Statistical Reliability Based Modeling of Series and Parallel Operating Systems using Extreme Value Theory
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.
[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.
[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.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:56333", author = "Mohamad Mahdavi and Mojtaba Mahdavi", title = "Statistical Reliability Based Modeling of Series and Parallel Operating Systems using Extreme Value Theory", 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", volume = "5", number = "7", pages = "1348-5", }
