Reliability function of \(k\)-out-of-\(n\) system equipped with two cold standby components. (English) Zbl 07532226
Summary: In this article, we focus on reliability properties of a k-out-of-n system equipped with two cold standby components. We describe the lifetime of the system and compute the reliability function of the system, which will allow us to investigate the risk of system failure. Further, three different mean residual life functions of the system are derived. We also provide some stochastic ordering results. The theoretical results established in this article addresses a problem proposed by Eryilmaz and Wang. We have presented a few numerical results to illustrate the importance of our analysis.
MSC:
| 62-XX | Statistics |
Keywords:
applied probability; coherent system; cold standby component; mean residual life; reliabilityReferences:
| [1] | Asadi, M.; Bayramoglu, I., A note on the mean residual life function of a parallel system, Communications in Statistics: Theory and Methods, 34, 2, 475-84 (2005) · Zbl 1062.62228 · doi:10.1081/STA-200047430 |
| [2] | Asadi, M.; Bayramoglu, I., The mean residual life function of a k-out-of-n structure at the system level, IEEE Transactions on Reliability, 55, 2, 314-8 (2006) · doi:10.1109/TR.2006.874934 |
| [3] | Bairamov, I.; Ahsanullah, M.; Akhundov, I., A residual life function of a system having parallel or series structures, Journal of Statistical Theory and Applications, 1, 2, 119-32 (2002) |
| [4] | Bairamov, I.; Arnold, B. C., On the residual lifelengths of the remaining components in an n- k+ 1- out of- n system, Statistics & Probability Letters, 78, 8, 945-52 (2008) · Zbl 1141.62081 · doi:10.1016/j.spl.2007.09.054 |
| [5] | Barlow, R. E.; Proschan, F., Statistical theory of reliability and life testing (1975), New York, NY, USA: Holt, Rinehart, & Winston, New York, NY, USA · Zbl 0379.62080 |
| [6] | Chen, J.; Zhang, Y.; Zhao, P.; Zhou, S., Allocation strategies of standby redundancies in series/parallel system, Communications in Statistics—Theory and Methods, 47, 3, 708-24 (2018) · Zbl 1394.90215 · doi:10.1080/03610926.2017.1313984 |
| [7] | David, H. A.; Nagaraja, H. N., Order statistics (2003), Hoboken, New Jersey: Wiley, Hoboken, New Jersey · Zbl 1053.62060 |
| [8] | Ding, W.; Zhao, P.; Zhou, S., On optimal allocation of active redundancies to multi-state r-out-of-n systems, Operations Research Letters, 45, 5, 508-12 (2017) · Zbl 1409.90076 · doi:10.1016/j.orl.2017.08.008 |
| [9] | Eryilmaz, S., Dynamic behavior of k-out-of-n: G systems, Operations Research Letters, 39, 2, 155-9 (2011) · Zbl 1219.93066 |
| [10] | Eryilmaz, S., On the mean residual life of a k-out-of-n: G system with a single cold standby component, European Journal of Operational Research, 222, 2, 273-7 (2012) · Zbl 1253.90089 · doi:10.1016/j.ejor.2012.05.012 |
| [11] | Eryilmaz, S., Reliability of a k-out-of-n system equipped with a single warm standby component, IEEE Transactions on Reliability, 62, 2, 499-503 (2013) · doi:10.1109/TR.2013.2259202 |
| [12] | Eryilmaz, S., A study on reliability of coherent systems equipped with a cold standby component, Metrika, 77, 3, 349-59 (2014) · Zbl 1297.90024 · doi:10.1007/s00184-013-0441-0 |
| [13] | Eryilmaz, S., The effectiveness of adding cold standby redundancy to a coherent system at system and component levels, Reliability Engineering & System Safety, 165, 331-5 (2017) · doi:10.1016/j.ress.2017.04.021 |
| [14] | Eryilmaz, S.; Erkan, T. E., Coherent system with standby components, Applied Stochastic Models in Business and Industry, 34, 3, 395-406 (2018) · Zbl 1410.90065 · doi:10.1002/asmb.2307 |
| [15] | Eryilmaz, S.; Tank, F., On reliability analysis of a two-dependent-unit series system with a standby unit, Journal of Applied Mathematics and Computing, 218, 7792-7 (2012) · Zbl 1238.62122 · doi:10.1016/j.amc.2012.01.046 |
| [16] | Esary, J. D.; Marshall, A. W., Coherent life functions, SIAM Journal on Applied Mathematics, 18, 4, 810-4 (1970) · Zbl 0198.24804 · doi:10.1137/0118073 |
| [17] | Franko, C.; Ozkut, M.; Kan, C., Reliability of coherent systems with a single cold standby component, The Journal of Computational and Applied Mathematics, 281, 230-8 (2015) · Zbl 1305.90136 · doi:10.1016/j.cam.2014.12.029 |
