Loading...
|
Please use this identifier to cite or link to this item:
https://nccur.lib.nccu.edu.tw/handle/140.119/124869
|
| Title: | 以階段型機率分佈表示異質生成衝擊系統
A System Subject to Non-Homogeneous Pure Birth Shocks with Phase-Type Distributions |
| Authors: | 劉宏展
Liu, Hong-Zhan |
| Contributors: | 陸行
Luh, Hsing
劉宏展
Liu, Hong-Zhan |
| Keywords: | 衝擊模型
階段型分佈
異質生成過程
再生過程
馬可夫過程
年齡置換策略
穩定機率
Shock model
Phase-type distribution
Non-homogeneous pure birth process
Renewal process
Markov process
Age replacement policy
Stationary probability |
| Date: | 2019 |
| Issue Date: | 2019-08-07 16:35:33 (UTC+8) |
| Abstract: | 考慮一個衝擊系統，它的衝擊依據異質生成過程而產生。這個系統有兩
種類型的損壞。類型一的損壞可以被修理消除。類型二的損壞可以被不定
期置換消除。假設兩個連續衝擊之間的時間間隔服從階段型分佈。例如，
在一個特殊的階段型分佈—亞指數分佈—之下，我們發現穩定機率存在的
條件。在這個模型下探討年齡置換策略，我們導出置換週期內的期望成本
率。為了找到最小化期望成本率的最佳定期置換年齡，我們提供一個有效
率的演算法並開發一個 MATLAB 工具來實現。一系列數值範例促使我們發
現新的定理，它比以前的定理更簡單，更實際，更直觀。該定理表明最佳定期置換年齡的存在性。
We consider a system subject to shocks which occur according to a non-homogeneous pure birth process. The system has two types of failures. Type-I failure can be removed by a repair. Type-II failure can be removed by an unplanned replacement. We assume that the inter-arrival time between consecutive shocks follows phase-type distributions. For example, under a special PH-distribution that is a hypo-exponential distribution, we find the conditions of the existence of stationary probability. Under this model we investigate the age replacement policy. We derive the expected cost rate of a replacement cycle. To find the optimal planned replacement age that minimizes the expected cost rate, we give an efficient algorithm and develop a MALAB tool for implementation. A series of numerical examples motivate us to write a new theorem. That is simpler, more practical, and more intuitive than a previous theorem. This theorem shows the existence of the optimal planned replacement age. |
| Reference: | [1] M. S. A-Hameed and F. Proschan. Nonstationary shock models. Stochastic Processes and their Applications, 1(4):383–404, 1973.
[2] M. S. A-Hameed and F. Proschan. Shock Models with Underlying Birth Process. Journal of Applied Probability, 12(1):18–28, 1975.
[3] S. Asmussen, O. Nerman, and M. Olsson. Fitting phase-type distributions via the em algorithm. Scandinavian Journal of Statistics, 23(4):419–441, 1996.
[4] R. Barlow and L. Hunter. Optimum preventive maintenance policies. Operations Research, 8(1):90–100, 1960.
[5] P. Buchholz, J. Kriege, and I. Felko. Input Modeling with Phase-Type Distributions and Markov Models: Theory and Applications. Springer, New York, 2014.
[6] D. R. Cox. Renewal Theory. Methuen, London, 1962.
[7] J. D. Esary, A. W. Marshall, and F. Proschan. Shock Models and Wear Processes. The Annals of Probability, 1(4):627–649, 1973.
[8] F. S. Hillier and G. J. Lieberman. Introduction To Operations Research. McGraw-Hill, New York, 10th edition, 2015.
[9] R. S. Maier and C. A. O’Cinneide. A Closure Characterisation of Phase-Type Distributions. Journal of Applied Probability, 29(1):92–103, 1992.
[10] D. Montoro-Cazorla, R. PérezOcón, and M. C. Segovia. Shock and wear models under policy N using phase-type distributions. Applied Mathematical Modelling, 33:543–554, 2009.
[11] M. F. Neuts. Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, 1981.
[12] B. F. Nielsen. Lecture notes on phase–type distributions for 02407 Stochastic Processes, 2017.
[13] S.H. Sheu, C.C. Chang, Z. G. Zhang, and Y.H. Chien. A note on replacement policy for a system subject to non-homogeneous pure birth shocks. European Journal of Operational Research, 216:503–508, 2012.
[14] H. M. Taylor and S. Karlin. An Introduction to Stochastic Modeling. Academic Press, Cambridge, Massachusetts, 3rd edition, 1998. |
| Description: | 碩士
國立政治大學
應用數學系
105751004 |
| Source URI: | http://thesis.lib.nccu.edu.tw/record/#G0105751004 |
| Data Type: | thesis |
| DOI: | 10.6814/NCCU201900384 |
| Appears in Collections: | [應用數學系] 學位論文
|
Files in This Item:
| File |
Size | Format | |
|---|
| 100401.pdf | 762Kb | Adobe PDF2 | 0 | View/Open |
|
All items in 政大典藏 are protected by copyright, with all rights reserved.
|
