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Reliability Shock Models: A Brief Excursion

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Applied Advanced Analytics

Part of the book series: Springer Proceedings in Business and Economics ((SPBE))

Abstract

We attempt to provide a brief introduction to the extensive area of shock model research in reliability theory. Possible connections with application areas such as risk analysis, inventory control and biometry are indicated. Important concepts and tools for proving shock model results such as total positivity and variation diminishing property (VDP) are introduced. Most of the important results concerning nonparametric ageing classes arising from shock models are summarized, and some typical techniques of proof are emphasized. A variety of scenarios with diverse arrival processes such as homogeneous Poisson process, nonhomogeneous Poisson process, stationary and nonstationary pure birth processes are considered. A few interesting results related to cumulative damage models are also discussed.

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References

  • A-Hameed, M., & Proschan, F. (1975). Shock models with underlying birth process. Journal of Applied Probability, 12(1), 18–28.

    Google Scholar 

  • A-Hameed, M. S., & Proschan, F. (1973). Nonstationary shock models. Stochastic Processes and their Applications, 1, 383–404.

    Google Scholar 

  • Abouammoh, A. M., & Ahmed, A. N. (1988). The new better than used failure rate class of life distributions. Advances in Applied Probability, 20, 237–240.

    Article  Google Scholar 

  • Abouammoh, A. M., Hendi, M. I., & Ahmed, A. N. (1988). Shock models with NBUFR and NBAFR survivals. Trabajos De Estadistica, 3(1), 97–113.

    Article  Google Scholar 

  • Akama, M., & Ishizuka, H. (1995). Reliability analysis of Shinkansen vehicle axle using probabilistic fracture mechanics. JSME international journal. Ser. A, Mechanics and Material Engineering, 38(3), 378–383.

    Google Scholar 

  • Anderson, K. K. (1987). Limit theorems for general shock models with infinite mean intershock times. Journal of Applied Probability, 24(2), 449–456.

    Article  Google Scholar 

  • Anis, M. (2012). On some properties of the IDMRL class of life distributions. Journal of Statistical Planning and Inference, 142(11), 3047–3055.

    Article  Google Scholar 

  • Barlow, R. E., & Proschan, F. (1965). Mathematical theory of reliability. New York: Wiley.

    Google Scholar 

  • Barlow, R. E., & Proschan, F. (1975). Statistical theory of reliability and life testing. Rinehart and Winston, New York: Holt.

    Google Scholar 

  • Belzunce, F., Ortega, E.-M., & Ruiz, J. M. (2007). On non-monotonic ageing properties from the Laplace transform, with actuarial applications. Insurance: Mathematics and Economics, 40(1), 1–14.

    Google Scholar 

  • Bogdanoff, J. L., & Kozin, F. (1985). Probabilistic models of cumulative damage. New York: Wiley.

    Google Scholar 

  • Bryson, M. C., & Siddiqui, M. M. (1969). Some criteria for aging. Journal of the American Statistical Association, 64, 1472–1483.

    Article  Google Scholar 

  • Campean, I. F., Rosala, G. F., Grove, D. M., & Henshall, E. (2005). Life modelling of a plastic automotive component. In Proceedings of Annual Reliability and Maintainability Symposium, 2005 (pages 319–325). IEEE.

    Google Scholar 

  • Dasgupta, A., & Pecht, M. (1991). Material failure mechanisms and damage models. IEEE Transactions on Reliability, 40(5), 531–536.

    Article  Google Scholar 

  • Durham, S., & Padgett, W. (1997). Cumulative damage models for system failure with application to carbon fibers and composites. Technometrics, 39(1), 34–44.

    Article  Google Scholar 

  • Ebrahimi, N. (1999). Stochastic properties of a cumulative damage threshold crossing model. Journal of Applied Probability, 36(3), 720–732.

    Article  Google Scholar 

  • Esary, J. D., Marshall, A. W., & Proschan, F. (1973). Shock models and wear processes. Annals of Probability, 1, 627–649.

    Article  Google Scholar 

  • Feller, W. (1968). An introduction to probability theory and its applications (Vol. 1). New York: Wiley.

    Google Scholar 

  • Garbatov, Y., & Soares, C. G. (2001). Cost and reliability based strategies for fatigue maintenance planning of floating structures. Reliability Engineering & System Safety, 73(3), 293–301.

    Article  Google Scholar 

  • Gertsbakh, I., & Kordonskiy, K. B. (2012). Models of failure. Springer Science & Business Media.

    Google Scholar 

  • Glaser, R. E. (1980). Bathtub and related failure rate characterizations. Journal of the American Statistical Association, 75(371), 667–672.

    Article  Google Scholar 

  • Guess, F., Hollander, M., & Proschan, F. (1986). Testing exponentiality versus a trend change in mean residual life. Annals of Statistics, 14(4), 1388–1398.

    Article  Google Scholar 

  • Hollander, M., & Proschan, F. (1984). Nonparametric concepts and methods in reliability. In P. R. Krishnaiah & P. K. Sen (Eds.), Handbook of statistics (Vol. 4, pp. 613–655). Amsterdam: Elsevier Sciences.

