Reliability Approximation through the Discretization of Random Variables using Reversed Hazard Rate Function
Sometime it is difficult to determine the exact reliability for complex systems in analytical procedures. Approximate solution of this problem can be provided through discretization of random variables. In this paper we describe the usefulness of discretization of a random variable using the reversed hazard rate function of its continuous version. Discretization of the exponential distribution has been demonstrated. Applications of this approach have also been cited. Numerical calculations indicate that the proposed approach gives very good approximation of reliability of complex systems under stress-strength set-up. The performance of the proposed approach is better than the existing discrete concentration method of discretization. This approach is conceptually simple, handles analytic intractability and reduces computational time. The approach can be applied in manufacturing industries for producing high-reliable items.
<p>[1] D. H. Evans, ”Statistical tolerancing: the state of the art: Part II, Methods
of estimating moments,” Journal of Quality Technology, 1975, Vol. 7,
pp. 1-12.
[2] G. Taguchi, ”Performance analysis and design,” Int. Journal of Production
Research, 1978, Vol. 16, pp. 521-530.
[3] J. R. Drrico and N. A. Zaino Jr., ”Statistical tolerancing using a
modification of Taguchi’s method,” Technometrics, 1988, Vol. 30, pp.
397-405.
[4] J. R. English, T. Sargent, and T. L. Landers, ”A discretizing approach
for stress/strength analysis,”IEEE Trans. on Reliability, 1996, Vol. 45,
pp. 84-89.
[5] T. Ghosh and D. Roy, ”Statistical tolerancing through discretization of
variables,” International Journal. of Quality and Reliability Management,
2011,Vol. 28, pp 220-232.
[6] D. Roy and R. P. Gupta ”Classification of discrete lives,” Microelectronics
and Reliability, 2000, Vol.52, pp. 1459-1473.
[7] D. Roy, ”Reliability measures in the discrete bivariate set- up and related
characterization results for a Bivariate Geometric distribution,” Journal
of Multivariate Analysis, 1993, Vol. 46, pp. 362-373.
[8] A.W. Kemp, ”Characterization of a discrete normal distribution,” 1997,
Vol. 63, pp. 223-229.
[9] S.Inusah and T. J. Kozubowski, ”A discrete analogue of the Laplace
distribution,” Journal of Statistical Planning and Inference, 2006, Vol.
136, pp. 1090-1102.
[10] D. Roy and T. Dasgupta, ”A discretizing approach for evaluating
reliability of complex systems under stress-strength model,” IEEE Trans.
on Reliability, 2001, Vol. 50, pp. 145-150.
[11] D. Roy, ”The discrete normal distribution,” Communications in Statistics
(Theory and Methods), 2003, Vol. 32, pp. 50-53.
[12] D. Roy, and T. Dasgupta, ”Evaluation of reliability of complex systems
by means of a discretizing approach: Weibull set-up,” Int. J. of Quality
and Reliability Management, 2002, Vol. 19, pp. 792-801.
[13] D. Roy, ”Discretization of continuous distributions with an application to
stress-strength analysis,” Calcutta Statistical Association Bulletin, 2002,
Vol. 52, pp. 297-313.
[14] D. Roy, ”Discrete Rayleigh distribution,” IEEE Trans. on Reliability,,
2004, Vol. 53 pp. 255-260.
[15] D. Roy, and T. Ghosh, ”A new discretization approach with application
in reliability estimation,” IEEE Trans. on Reliability, 2009, Vol. 58, pp.
456-461.
[16] T. Ghosh, D. Roy and N. K. Chandra, ”Discretization of Random
Variables with Applications”, Calcutta Statistical Association Bulletin,
2011, Vol. 63, pp. 323-338.
[17] Kapur K.C and Lamberson L.R., Reliability in Engineering Design, New
York: John Wiley & Sons, 1977.</p>
<p>[1] D. H. Evans, ”Statistical tolerancing: the state of the art: Part II, Methods
of estimating moments,” Journal of Quality Technology, 1975, Vol. 7,
pp. 1-12.
