The Weibull Distribution: Reliability Characterization Based on Linear and Circular Consecutive Systems
Abstract
Linear and circular consecutive models play a vital role to study the mechanical systems emerging in various fields including survival analysis, reliability theory, biological disciplines, and other lifetime sciences. As a result, analysis of reliability properties of consecutive k − out − of − n : F systems has gained a lot of attention in recent years from a theoretical and practical point of view. In the present article, we have studied some important stochastic and aging properties of residual lifetime of consecutive k − out − of − n : F systems under the condition n − k + 1, k ≤ n and all components of the system are working at time t. The mean residual lifetime (MRL) and its hazard rate function are proposed for the linear consecutive k − out − of − n : F (lin/con/k/n:F) and circular consecutive k − out − of − n : F (cir/con/k/n:F) systems. Furthermore, several mathematical properties of the proposed MRL are examined. Finally, the Weibull distribution with two parameters is used as an example to explain the theoretical results.References
Almamy, J. A., Ibrahim, M., Eliwa, M. S., Al-mualim, S., & Yousof, H. M. (2018). The two-parameter odd Lindley Weibull lifetime model with properties and applications. International journal of statistics and probability, 7(4), 1927-7040.
Aki, S., & Hirano, K. (1996). Lifetime distribution and estimation problems of consecutive-k-out-of-n: F systems. Annals of the Institute of Statistical Mathematics, 48, 185-199.
Aki, S., & Hirano, K. (1996). Lifetime distribution and estimation problems of consecutive-k-out-of-n: F systems. Annals of the Institute of Statistical Mathematics, 48(1), 185-199.
Alizadeh, M., Afify, A. Z., Eliwa, M. S., & Ali, S. (2020). The odd log-logistic Lindley-G family of distributions: properties, Bayesian and non-Bayesian estimation with applications. Computational Statistics, 35(1), 281-308.
Alizadeh, M., Khan, M. N., Rasekhi, M., & Hamedani, G. G. (2021). A New Generalized Modified Weibull Distribution. Statistics, Optimization & Information Computing, 9(1), 17-34.
Asadi, M., & Bayramoglu, I. (2006). The mean residual life function of a k-out-of-n structure at the system level. IEEE Transactions on Reliability, 55(2), 314-318.
Asadi, M., & Goliforushani, S. (2008). On the mean residual life function of coherent systems. IEEE Transactions on Reliability, 57(4), 574-580.
Bairamov, I., Ahsanullah, M., & Akhundov, I. (2002). A residual life function of a system having parallel or series structures. Journal of Statistical Theory and Applications, 1(2), 119-132.
Benkhelifa, L. (2021). The Weibull Birnbaum-Saunders distribution and its applications. Statistics, Optimization & Information Computing, 9(1), 61-81.
Boland, P. J., & Samaniego, F. J. (2004). Stochastic ordering results for consecutive k-out-of-n: F systems. IEEE Transactions on Reliability, 53(1), 7-10.
Bollinger, R. C., & Salvia, A. A. (1982). Consecutive-k-out-of-n: F networks. IEEE Transactions on Reliability, 31(1), 53-56.
Chang, G. J. L. R., In-Hang, C., Cui, L., & Hwang, F. K. (2000). Reliabilities of consecutive-k systems (Vol. 4). Springer Science & Business Media.
Chiang, D. T., & Niu, S. C. (1981). Reliability of consecutive-k-out-of-n: F system. IEEE Transactions on Reliability, 30(1), 87-89.
Chen, R. W., & Hwang, F. K. (1985). Failure distributions of consecutive-k-out-of-n: F systems. IEEE Transactions on Reliability, 34(4), 338-341.
Elgohari, H.,& Yousof, H. (2020). New extension of weibull distribution: copula, mathematical properties and data modeling. Statistics, Optimization & Information Computing, 8(4), 972-993.
Eliwa, M. S., Alshammari, F. S. S., Abualnaja, K. M., & El-Morshedy, M. (2021). A Flexible Extension to an Extreme Distribution. Symmetry, 13(5), 745.
Eliwa, M. S., El-Morshedy, M., & Ali, S. (2020). Exponentiated odd Chen-G family of distributions: Statistical properties, Bayesian and non-Bayesian estimation with applications. Journal of Applied Statistics, 1-27. https://doi.org/10.1080/02664763.2020.1783520
Eliwa, M. S., & El-Morshedy, M. (2020a). Bayesian and non-Bayesian estimation of four-parameter of bivariate discrete inverse Weibull distribution with applications to model failure times, football and biological data. Filomat, 34(8), 2511-2531.
Eliwa, M. S., & El-Morshedy, M. (2020b). Bivariate odd Weibull-G family of distributions: properties, Bayesian and non-Bayesian estimation with bootstrap confidence intervals and application. Journal of Taibah University for science, 14(1), 331-345.
