, pp. 360–364.