臺灣博碩士論文加值系統

此篇論文分析研究了< p, N >-方策M/G/1排隊系統，含單一可移動及服務者會故障的情形。其中服務站的故障為一卜瓦松過程，而服務時間的分配則假設為任意分配。

我們假設當系統中的顧客數到達N時，系統將有p的機率開啟服務站，並且有(1-p)的機率保持關閉服務站。

最大熵値法在此篇論文的用處主要是去推導出系統中顧客數的估計機率分配，以及系統中顧客的估計等待時間。我們把導出的近似結果在四種不同服務時間及修
理時間下(指數、均勻、珈瑪、常數)，與精確的結果作一比較分析。

由數值結果可以發現，最大熵値法在實務應用上是足夠精確的，並且在不同的服務時間及修理時間分配下，最大熵値法的表現也是夠穩定的。

This thesis analyzes a single removable and unreliable server in the <p,N >-policy M/G/1 queueing system in which the server breaks down according to a Poisson process and the repair time obeys an arbitrary distribution.

We assume that when the number of customers in the system
reaches N, turn the server on with probability p and leave it off with probability (1 − p). The use of maximum entropy approach is to develop the approximate formulae for the probability distributions of the number of
customers and the expected waiting time in the system.
We perform the comparative analysis between approximate results and exact results with four different service time and repair time distributions, including exponential,uniform, gamma, and deterministic.

It appears from numerical results that the maximum entropy approach is sufficiently accurate for practical use and based on the maximum entropy approach , we demonstrate that
the < p,N >-policy M/G(G)/1 queueing system is sufficiently robust to the variations of service time distribution and repair time distribution functions.

1 Introduction 1
1.1 Problem Statement ...............................1
1.2 Literature Review ...............................3
1.3 The purpose of this paper .......................4
1.4 Notations .......................................5
2 Steady-State Results 9
2.1 The < p,N >-policy M/G/1 queue with a removable and
unreliable server ...................................9
2.2 The Maximum entropy principle ..................11
2.3 The Maximum entropy solution ...................12
2.3.1 The four basic known constraints .............12
2.3.2 The maximum entropy model ....................13
2.3.3 The maximum entropy solutions ................14
3 Comparative analysis between the exaxt and maximum entropy results 18
3.1 The exact and approximate expected waiting time in the system 18
3.1.1 The exact waiting time in the system .........18
3.1.2 The aproximate waiting time in the system ....19
3.2 Comparative analysis between the exact and the maximum
entropy results ....................................19
3.2.1 Comparative analysis betweenWs1(p,N) andW s1(p,N)
to the < p,N >-policy M/M(M)/1 queueing system .....20
3.2.2 Comparative analysis betweenWs2(p,N) andW s2(p,N)
to the < p,N >-policy M/U[a1, b1](U[a2, b2])/1 queueing
system .............................................21
3.2.3 Comparative analysis betweenWs3(p,N) andW s3(p,N)
to the < p,N >-policy M/Gam(r1)(Gam(r2))/1 queueing
system .............................................26
3.2.4 Comparative analysis betweenWs4(p,N) andW s4(p,N)
to the < p,N >-policy M/D(D)/1 queueing system .....29
4 Conclusions and Future Research 32
4.1 Conclusions ....................................32
4.2 Future Research ................................32

References .........................................33

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