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研究生:張英仲
研究生(外文):YING CHUNG CHANG
論文名稱:M/M/R有限容量排隊系統含阻礙,放棄,以及服務者故障之效能測度分析
論文名稱(外文):Performance Measures Analysis of the M/M/R Queueing System with Finite Capacity plus Balking, Reneging, and Server Breakdowns
指導教授:王國雄王國雄引用關係
指導教授(外文):K. H. Wang
學位類別:碩士
校院名稱:國立中興大學
系所名稱:應用數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2001
畢業學年度:89
語文別:英文
論文頁數:22
中文關鍵詞:阻礙成本放棄服務者故障
外文關鍵詞:balkcostrenegeserver breakdowns
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在本篇論文,我們研究 M/M/R 有限容量排隊系統,包含阻礙、放棄,以及服務者故障。顧客到達服從卜瓦松過程,參數是 。而顧客被服務的時間則是服從指數分配,參數是 。顧客到達具有 (1-bn) 的機率遇到阻礙 (不進入排隊) 以及放棄 (進入排隊後卻離開) 的時間依照指數分配。服務者在任何時間都可能會故障,甚至沒有顧客在這排隊系統中,故障率是 。當服務者故障時,它會馬上被送去修理,修理率是 。假設服務者的故障時間跟修理時間都是服從指數分配。我們使用matrix- geometric method 去推導穩態的機率,而多種系統效能測度可以因而獲得。接著發展一個成本模型,用以決定最佳的服務者個數。在給定的系統參數值之下,我們計算出一些系統測度分析的數值結果。同時,我們也研究了敏感度分析。

In this thesis, we study the M/M/R queueing system with finite capacity plus balking, reneging, and server breakdowns. Customers arrive following a Poisson process with parameter . The service times of the customers according to a negative exponential distribution with parameter . Arriving customers balk (do not enter) with a probability (1-bn) and renege (leave the queue after entering) according to a negative exponential distribution. The server can break down at any time with breakdown rate even if no customers are in the system. When the server fails, he is immediately repaired at a repair rate . Breakdown times and repair times of the servers are assumed to follow a negative exponential distribution. We use a matrix- geometric method to derive the steady-state probabilities, using which various system performance measures that can be obtained. A cost model is developed to determine the optimum number of servers. Numerical results are presented in which several system performance measures are evaluated based on assumed numerical values given to the system parameters. Sensitivity analysis is also investigated.

CONTENTS
Abstract
1Introduction………
………………………………………………...1
1.1 Problem Statement ……………………………………………1
1.2 Literature Review …………………………………………….2
1.3 the Scope of the Study ………………………………………..3
2Steady-State results ………………………………………………..4
2.1 Problem Statement ……………………………………………4
2.2 Steady-State Equations ……………………………………….6
2.3 Matrix-Geometric Solutions ………………………………...10
2.4 System Performance Measures ……………………………...13
3Cost Analysis and Sensitivity Analysis …………...……………..15
3.1 Cost Analysis ………………………………………………..15
3.2 Sensitivity Analysis …………………………………………16
4Conclusions and Future Research ……………………………….20
4.1 Conclusions …………………………………………………20
4.2 Future Research ……………………………………………..20
References ……………………………………………………………..22

REFFERENCES
[1]F. A. Haight (1957). Queueing with balking. Biometrika 44: 360-369.
[2]F. A. Haight (1959). Queueing with reneging. Metrika 2: 186-197.
[3]P. D. Finch (1960). Deterministic customer impatience in the queueing system GI/M/1. Biometrika 47: 45-52.
[4]C. J. Ancker (1963), Jr. and A. V. Gafarian. Some queueing problem with balking and reneging:Ⅰ. Operations Research 11: 88 - 100.
[5]C. J. Ancker, Jr. and A. V. Gafarian (1963). Some queueing problems with balking and reneging:Ⅱ. Operations Research 11: 928 - 937.
[6]M. O. Abou-El-Ata and A. I. Shawky (1992). The single-server Markovian overflow queue with balking, reneging and an additional server for longer queues. Microelectronics and Reliability 32: 1389 -1394.
[7]N. K. Jaiswal (1965). On some waiting line problems. Opsearch (India) 2: 27-43.
[8]S. S. Rao (1968). Queueing with balking and reneging in M/G/1 systems. Metrika 12: 173 — 188.
[9]S. S. Rao (1965). Queueing models with balking, reneging, and interruptions. Operations Research 13: 596-608.
[10]S. S. Rao and N. K. Jaiswal (1969). On a class of queueing problems and discrete transforms. Operations Research 17: 1063-1076.
[11]M. O. Abou-El-Ata and A. M. A. Hariri (1992). The M/M/c/N queue with balking and reneging. Computers & Operations Research 19, No. 8: 713-716.
[12]M. F. Neuts (1981). Matrix Geometric Solutions in Stochastic Models: an Algorithmic Approach. The Johns Hopkins University Press: Baltimore.

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