M/G/2/3的系統服務分配及機率的數值解

Abstract

藉由研究M/G/2/3服務系統的子密度分配(sub - density)， $f_1(s)$、$f_2(s,t)$、$f_3(s,t)$，分別代表系統在穩定狀態時系統有1、2、3人並且服務員已服務時間為s、(s,t)、(s,t)的密度函數，能較有效的求系統的穩定機率及其他特殊值。在這篇研究中我們找到M/M/2/3密度函數的解析解，和M/G/2/3的數值解以及近似解，其中近似解可表現為三個已知函數的線性組合，並且有不錯的效率和近似。之後我們試著將演算法推廣至M/G/2/K，並討論M/G/C/K計算上的可能方法。這篇論文的架構如下，第一章回顧相似的文獻並介紹這篇研究所使用的方法，第二章中探討M/M/2/3的情況，以矩陣運算的方式得到系統的密度函數及機率，第三章中探討M/G/2/3的情況，並列出數值演算法和近似演算法，第四章中列出實驗結果，第五章將演算法推廣至M/G/2/K並討論M/G/C/K的情況，第六章是結論。

By studying the sub-density of the M/G/2/3 queuing system,$f_1(s)$、$f_2(s,t)$、$f_3(s,t)$,which respectively stand for the density function of the system in a steady state when the system has 1,2,3 people and they are has been serving for s, (s ,t), (s, t) unit of time, we can find the density function of the system and other special values (e.g.stationary probability). In this study, we find the analytical solution of the M/M/2/3, the numerical solution and the approximate solution of M/G/2/3 where the approximate solution can be expressed as the linear combination of several known functions and have good efficiency and approximation. We then try to extend the algorithm to M/G/2 /K and discuss possible approaches to M/G/C/K calculations. The structure of this paper is as follows. In the first chapter, we review the similar literature and introduce the method used in this study. In chapter 2, we discuss the situation of M/M/2/3, and solve the density function and the stationary probability. The third chapter to explore the M/G/2/3 situation, and lists the numerical algorithm and approximate algorithm. The fourth chapter lists the experimental results. The fifth chapter will be extended to M/G/2/K and discuss the case of M/G/C/K. In the end, the chapter sixth is the conclusion.