臺灣博碩士論文加值系統

　　考慮在M/G/2/3系統中的密度函數，可以利用系統平衡圖找出的系統平衡式，知道狀態之間的關係與狀態變化，得到一偏微分方程組。在M/M/2/3系統，我們可以找出其系統各狀態的密度函數，並觀察M/M/2/3系統的結果，得知其密度函數為三個指數項的線性組合，考慮M/G/2/3系統當服務分配的風險函數為二階階梯函數時滿足假設條件下的密度函數，以假設的密度函數與模擬的結果比較其分配和極限機率，最後探討其結果與假設的限制。本論文的架構如下，第一章對於系統模型介紹、文獻回顧及研究所使用的方法，第二章考慮M/M/2/3系統的解析與對f_2 (0,t)在M/G/2/3時的推測，第三章探討M/G/2/3服務分配的風險函數為階梯函數時假設下密度函數，及考慮各狀態的分配，第四章為數值實驗，比較其分配與極限機率在利用數值演算法及模擬的結果，並與模擬實驗做比較，第五章探討假設的限制，第六章為結論。

　　Consider the density function of the M/G/2/3 queueing system, we can use the balance diagram to find the balance equations of the system, we can understand the relationship between states and changes of states, obtaining a system of partial differential equations. In the M/M/2/3 system, we can find the density functions of each state, and observe the result. We know the density function of each state is a linear combination of three exponential terms. Therefore, we consider the M/G/2/3 system when the hazard function of the service distribution is a two values step function, and the density functions of this system must meet some hypotheses, comparing the distribution and limiting probabilities with simulation. Finally, we discuss the experimental result and the limitations of hypotheses. The paper is organized as follows. Section 1 introduces the system models, literature review, and methods used in this paper. Section 2 consider the analysis of the M/M/2/3 system, and speculation of the f_2 (0,t) in the M/G/2/3 system. Section 3, we discuss the hypothetical density functions for M/G/2/3 when the hazard function of the service distribution is a step function, considering their distributions. Section 4 is the numerical experiments, comparing the distributions and limiting probability between the approximation method and simulation. Section 5, we discuss the limitations of hypotheses. Section 6 is the conclusion.

摘要 i
Abstract ii
誌謝 iii
目錄 iv
圖目錄 v
表目錄 vi
一、緒論 1
1.1 研究動機 1
1.2 文獻回顧 2
1.3 研究方法 3
二、M/M/2/3解析 9
2.1 系統平衡式 9
2.2 M/M/2/3理論解 10
2.3 f_2 (0,t)在M/G/2/3時之推測 12
三、M/G/2/3 服務分配的風險函數為階梯函數 14
3.1 密度函數及假設 14
3.2 系統中各狀態的密度函數 15
3.3 數值演算法 18
四、數值實驗 20
4.1 實驗概述 20
4.2 模擬實驗、演算法逼近及迴歸線配適 21
4.3 實驗結果分析 27
4.4 觀察不同條件下的f_2 (0,t) 27
五、討論與假設限制 29
六、結論 32
參考文獻 33
附錄一 固定μ_1變動μ_2的四種極限機率算法比較 34

[1]D. Gross and C. Harris. (1998). Fundamental of queueing theory (3rd). John Wiley.
[2]Cohen, J. W. (1982). On the M/G/2 queueing model. Stochastic Processes and their Applications, 12(3), 231-248.
[3]Filipowicz, B., & Kwiecień, J. (2008). Queueing systems and networks. Models and applications. Bulletin of the Polish Academy of Sciences. Technical Sciences, 56(4).
[4]鄭冠宏，「M/M/2/3平衡狀態下的機率密度函數」，國立交通大學統計學研究所，碩士論文，民國104年
[5]顏天保，「M/G/2/3的系統服務分配及機率的數值解」，國立交通大學統計學研究所，碩士論文，民國106年
[6]Gelenbe, E., Pujolle, G., Gelenbe, E., & Pujolle, G. (1998). Introduction to queueing networks.
[7]Harron, R. (2006). MAT-203: The Leibniz Rule.
[8]Moya-Cessa, H. M., & Soto-Eguibar, F. (2011). Differential equations: an operational approach. Rinton Press, Incorporated.
[9]Wolff, R. W. (1982). Poisson arrivals see time averages. Operations Research, 30(2), 223-231.