    Google Scholar 

  • Karlin, S. (1968). Total positivity, (Vol. 1). Stanford University Press.

    Google Scholar 

  • Karlin, S., & Proschan, F. (1960). Pólya type distributions of convolutions. The Annals of Mathematical Statistics, 721–736.

    Google Scholar 

  • Khan, R. A., Bhattacharyya, D., & Mitra, M. (2021). On classes of life distributions based on the mean time to failure function. Journal of Applied Probability, 58(2), in-press.

    Google Scholar 

  • Klefsjö, B. (1981). HNBUE survival under some shock models. Scandinavian Journal of Statistics, 8, 39–47.

    Google Scholar 

  • Klefsjö, B. (1983). A useful ageing property based on the Laplace transformation. Journal of Applied Probability, 20, 615–626.

    Article  Google Scholar 

  • Kochar, S. C. (1990). On preservation of some partial orderings under shock models. Advances in Applied Probability, 22(2), 508–509.

    Article  Google Scholar 

  • Lai, C. D., & Xie, M. (2006). Stochastic ageing and dependence for reliability. New York: Springer.

    Google Scholar 

  • Loh, W. Y. (1984). A new generalization of the class of NBU distributions. IEEE Transactions on Reliability, R-33, 419–422.

    Google Scholar 

  • Lukić, M., & Cremona, C. (2001). Probabilistic optimization of welded joints maintenance versus fatigue and fracture. Reliability Engineering & System Safety, 72(3), 253–264.

    Article  Google Scholar 

  • Marshall, A. W., & Olkin, I. (2007). Life distributions. New York: Springer.

    Google Scholar 

  • Mitra, M., & Basu, S. (1994). On a nonparametric family of life distributions and its dual. Journal of Statistical Planning and Inference, 39, 385–397.

    Article  Google Scholar 

  • Mitra, M., & Basu, S. K. (1996). Shock models leading to non-monotonic ageing classes of life distributions. Journal of Statistical Planning and Inference, 55, 131–138.

    Article  Google Scholar 

  • Nakagawa, T. (2007). Shock and damage models in reliability theory. Springer Science & Business Media.

    Google Scholar 

  • Padgett, W. (1998). A multiplicative damage model for strength of fibrous composite materials. IEEE Transactions on Reliability, 47(1), 46–52.

    Article  Google Scholar 

  • Pellerey, F. (1993). Partial orderings under cumulative damage shock models. Advances in Applied Probability, 25(4), 939–946.

    Article  Google Scholar 

  • Pérez-Ocón, R., & Gámiz-Pérez, M. L. (1995). On the HNBUE property in a class of correlated cumulative shock models. Advances in Applied Probability, 27(4), 1186–1188.

    Article  Google Scholar 

  • Petryna, Y. S., Pfanner, D., Stangenberg, F., & Krätzig, W. B. (2002). Reliability of reinforced concrete structures under fatigue. Reliability Engineering & System Safety, 77(3), 253–261.

    Article  Google Scholar 

  • Rolski, T. (1975). Mean residual life. Bulletin of the International Statistical Institute, 46, 266–270.

    Google Scholar 

  • Satow, T., Teramoto, K., & Nakagawa, T. (2000). Optimal replacement policy for a cumulative damage model with time deterioration. Mathematical and Computer Modelling, 31(10–12), 313–319.

    Article  Google Scholar 

  • Shanthikumar, J. G., & Sumita, U. (1983). General shock models associated with correlated renewal sequences. Journal of Applied Probability, 20(3), 600–614.

    Article  Google Scholar 

  • Sobczyk, K., & Spencer, B, Jr. (2012). Random fatigue: From data to theory. Academic Press.

    Google Scholar 

  • Sobczyk, K., & Trebicki, J. (1989). Modelling of random fatigue by cumulative jump processes. Engineering Fracture Mechanics, 34(2), 477–493.

    Article  Google Scholar 

  • Yamada, K. (1989). Limit theorems for jump shock models. Journal of Applied Probability, 26(4), 793–806.

    Article  Google Scholar 

  • Zacks, S. (1991). Introduction to reliability analysis. New York: Springer.

    Google Scholar 

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Acknowledgements

The authors are indebted to Professor Arnab K. Laha for his constant encouragement in making this article a reality and to Mr. Dhrubasish Bhattacharyya for some helpful discussions during the preparation of this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Engineering Science and Technology, Howrah, India

    Murari Mitra & Ruhul Ali Khan

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  1. Murari Mitra
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  2. Ruhul Ali Khan
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Correspondence to Murari Mitra .

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Editors and Affiliations

  1. Production and Quantitative Methods, Indian Institute of Management Ahmedabad, Ahmedabad, Gujarat, India

    Arnab Kumar Laha

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Mitra, M., Khan, R.A. (2021). Reliability Shock Models: A Brief Excursion. In: Laha, A.K. (eds) Applied Advanced Analytics. Springer Proceedings in Business and Economics. Springer, Singapore. https://doi.org/10.1007/978-981-33-6656-5_3

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