Please use this identifier to cite or link to this item: `https://ah.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 modelPhase-type distributionNon-homogeneous pure birth processRenewal processMarkov processAge replacement policyStationary 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 Appears in Collections: [應用數學系] 學位論文

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