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Bootstrapping computation of availability for a repairable system with standby subject to imperfect switching
|關鍵字:||Availability;可用度;Bootstrap method;Recursive method;Repairable system;Simulation;Supplementary variable technique;拔靴法;遞迴法;可修理系統;模擬;輔助變數技術||出版社:||應用數學系所||引用:||References  E.E. Lewi (1996), Introduction to Reliability Engineering, 2nd Edition, John Wiley & Sons, New York.  K.H. Wang, J.B. Ke, and J.C. Ke (2007), Profit analysis of the M/M/R machine repair problem with balking, reneging, and standby switching failures, Computers and Operations Research Volume: 34, Issue: 3, March, 2007, pp. 835-847.  D.R. Cox (1955), The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables, Proceeedings of the Combridge Philosophical Society 5 433-44.  U.C. Gupta, and T.S.S.S. Rao (1994), A recursive method compute the steady state probabilities of the machine interference model: (M/G/1), K. Comput. Oper. Res. 21 597-605.  U.C. Gupta, and T.S.S.S. Rao (1996), On the M/G/1 machine interference model with spares, European Journal of Operational Research 89 164-171.  J.C. Ke, and Wang, K.H (2002). A recursive method for the N policy G/M/1 queueing system with finite capacity, European Journal of Operational Research, 142(3): 577-594.  J.C. Ke (2003), The analysis of general input queue with N policy and exponential vacations, Queueing Systems, 45: 135-160.  M. Jie (1991), Interval estimation of availability of a series system, IEEE Transactions in Reliability R-40(5) 541-546.  B.N. Masters, T.O. Lewis and W.J. Kolarik (1992), A confidence interval availability for systems with Weibull operating time and lognormal repair time, Microelectronics and Reliability 32 89-99.  V. Sridharan and P. Mohanavadivu (1998), Some statistical characteristics of a repairable, standby, human & machine system, IEEE Transactions on Reliability 47(4) 431-435.  V.S.S. Yadavalli, M. Botha and A.Bekker (2002), Asymptotic confidence limits for the steady-state availability of a two-unit parallel system with preparation time for the repair facility, Asia-Pacific Journal of Operational Research 19 249-256.  P. Chandrasekhar, R. Natarajan, and V.S.S. Yadavalli (2004), A study on a two unit standby system with Erlangian repair time, Asia-Pacific Journal of Operation Research 42(3), 401-407.  Y.H.Chien, J.-C. Ke, and S.L. Lee (2006), Asymptotic confidence limits for performance measures of a repairable system with imperfect service station, Communication in Statistics-Simulation and Computation 35: 813-830.  G.G. Roussas (1997), A Course in Mathematical Statistics, second ed., Academic Press.  G. Casella and R.L. Berger (2002), Statistical Inference, second ed., Duxbury.  B. Efron (1979), Bootstrap methods: another look at the jackknife, Annals of Statistics 7 1-26.  B. Efron (1982), The jackknife, the bootstrap, and other resampling plans, SIAM Monograph #38.  B. Efron (1983), G. Gong, A leisurely look at the boot strap, the jackknife and cross-validation, American Statistician 37 36-48.  B. Efron (1986), R.J. Tibshirani, Bootstrap methods for standard errors, confidence intervals, and other measures of statistical accuracy, Statistical Science 1 54-77.  B. Efron (1987), Better bootstrap confidence intervals, Journal of the American Statistical Association 82 171-200.  B. Gunter (1991), Bootstrapping: How to make something from almost nothing and get statistically valid answers, Part I, Quality Progress 97-103.  B. Gunter (1992), Bootstrapping: How to make something from almost nothing and get statistically valid answers, Part III, Quality Progress 119-122.  Z. Mooney, R.D. Duval (1993), Bootstrapping: A Nonparametric Approach to Statistical Inference, Sage Publications, Newbury Park, CA.  A. Young (1994), Bootstrap: more than a stab in the dark, Statistical Science 9 382-415.  L. Wasserman (1990), Belief functions and statistical inference, Can. J. Statist. 18, 193-196.||摘要:||
我們考慮一個具有不完美備元可修理系統，此系統包含四個主要和二個備用可修理機器。本文主要討論以五種拔靴法為基礎建構其可用度的信賴區間。我們假設主要和備用機器損壞的時間間隔服從指數分配，修護時間服從四種不同的分配：指數、伽瑪、均勻、混和分配，接著使用遞迴法和輔助變數技巧求得系統在穩態時的可用度，下一步則採用以拔靴法為基礎，利用標準拔靴法(standard bootstrap)、百分比拔靴法(percentile bootstrap)、誤差修正百分比拔靴法(bias-corrected percentile bootstrap)、加速並修正誤差百分比拔靴法(bias-corrected and accelerated percentile bootstrap) 以及樞
紐拔靴法(bootstrap pivotal) 等五種方法建構並計算出其信賴區間、覆蓋率和信賴長度以便探討它們的表現，最後，以一些實驗數據來說明結論。
This paper deals with the availability behavior of a repairable system in which standby switched to primary is subject to breakdowns. The time-to-failure of the four primary and two standby units are assumed to be exponentially and generally distributed. In addtion, the repair time of service station follow four common distributions:
exponential(EXP), Gamma(G), Uniform(U), Mixture(M). We use a
recursive method, and the supplementary variable technique to develop the steadystate availability, Av. The estimator Aˆv is strongly consistent and asymptotically normal. The interval estimations of Av are constructed by five bootstrap approaches: standard bootstrap confidence interval(SB), the percentile bootstrap confidence interval(PB), the bias-corrected percentile bootstrap confidence interval(BCPB), the bias-corrected and accelerated confidence interval(BCa) and bootstrap pivot confidence interval(BP). Finally, some simulation computations are conducted in order
to describe the performances of Aˆv on various interval estimation by calculating the coverage percentage and the average length of intervals.
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