本論文延續前人 (林程富~2022) 有關 n 個連續網格點的線段上之新型 k-服務器問題的研究。
與前人不同的是，我們考慮每個時間點有兩個不同位置的請求同時出現（而非一個位置），
並設計了確定性演算法以最小化錯失率。

針對請求的出現，我們討論最壞情況和隨機輸入情況。
在最壞情況中，演算法必須良好地處理對所有可能的請求序列。
而隨機輸入情況中，我們專注在分析演算法的期望錯失率。

This thesis continues Lin's work of ``A New $k$-server Problem on a Line Segment with $n$ Contiguous Grid points" (2022). Different from Lin's work, we consider scenarios where requests from two different locations appear simultaneously at each time (instead of one location),
and design deterministic algorithms to minimize the miss rate.

We discuss both the worst-case and the randomized input scenarios for the occurrence of requests.
In the worst-case scenario, the algorithm must handle all possible request sequences effectively.
For the randomized input scenario, we focus on analyzing the expected miss rate of the algorithm.

摘要 i
Abstract ii
Acknowledgment iii
Contents v
1 Introduction 1
2 Preliminaries 4
2.1 RangeofNumberofGridPoints 4
2.2 DistancetoMiss 5
3 The Worst-Case Bounds 8
3.1 TheLowerBound 8
3.2 TheUpperBound 9
4 Randomized Inputs 15
4.1 TheLowerBound 16
4.2 TheUpperBound 17
5 Conclusion 21
A Missing Proof and Enumeration 22
A.1 ClaiminsideProofofLemma1 22
A.2 EnumerationforMaximumPotentialValue 22
A.3 EnumerationforOtherPotentialValues 28

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