Abstract
We consider in this paper a parallel system consisting of \(\eta \) identical components. Each component works independently of the others and has a Weibull distributed inter-failure time. When the system fails, we assume that the repair maintenance is imperfect according to the Arithmetic Reduction of Age models (\(ARA_{m}\)) proposed by Doyen and Gaudoin. The purpose of this paper is to generate a simulated failure data of the whole system in order to forecast the behavior of the failure process. Besides, we estimate the maintenance efficiency and the reliability parameters of an imperfect repair following \(ARA_{m}\) models using maximum likelihood estimation method. Our method is tested with several data sets available from related sources. The real data set corresponds to the time between failures of a compressor which is tested by Likelihood Ratio Test (LR). An analysis of the importance and the effect of the memory order of imperfect repair classes (\(ARA_{m}\)) will be discussed using LR test.
Similar content being viewed by others
References
Brown M, Proshan F (1982) Imperfect maintenance. In: Crowley J, Johnson RA (eds) IMS lecture notes-monograph ser, 2, survival analysis. Institute of Mathematical Statistics, Hayward, pp 179–188
Dorado C, Hollander M, Sethuraman J (1997) Nonparametric estimation for a general repair model. Ann Stat 25(3):1140–1160
Doyen L, Gaudoin O (2004) Classes of imperfect repair models based on reduction of failure intensity or virtual age. Reliab Eng Syst Saf 84(1):45–56
Doyen L, Gaudoin O (2006) Imperfect maintenance in a generalized competing risks framework. J Appl Probab 43(3):825–839
Gasmi S, Love CE, Kahle W (2003) A general repair, proportional-hazards, framework to model complex repairable systems. IEEE Trans Reliab 52(1):26–32
Ghnimi S, Gasmi S (2014) Parameter estimations for some modifications of the Weibull distribution. Open J Stat 4:597–610
Gupta RD, Kundu D (2001) Exponentiated exponential family: an alternative to gamma and Weibull distributions. Biom J 33(1):117–130
Kijima M (1989) Some results for repairable systems with general repair. J Appl Probab 26(1):89–102
Kijima M, Sumita U (1986) A useful generalization of renewal theory: counting process governed by non-negative Markovian increments. J Appl Probab 23(1):71–88
Malik M (1979) Reliable preventive maintenance scheduling. AIIE Trans 11(3):221–228
Mudholkar GS, Hutson AD (1996) The exponentiated Weibull family: some properties and a flood data application. Commun Stat Theory Methods 25(12):3059–3083
Mudholkar GS, Srivastava DK (1993) Exponentiated Weibull family for analyzing bathtub failure rate data. IEEE Trans Reliab 42(2):299–302
Mudholkar GS, Srivastava DK, Friemer M (1995) The exponentiated Weibull family: a reanalysis of the bus-motor-failure data. Technometrics 37(4):436–445
Wang H, Pham H (1996) Optimal maintenance policies for several imperfect repair models. Int J Syst Sci 27(6):543–549
Yañez M, Joglar F, Modarres M (2002) Generalized renewal process for analysis of repairable systems with limited failure experience. Reliab Eng Syst Saf 77(2):167–180
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ghnimi, S., Gasmi, S. & Nasr, A. Reliability parameters estimation for parallel systems under imperfect repair. Metrika 80, 273–288 (2017). https://doi.org/10.1007/s00184-016-0603-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-016-0603-y
Keywords
- Repairable systems reliability
- Exponentiated Weibull distribution
- Imperfect repair
- Maximum likelihood estimation
- Virtual age