在本論文中，我們提出一種新型的 k-服務器問題。與原始的 k-服務器問題不同的是，在我們的問題中，由於服務器的移動有限制，請求有可能會被錯過。因此，演算法的目標在於最小化錯過率。
我們討論了這個問題的兩種不同情境。一種是最壞情況，其中演算法必須對所有可能的請求序列表現良好。另一種是隨機情況，其中請求出現的位置是隨機決定的，並且我們關注的是演算法的期望錯過率。
對於最壞情況，我們得到了於 n = 2k + 1 和 n = 3k 情況下的最佳演算法，其中 n 為線段的長度。而對於隨機情況，我們則設計了兩種各有優缺的演算法。

In this thesis, we propose a new k-server problem. In contrast to the original k-server problem, a request may be missed due to the constrained movement of servers in our problem. Therefore, the goal of an algorithm is to minimize the missing rate.
We discuss the problem in two different settings. One is the worst case where an algorithm must perform well for all possible request sequences. The other is the randomized case where the position of a request to appear is chosen at random, and we focus on the expected missing rate of an algorithm.
For the worst case, we obtain an optimal algorithm for the case n = 2k + 1 and n = 3k where n is the length of the line segment. For the randomized case, we devise two algorithms where both have their advantages and disadvantages.

Abstract (Chinese) I
Abstract II
Contents III
List of Figures V
1 Introduction 1
1.1 Related Work 3
1.2 Our Results 5
1.3 Thesis Organization 5
2 Preliminaries 6
3 Distance to Miss 8
4 The Worst Case 12
4.1 Lower Bound 13
4.2 Upper Bound: The 2k + 1 Case 14
4.3 Upper Bound: The General Case 17
5 Randomness 19
5.1 Lower Bound 20
5.2 Upper Bound: The General Case 21
5.3 Upper Bound: The Case with Small d 30
6 Conclusion 41
Bibliography 43

[Bei+05] Wolfgang W. Bein et al. “The Delayed k-Server Problem”. In: Proceedings of Fundamentals of Computation Theory (FCT). 2005, pp. 281–292.
[BIN14] E. Bittner, Cs. Imreh, and J. Nagy-Gy¨orgy. “The Online k-Server Problem with Rejection”. In: Discrete Optimization 13 (2014), pp. 1–15.
[Cas+22] Jannik Castenow et al. “The k-Server with Preferences Problem”. In: Proceedings of Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA). 2022, pp. 345–356.
[CHM98] Moses Charikar, Dan Halperin, and Rajeev Motwani. “The Dynamic Servers Problem”. In: Proceedings of Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). 1998, pp. 410–419.
[FR94] Amos Fiat and Moty Ricklin. “Competitive Algorithms for the Weighted Server Problem”. In: Theoretical Computer Science 130.1 (1994), pp. 85–99.
[KT04] Elias Koutsoupias and David Scot Taylor. “The CNN Problem and Other k-Server Variants”. In: Theoretical Computer Science 324.2 (2004), pp. 347–359.
[MMS88] Mark Manasse, Lyle McGeoch, and Daniel Sleator. “Competitive Algorithms for On-Line Problems”. In: Proceedings of Annual ACM Symposium on Theory of Computing (STOC). 1988, pp. 322–333.
[Wik] Wikipedia contributors. Birth–Death Process. [Online; accessed 30-July2023]. url: https://en.wikipedia.org/w/index.php?title=Birth%E2%80%93death_process.