臺灣博碩士論文加值系統

此篇論文研究處理了含有假期,啟動時間及會故障的服務者在< p,N >-方策M/G/1排隊系統之最佳控制. 顧客的到達假設為一卜瓦松過程. 服務時間, 啟動時間及修理時間假設為一般分配. 當系統是空的時候, 服務者會被關閉並旅遊. 當旅遊返回後, 發現在等候線上的顧客人數少於N人, 它會繼續另一個旅遊. 一旦在系統累積的顧客人數達到N人或更多, 系統開啟服務者的機率為p並且保持關閉服務者的機率為(1-p). 如果服務者立刻被開啟, 它暫時不能服務等待的顧客. 它需要一個啟動時間後才開始服務顧客直到系統中沒有顧客. 我們研究各種系統性能測度和發展每單位時間每位顧客的總期望花費, 其中p and N是決策變數. 我們使用了一個有效率的程序來決定聯合最佳的開端值(p*,N*),使得花費函數有最小值.

關鍵詞: < p,N >-方策, 有假期的服務者, 啟動時間, 聯合最佳的開端值,花費函數.

This thesis deals with the optimal control of the < p,N >-policy M/G/1 queue with server vacations, startup and breakdowns. It is assumed that customers arrive following a Poisson process. Service times, startup times and repair times obey general distribution. The server is turned off and takes a vacation when the system is empty. When the server returns from a vacation and finds the number of customers in the waiting line is less than N, he will go on another vacation. Once N or more customers are accumulated in the system, turn the server on with probability p and leave the server off with probability (1-p). If the server is immediately turned on, it is temporarily unavailable to serve the waiting customers. He needs a startup time before providing service until there are no customers in the system. We develop various system performance measures and the total expected cost per unit time, in which p and N are decision variables. An efficient procedure is used to determine the joint optimal threshold values (p*,N*) , so as to minimize the cost function.

Keywords: < p,N >-policy, server vacations, a startup time, the joint optimal threshold values, the cost function.

Contents

1 Introduction…………………………………1

1.1 Problem Statement…………………………1
1.2 Literature Review………………………………3
1.3 The aim of the thesis…………………………4
1.4 Notations……………………5

2 System Performance Measures…………………………8

2.1 The N-policy M/G/1 queue –system performance measures……8

2.1.1 Expected length of the vacation, startup, breakdown, busy periods and busy cycle…………8

2.1.2 Expected number of customers in the system…………10

2.2 The < p,N >-policy M/G/1 queue –system performance measures.........11

2.2.1 Expected length of the vacation, startup, breakdown, busy periods and busy cycle…………………11

2.2.2 Expected number of customers in the system…………12

2.2.3 Numerical experiments………………………………13

3 Cost Analysis………………………………18

3.1 The long-run fraction of time measures………………18

3.2 Optimal < p,N >-policy ………………………………19

3.2.1 The cost function………………………………19

3.2.2 The joint optimal values of p and N………………20

3.3 Numerical examples………………………………22

4 Conclusions and Future Research………………………26

4.1 Conclusions…………………………………………………26

4.2 Future research…………………………………………26

References…………………………………………28

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