Statistical Reliability Based Modeling of Series and Parallel Operating Systems using Extreme Value Theory

Authors: Mohamad Mahdavi, Mojtaba Mahdavi

Abstract:

This paper tries to represent a new method for computing the reliability of a system which is arranged in series or parallel model. In this method we estimate life distribution function of whole structure using the asymptotic Extreme Value (EV) distribution of Type I, or Gumbel theory. We use EV distribution in minimal mode, for estimate the life distribution function of series structure and maximal mode for parallel system. All parameters also are estimated by Moments method. Reliability function and failure (hazard) rate and p-th percentile point of each function are determined. Other important indexes such as Mean Time to Failure (MTTF), Mean Time to repair (MTTR), for non-repairable and renewal systems in both of series and parallel structure will be computed.

Keywords: Reliability, extreme value, parallel, series, lifedistribution

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1333460

References:

[1] Gumbel, E. J., "Statistical Theory of Extreme values and Some Practical Applications", Applied Mathematics Series Publication No. 33, 1954.
[2] Embrechts, P., Kl¨uppelberg, C., and Mikosch, T. (1999). Modelling ExtremalEvents for Insurance and Finance. Applications of Mathematics. Springer.2nd ed.(1st ed., 1997).
[3] Reiss, R. D. and Thomas, M. (1997). Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields.Birkh¨auser Verlag, Basel.
[4] Koedijk, K. G., Schafgans, M., and de Vries, C. (1990). The Tail Index of Exchange Rate Returns. Journal of International Economics, 29:93- 108.
[5] Dacorogna, M. M., M¨uller, U. A., Pictet, O. V., and de Vries, C. G. (1995). 22 The distribution of extremal foreign exchange rate returns in extremelylarge data sets. Preprint, O&A Research Group.
[6] Loretan, M. and Phillips, P. (1994). Testing the covariance stationarity of heavy-tailed time series. Journal of Empirical Finance, 1(2):211-248.
[7] Neftci, S. N. (2000). Value at risk calculations, extreme events, and tail esti-mation. Journal of Derivatives, pages 23-37.
[8] McNeil, A. J. (1999). Extreme value theory for risk managers. In InternalModelling and CAD II, pages 93-113. RISK Books.
[9] Diebold, F. X., Schuermann, T., and Stroughair, J. D. (1998). Pitfalls and opportunities in the use of extreme value theory in risk management. In Refenes, A.-P., Burgess, A., and Moody, J., editors, Decision Technologiesfor Computational Finance, pages 3-12. Kluwer Academic Publishers.
[10] Fisher, R. A., Tippett, L., Limiting forms of the frequency distribution of the largest and smallest member of a sample, Proc Cambridge Phil Soc, No.24, 1928, pp.180-190.
[11] Gumbel, E. J., statistical Theory of Extreme values and Some Practical Applications, Applied Mathematics Series Publication, No. 33, 1954.
[12] Ang, AH-S, Tang, WH., Probability concepts in engineering planning and design, Vol. 2, New York, John Wiley and Sons, 1984.
[13] Kotz, S., Nagarajah, S., Extreme Value Distribution: Theory and Applications, London, Imperial College Press, 2000.
[14] Elkahlout, G. R., Bayes Estimators for the Extreme-Value Reliability Function, Computers and Mathematics with Applications, No.51, 2006, pp.673-679.
[15] Lye, L. M., Hapuarachchi, K. P., Ryan, S., Bayes estimation of the extreme-value reliability function, IEEE Trans. Reliability, Vol.4, No.42, 1993, pp.641-644.
[16] Crandall, S. H., First-crossing probability of the linear oscillator, J Sound Vib, No.12, 1970, pp.285-299.
[17] Kawano, K., Venkataramana, K., Dynamic response and reliability analysis of large offshore structures, Comput Methods Appl Mech Eng, No.168, 1999, pp.255-272.
[18] Chen, J. J., Duan, B.Y., Zeng, Y.G., Study on dynamic reliability analysis of the structures with multi-degree-of-freedom, Comput Struct, Vol.5, No.62, 1997, pp.877-881.
[19] Jian-Bing, Chen, Jie, Li, The extreme value distribution and dynamic reliability analysis of nonlinear structures with uncertain parameters, Structural Safety, No.29, 2007, pp.77-93.
[20] Hideo, Hirose, More accurate breakdown voltage estimation for the new step-up test method in the gumbel distribution model, European Journal of Operational Research, No.177, 2007, pp. 406-419.
[21] Jie, Li, Jian-bing, Chen, Wen-liang, Fan, The equivalent extreme-value event and evaluation of the structural system reliability, Structural Safety, No.29, 2007, pp.112-131.
[22] Nancy R., Methods for Statistical Analysis of Reliability and Life Data (Probability & Mathematical Statistics), John Wiley, 1974.
[23] Jardine, A. K. S., Maintenance, Replacement, and Reliability, Department of Engineering Production, University of Birmingham, 1973.
[24] Zacks, Shelemyuhu; Introduction to Reliability Analysis, New York: Springer-Verlag, 1992.