Models for Minimal Repair

In this chapter, we consider systems that, once the failure has occurred and the repair has been completed, exhibit identical conditions to that just before the failure. In this case, minimal repair maintenance actions are being carried out in the system environment. The model considered is a non-homogeneous Poisson process (NHPP) and the main objective is a nonparametric estimation of the intensity function of the process or equivalently, the rate occurrence of failures (rocof). To achieve this, we will consider one or multiple realizations of a NHPP, which means observing a single system or, alternatively, a population of systems of the same characteristics. No assumption is adopted for the functional form of the rocof except concerning the smoothness, which is understood in terms of some properties of derivability. We are also interested in estimating this function under some restrictions of monotonicity.

[1]  Michael J. Phillips.  Bootstrap confidence regions for the expected ROCOF of a repairable system , 2000, IEEE Trans. Reliab..

[2]  Maxim Finkelstein,et al.  Failure Rate Modelling for Reliability and Risk , 2008 .

[3]  M. Rudemo.  Empirical Choice of Histograms and Kernel Density Estimators , 1982 .

[4]  Ron S. Kenett,et al.  Encyclopedia of statistics in quality and reliability , 2007 .

[5]  Terje Aven,et al.  General Minimal Repair Models , 2008 .

[6]  Niels Keiding,et al.  Statistical Models Based on Counting Processes , 1993 .

[7]  W. Nelson.  Statistical Methods for Reliability Data , 1998 .

[8]  Marvin Rausand,et al.  System Reliability Theory: Models, Statistical Methods, and Applications , 2003 .

[9]  Harold E. Ascher,et al.  Repairable Systems Reliability: Modelling, Inference, Misconceptions and Their Causes , 1984 .

[10]  H. Ascher.  Evaluation of Repairable System Reliability Using the ``Bad-As-Old'' Concept , 1968 .

[11]  Hoang Pham,et al.  NHPP software reliability and cost models with testing coverage , 2003, Eur. J. Oper. Res..

[12]  A. Bowman.  An alternative method of cross-validation for the smoothing of density estimates , 1984 .

[13]  O. Aalen.  Nonparametric Inference for a Family of Counting Processes , 1978 .

[14]  M. J. Phillips,et al.  Estimation of the expected ROCOF of a repairable system with bootstrap confidence region , 2001 .

[15]  B. Silverman.  Density estimation for statistics and data analysis , 1986 .

[16]  Uday Kumar,et al.  Reliability analysis of hydraulic systems of LHD machines using the power law process model , 1992 .

[17]  J. Marron,et al.  Equivalence of Smoothing Parameter Selectors in Density and Intensity Estimation , 1988 .

[18]  Asit P. Basu,et al.  Statistical Methods for the Reliability of Repairable Systems , 2000 .

[19]  Wayne B. Nelson,et al.  Recurrent Events Data Analysis for Product Repairs, Disease Recurrences, and Other Applications , 2002 .

[20]  Harold Ascher.  Different insights for improving part and system reliability obtained from exactly same DFOM "failure numbers" , 2007, Reliab. Eng. Syst. Saf..

[21]  Shane G. Henderson,et al.  Estimation for nonhomogeneous Poisson processes from aggregated data , 2003, Oper. Res. Lett..

[22]  J. Marron,et al.  Asymptotic Optimality of the Least-Squares Cross-Validation Bandwidth for Kernel Estimates of Intensity Functions , 1991 .

[23]  Min Xie,et al.  Software Reliability Modelling , 1991, Series on Quality, Reliability and Engineering Statistics.

[24]  M. T. Boswell.  Estimating and Testing Trend in a Stochastic Process of Poisson Type , 1966 .

[25]  Isotonic estimation of the intensity of a nonhomogeneous Poisson process: The multiple realization setup , 1993 .

[26]  Mei-Cheng Wang,et al.  Kernel Estimation of Rate Function for Recurrent Event Data , 2005, Scandinavian journal of statistics, theory and applications.

[27]  Vasiliy V. Krivtsov.  Practical extensions to NHPP application in repairable system reliability analysis , 2007, Reliab. Eng. Syst. Saf..

[28]  H. Ramlau-Hansen.  Smoothing Counting Process Intensities by Means of Kernel Functions , 1983 .

[29]  Zhiguo Wang,et al.  Non-parametric Estimation for NHPP Software Reliability Models , 2007 .

[30]  James R. Thompson,et al.  Some Nonparametric Techniques for Estimating the Intensity Function of a Cancer Related Nonstationary Poisson Process , 1981 .

[31]  Peter Hall,et al.  Bootstrap Confidence Regions for the Intensity of a Poisson Point Process , 1996 .