Statistical inference of the stress-strength reliability and mean remaining strength of series system with cold standby redundancy at system and component levels

Year 2021, , 1793 - 1821, 14.12.2021

Gülce Cüran , Fatih Kızılaslan

https://doi.org/10.15672/hujms.804070

Cited By: 2

Abstract

In this study, we consider the stress-strength reliability and mean remaining strength of a series system with cold standby redundancy at the component and system levels. Classical and Bayesian approaches are studied in order to obtain the estimates when the underlying stress, strength and standby components follow the exponential distribution with different parameters. Bayes estimates are approximated by using Lindley’s approximation and Markov Chain Monte Carlo methods. Asymptotic confidence intervals and highest probability density credible intervals are constructed. We perform Monte Carlo simulations to compare the performance of proposed estimates. A real data set is analyzed for the purpose of illustration.

Keywords

Stress-strength reliability, mean remaining strength, cold standby, series system

References

• [1] K. Ahmadi and S. Ghafouri, Reliability estimation in a multicomponent stress– strength model under generalized half-normal distribution based on progressive type-II censoring, J. Stat. Comput. Simul. 89 (13), 2505-2548, 2019.
• [2] F.G. Akgül, Reliability estimation in multicomponent stress–strength model for ToppLeone distribution, J. Stat. Comput. Simul. 89 (15), 2914-2929, 2019.
• [3] E.K. AL-Hussaini, M.A.M.A. Mousa and K.S. Sultan, Parametric and nonparametric estimation of P(Y < X) for finite mixtures of lognormal components, Comm. Statist. Theory Methods 26 (5), 1269-1289, 1997.
• [4] M.M. Ali, M. Pal and J. Woo, Estimation of P(Y < X) in a four-parameter generalized gamma distribution, Austrian J. Stat. 41 (3), 197-210, 2012.
• [5] A. Baklizi and O. Eidous, Nonparametric estimation of P(X < Y ) using kernel methods, Metron 64 (1), 47-60, 2006.
• [6] R.E. Barlow and F. Proschan, Statistical Theory of Reliability and Life Testing, International Series in Decision Processes, Holt, Rinehart and Winston Inc, 1975.
• [7] M. Basirat, S. Baratpour and J. Ahmadi, Statistical inferences for stress-strength in the proportional hazard models based on progressive Type-II censored samples, J. Stat. Comput. Simul. 85 (3), 431-449, 2015.
• [8] Z.W.Birnbaum, On a use of Mann-Whitney statistics, in: Proceedings of the 3rd Berkley Symposium in Mathematics, Statistics and Probability, 3 (1), 13-17, 1956.
• [9] Z.W. Birnbaum and B.C. McCarty, A distribution-free upper confidence bounds for P r(Y < X) based on independent samples of X and Y , Ann. Math. Statist. 29 (2), 558-562, 1958.
• [10] P.J. Boland and E. El-Neweihi E, Component redundancy vs system redundancy in the hazard rate ordering, IEEE Trans. Rel. 44 (4), 614-619, 1995.
• [11] C. Cetinkaya and A.I. Genc, Stress-strength reliability estimation under the standard two-sided power distribution, Appl. Math. Model. 65, 72-88, 2019.
• [12] J.H. Cha, J. Mi and W.Y. Yun, Modelling a general standby system and evaluation of its performance, Appl. Stoch. Models Bus. Ind. 24 (2), 159-169, 2008.
• [13] J. Chen, Y. Zhang, P. Zhao and S. Zhou, Allocation strategies of standby redundancies in series/parallel system, Comm. Statist. Theory Methods 47 (3), 708-724, 2018.
• [14] M.H. Chen and Q.M. Shao, Monte Carlo estimation of Bayesian credible and HPD intervals, J. Comput. Graph. Statist. 8 (1), 69-92, 1999.
• [15] S. Eryilmaz, Dynamic reliability and performance evaluation of multi-state systems with two components, Hacet. J. Math. Stat. 62 (1), 125-133, 2011.
• [16] S. Eryilmaz, Reliability of a k-out-of-n system equipped with a single warm standby component, IEEE Trans. Rel. 62 (2), 499-503, 2013.