[2] G. Taguchi, ”Performance analysis and design,” Int. Journal of Production
Research, 1978, Vol. 16, pp. 521-530.
[3] J. R. Drrico and N. A. Zaino Jr., ”Statistical tolerancing using a
modification of Taguchi’s method,” Technometrics, 1988, Vol. 30, pp.
397-405.
[4] J. R. English, T. Sargent, and T. L. Landers, ”A discretizing approach
for stress/strength analysis,”IEEE Trans. on Reliability, 1996, Vol. 45,
pp. 84-89.
[5] T. Ghosh and D. Roy, ”Statistical tolerancing through discretization of
variables,” International Journal. of Quality and Reliability Management,
2011,Vol. 28, pp 220-232.
[6] D. Roy and R. P. Gupta ”Classification of discrete lives,” Microelectronics
and Reliability, 2000, Vol.52, pp. 1459-1473.
[7] D. Roy, ”Reliability measures in the discrete bivariate set- up and related
characterization results for a Bivariate Geometric distribution,” Journal
of Multivariate Analysis, 1993, Vol. 46, pp. 362-373.
[8] A.W. Kemp, ”Characterization of a discrete normal distribution,” 1997,
Vol. 63, pp. 223-229.
[9] S.Inusah and T. J. Kozubowski, ”A discrete analogue of the Laplace
distribution,” Journal of Statistical Planning and Inference, 2006, Vol.
136, pp. 1090-1102.
[10] D. Roy and T. Dasgupta, ”A discretizing approach for evaluating
reliability of complex systems under stress-strength model,” IEEE Trans.
on Reliability, 2001, Vol. 50, pp. 145-150.
[11] D. Roy, ”The discrete normal distribution,” Communications in Statistics
(Theory and Methods), 2003, Vol. 32, pp. 50-53.
[12] D. Roy, and T. Dasgupta, ”Evaluation of reliability of complex systems
by means of a discretizing approach: Weibull set-up,” Int. J. of Quality
and Reliability Management, 2002, Vol. 19, pp. 792-801.
[13] D. Roy, ”Discretization of continuous distributions with an application to
stress-strength analysis,” Calcutta Statistical Association Bulletin, 2002,
Vol. 52, pp. 297-313.
[14] D. Roy, ”Discrete Rayleigh distribution,” IEEE Trans. on Reliability,,
2004, Vol. 53 pp. 255-260.
[15] D. Roy, and T. Ghosh, ”A new discretization approach with application
in reliability estimation,” IEEE Trans. on Reliability, 2009, Vol. 58, pp.
456-461.
[16] T. Ghosh, D. Roy and N. K. Chandra, ”Discretization of Random
Variables with Applications”, Calcutta Statistical Association Bulletin,
2011, Vol. 63, pp. 323-338.
[17] Kapur K.C and Lamberson L.R., Reliability in Engineering Design, New
York: John Wiley & Sons, 1977.</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:65371", author = "Tirthankar Ghosh and Dilip Roy and Nimai Kumar Chandra", title = "Reliability Approximation through the Discretization of Random Variables using Reversed Hazard Rate Function", abstract = "Sometime it is difficult to determine the exact reliability for complex systems in analytical procedures. Approximate solution of this problem can be provided through discretization of random variables. In this paper we describe the usefulness of discretization of a random variable using the reversed hazard rate function of its continuous version. Discretization of the exponential distribution has been demonstrated. Applications of this approach have also been cited. Numerical calculations indicate that the proposed approach gives very good approximation of reliability of complex systems under stress-strength set-up. The performance of the proposed approach is better than the existing discrete concentration method of discretization. This approach is conceptually simple, handles analytic intractability and reduces computational time. The approach can be applied in manufacturing industries for producing high-reliable items.
", keywords = "Discretization, Reversed Hazard Rate, Exponential
distribution, reliability approximation, engineering item.", volume = "7", number = "4", pages = "715-5", }