El-Morshedy, M., Alshammari, F. S., Hamed, Y. S., Eliwa, M. S., & Yousof, H. M. (2021b). A new family of continuous probability distributions. Entropy, 23(2), 194.
El-Morshedy, M., Alshammari, F. S., Tyagi, A., Elbatal, I., Hamed, Y. S., & Eliwa, M. S. (2021c). Bayesian and frequentist inferences on a type I half-logistic odd Weibull generator with applications in engineering. Entropy, 23(4), 446.
El-Morshedy, M., El-Faheem, A. A., & El-Dawoody, M. (2020). Kumaraswamy inverse Gompertz distribution: Properties and engineering applications to complete, type-II right censored and upper record data. Plos one, 15(12), e0241970.
El-Morshedy, M., Eliwa, M. S., El-Gohary, A., Almetwally, E. M., & EL-Desokey, R. (2021a). Exponentiated Generalized Inverse Flexible Weibull Distribution: Bayesian and Non-Bayesian Estimation Under Complete and Type II Censored Samples with Applications. Communications in Mathematics and Statistics, 1-22.
El-Morshedy, M., & Eliwa, M. S. (2019). The odd flexible Weibull-H family of distributions: Properties and estimation with applications to complete and upper record data. Filomat, 33(9), 2635-2652.
El-Morshedy, M., Eliwa, M. S., & Afify, A. Z. (2020a). The odd Chen generator of distributions: Properties and estimation methods with applications in medicine and engineering. J. Natl. Sci. Found. Sri Lanka, 48, 113-130.
El-Morshedy, M., Eliwa, M. S., El-Gohary, A., & Khalil, A. A. (2020b). Bivariate exponentiated discrete Weibull distribution: statistical properties, estimation, simulation and applications. Mathematical Sciences, 14(1), 29-42.
El-Morshedy, M., Eliwa, M. S., & Tyagi, A. (2021d). A discrete analogue of odd Weibull-G family of distributions: properties, classical and Bayesian estimation with applications to count data. Journal of Applied Statistics, 1-25.
Eryilmaz, S. (2013). Component importance for linear consecutive-k-out-of-n and m-consecutive-k-Out-of-n systems with exchangeable components. Naval Research Logistics (NRL), 60(4), 313-320.
Eryilmaz, S. (2007). On the lifetime distribution of consecutive k-out-of-n:F system. IEEE Transactions on Reliability, 56, 35-39.
Eryılmaz, S. (2009). Reliability properties of consecutive k-out-of-n systems of arbitrarily dependent components. Reliability Engineering & System Safety, 94(2), 350-356.
Goliforushani, S., Xie, M., & Balakrishnan, N. (2018). On the mean residual life of a generalized k-out-of-n system. Communications in Statistics-Theory and Methods, 47(10), 2362-2372.
Khaledi,B.E.,&Shaked,M (2007).Ordering conditional lifetimes of coherent systems. Journal of Statistical Planning and Inference, 137(4), 1173-1184.
Khan, R., Khan, M. A., & Khan, M. A. R. (2017). Relations for moments of generalized record values from additive Weibull distribution and associated inference. Statistics, Optimization& Information Computing, 5(2), 127-136.
Kotz, S., & Shanbhag, D. N. (1980). Some new approaches to probability distributions. Advances in Applied Probability, 903-921.
Kuo, W., & Zuo, M. J. (2003). Optimal reliability modeling: principles and applications. John Wiley & Sons.
Navarro, J., & Eryilmaz, S. (2007). Mean residual lifetimes of consecutive-k-out-of-n systems. Journal of Applied Probability, 44(1), 82-98.
Papastavridis, S. G. (1989). Lifetime distribution of circular consecutive-k-out-of-n: F systems with exchangeable lifetimes. IEEE Transactions on Reliability, 38(4), 460-461.
Salehi, E. T., Asadi, M., & Eryılmaz, S. (2011). Reliability analysis of consecutive k-out-of-n systems with non-identical components lifetimes. Journal of Statistical Planning and Inference, 141(8), 2920-2932.
Samaniego, F. J. (2007). System signatures and their applications in engineering reliability (Vol. 110). Springer Science & Business Media.
Shanthikumar, J. G. (1985). Lifetime distribution of consecutive-k-out-of-n: F systems with exchangeable lifetimes. IEEE Transactions on Reliability, 34(5), 480-483.
Tahir, M. H., Hussain, M. A., Cordeiro, G. M., El-Morshedy, M., & Eliwa, M. S. (2020). A new Kumaraswamy generalized family of distributions with properties, applications, and bivariate extension. Mathematics, 8(11), 1989.
Triantafyllou, I. S., & Koutras, M. V. (2008). On the signature of coherent systems and applications. Probability in the Engineering and Informational Sciences, 22(1), 19.
Zaidi, S. M., Sobhi, M. M. A., El-Morshedy, M., & Afify, A. Z. (2021). A new generalized family of distributions: Properties and applications. Aims Math, 6, 456-476.
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
