封閉式等候網路機率分配之估計與分析

Abstract

在這一篇論文裡，我們討論兩個階段的封閉式等候線網路，其中服務時間的機率分配都是Phase type分配。我們猜測服務時間的機率分配和離開時間間隔的機率分配滿足一組聯立方程組。然後，我們推導出非邊界狀態的穩定機率可以被表示成 product-form的線性組合，而每個product-form可以用聯立方程組的根來構成。利用非邊界狀態的穩定機率， 我們可以求出邊界狀態的機率。最後我們建立一個求穩定機率的演算過程。利用這個演算方法，可以簡化求穩定機率的複雜度。

In this thesis, we are concerned with the property of a two-stage closed system in which the service times are identically of phase type. We first conjecture that the Laplace-Stieltjes Transforms (LST) of service time distributions may satisfy a system of equations. Then we present that the stationary probabilities on the unboundary states can be written as a linear combination of product-forms. Each component of these products can be expressed in terms of roots of the system of equations. Finally, we establish an algorithm to obtain all the stationary probabilities. The algorithm is expected to work well for relatively large customers in the system.

封面頁\r\n證明書\r\n致謝詞\r\n論文摘要\r\n目錄\r\n1 Introduction\r\n1.1 Background\r\n1.2 Literature Review\r\n1.3 Organization of the thesis\r\n2 The Model\r\n2.1 Phase type distribution\r\n2.2 Problem statement and assumptions\r\n2.3 Preliminaries results\r\n2.4 Propositions\r\n3 A two stage closed network with ρ1 ≧ ρ2\r\n3.1 Transition rate matrix\r\n3.2 Balance equation\r\n3.3 Product form solutions\r\n3.4 Algorithm for the boundary probabilities\r\n3.5 A summary of the algorithm\r\n4 A two stage closed network with ρ1 ＜ ρ2\r\n4.1 Transition rate matrix\r\n4.2 Balance equation\r\n4.3 Product form solutions\r\n4.4 Algorithm for the unboundary probabilities\r\n4.5 A summary of the algorithm\r\n5 Examples\r\n5.1 Example for -/M/1 → /M/1 system\r\n5.2 Example for -/E2/1 → /E2/1 system\r\n6 Conclusions and future research\r\n6.1 Conclusion\r\n6.2 Future research\r\nReferences\r\nAppendix