• [17] S. Eryilmaz, The effectiveness of adding cold standby redundancy to a coherent system at system and component levels, Reliab. Eng. Syst. Saf. 165, 331-335, 2017.
• [18] A. Gelman, J.B. Carlin, H.S. Stern and D.B. Rubin, Bayesian Data Analysis, Chapman Hall, 2003. [19] S. Gurler, The mean remaining strength of systems in a stress-strength model, Hacet. J. Math. Stat. 42 (2), 181-187, 2013.
• [20] S. Gurler, B.H. Ucer and I. Bairamov, On the mean remaining strength at the system level for some bivariate survival models based on exponential distribution, J. Comput. Appl. Math. 290, 535-542, 2015.
• [21] B. Hasselman, Package "nleqslv", R package version: 3.3.2, 2018.
• [22] N.K. Hazra and A.K. Nanda, Component redundancy versus system redundancy in different stochastic orderings, IEEE Trans. Rel. 63 (2), 567-582, 2014.
• [23] M. Jovanovıć, B. Milošević and M. Obradović, Estimation of stress-strength probability in a multicomponent model based on geometric distribution,Hacet. J. Math. Stat. 49 (4), 1515-1532, 2020.
• [24] T. Kayal, Y.M. Tripathi, S. Dey and S. Wu, On estimating the reliability in a multicomponent stress-strength model based on Chen distribution, Comm. Statist. Theory Methods 49 (10), 2429-2447, 2019.
• [25] F. Kizilaslan, Classical and Bayesian estimation of reliability in a multicomponent stress-strength model based on a general class of inverse exponentiated distributions, Statist. Papers 59 (3), 1161-1192, 2018.
• [26] F. Kizilaslan, The mean remaining strength of parallel systems in a stress-strength model based on exponential distribution, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 68 (2), 1435-1451, 2019.
• [27] C. S. Kumar, Standby redundancy at system and component levels-A comparison, Microelectron. Reliab 35 (4), 751-752, 1995.
• [28] J. Lawless, Statistical Models and Methods for Lifetime Data, 2nd ed., Wiley, New York, 2003.
• [29] D.V. Lindley, Approximate Bayes method, Trabajos de Estadistica 3, 281-288, 1980.
• [30] Y. Liu, Y. Shi, X. Bai and P. Zhan, Reliability estimation of a N-M-cold-standby redundancy system in a multicomponent stress-strength model with generalized halflogistic distribution, Physica A 490, 231-249, 2018.
• [31] M. Mahdizadeh, On estimating a stress-strength type reliability, Hacet. J. Math. Stat. 47 (1), 243-253, 2018.
• [32] M. Mahdizadeh and E. Zamanzade, Kernel-based estimation of P(X > Y ) in ranked set sampling, SORT 40 (2), 243-266, 2016.
• [33] R. Nojosa and P.N. Rathie, Stress-strength reliability models involving generalized gamma and Weibull distributions, Int. J. Qual. Reliab. 37 (4), 538-551, 2020.
• [34] R Core Team R, A language and environment for statistical computing, Vienna, Austria, R Foundation for Statistical Computing, 2020.
• [35] C.R. Rao, Linear Statistical Inference and Its Applications, Wiley, 1965.
• [36] T.R. Rasethuntsa and M. Nadar, Stress-strength reliability of a non-identicalcomponent-strengths system based on upper record values from the family of Kumaraswamy generalized distributions, Statistics 52 (3), 684-716, 2018.
• [37] K. Shen and M. Xie, The effectiveness of adding standby redundancy at system and component levels, IEEE Trans. Rel. 40 (1), 53-55, 1991.
• [38] K.C. Siju and M. Kumar, Estimation of stress-strength reliability of a parallel system with active, warm and cold standby components, J. Ind. Prod. Eng. 34 (8), 590-610, 2017.
• [39] L. Tierney, Markov chains for exploring posterior distributions, Ann. Statist. 22 (4), 1701-1728, 1994.
• [40] R. Yan, B. Lu and X. Li, On Redundancy allocation to series and parallel systems of two components, Comm. Statist. Theory Methods 48 (18), 4690-4701, 2019.
• [41] P. Zhao, Y. Zhang and L. Li, Redundancy allocation at component level versus system level, European J. Oper. Res. 241 (2), 402-411, 2015.