| [18] | Franko, C.; Tütüncü, G. Y.; Eryilmaz, S., Reliability of weighted k-out-of-n: G systems consisting of two types of components and a cold standby component, Communications in Statistics—Simulation and Computation, 46, 5, 4067-81 (2017) · Zbl 1368.90054 · doi:10.1080/03610918.2015.1096377 |
| [19] | Khaledi, B.-E.; Shaked, M., Ordering conditional lifetimes of coherent systems, Journal of Statistical Planning and Inference, 137, 4, 1173-84 (2007) · Zbl 1111.60012 · doi:10.1016/j.jspi.2006.01.012 |
| [20] | Kochar, S.; Xu, M., On residual lifetimes of k-out-of-n systems with nonidentical components, Probability in the Engineering and Informational Sciences, 24, 1, 109-27 (2010) · Zbl 1190.90060 · doi:10.1017/S0269964809990167 |
| [21] | Kundu, P.; Hazra, N. K.; Nanda, A. K., Reliability study of a coherent system with single general standby component, Statistics & Probability Letters, 110, 25-33 (2016) · Zbl 1337.60225 · doi:10.1016/j.spl.2015.11.023 |
| [22] | Leung, K. N. F.; Zhang, Y. L.; Lai, K. K., Analysis for a two-dissimilar-component cold standby repairable system with repair priority, Reliability Engineering & System Safety, 96, 11, 1542-51 (2011) · doi:10.1016/j.ress.2011.06.004 |
| [23] | Levitin, G.; Xing, L.; Dai, Y., Cold vs. hot standby mission operation cost minimization for 1-out-of-N systems, European Journal of Operational Research, 234, 1, 155-62 (2014) · Zbl 1305.90141 · doi:10.1016/j.ejor.2013.10.051 |
| [24] | Li, X.; Zhao, P., Some aging properties of the residual life of k-out-of-n systems, IEEE Transactions on Reliability, 55, 3, 535-41 (2006) · doi:10.1109/TR.2006.879652 |
| [25] | Mirjalili, A.; Sadegh, M. K.; Rezaei, M., A note on the mean residual life of a coherent system with a cold standby component, Communications in Statistics—Theory and Methods, 46, 20, 10348-58 (2017) · Zbl 1380.62241 · doi:10.1080/03610926.2016.1235194 |
| [26] | Raqab, M. Z.; Rychlik, T., Bounds for the mean residual life function of a k-out-of-n system, Metrika, 74, 3, 361-80 (2011) · Zbl 1226.62100 · doi:10.1007/s00184-010-0307-7 |
| [27] | Shaked, M.; Shanthikumar, J. G., Stochastic orders (2007), New York, NY: Springer, New York, NY · Zbl 1111.62016 |
| [28] | Shen, Z.; Dai, X.; Huang, X., A supplemental algorithm for the repairable system in the GO methodology, Reliability Engineering & System Safety, 91, 8, 940-4 (2006) · doi:10.1016/j.ress.2005.09.008 |
| [29] | Van Gemund, A. J. C.; Reijns, G. L., Reliability analysis of k-out-of-n systems with single cold standby using Pearson distributions, IEEE Transactions on Reliability, 61, 2, 526-32 (2012) · doi:10.1109/TR.2012.2192587 |
| [30] | Wang, Y., Conditional k-out-of-n systems with a cold standby component, Communications in Statistics—Theory and Methods, 45, 21, 6253-62 (2016) · Zbl 1378.62123 · doi:10.1080/03610926.2014.958354 |
| [31] | Yi, X. J.; Dhillon, B. S.; Shi, J.; Mu, H. N.; Zhang, Z., A new reliability analysis method for vehicle systems based on goal-oriented methodology, Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 231, 8, 1066-95 (2017) · doi:10.1177/0954407016671276 |
| [32] | Zhang, Y. J.; Sun, Y. C.; Zhong, L., Copula function-based reliability analysis of a series system with a single cold standby unit, Acta Aeronautica Et Astronautica Sinica, 35, 8, 2207-16 (2014) |
| [33] | Zhang, Y. J.; Sun, Y. C.; Li, L.; Zhao, M., Copula-based reliability analysis for a parallel system with a cold standby, Communications in Statistics-Theory and Methods, 47, 3, 562-82 (2018) · Zbl 1388.90047 |
| [34] | Zhao, P.; Li, X.; Balakrishnan, N., Conditional ordering of k-out-of-n systems with independent but nonidentical components, Journal of Applied Probability, 45, 4, 1113-25 (2008) · Zbl 1155.60309 · doi:10.1017/S0021900200005015 |
| [35] | Zhao, P.; Chan, P.; Li, L.; Ng, H. K. T., Optimal allocation of redundancies in series systems, European Journal of Operational Research, 220, 3, 673-83 (2012) · Zbl 1253.90099 · doi:10.1016/j.ejor.2012.02.024 |
| [36] | Zhao, P.; Chan, P.; Li, L.; Ng, H. K. T., Allocation of two redundancies in two-component series systems, Naval Research Logistics (NRL), 60, 7, 588-98 (2013) · Zbl 1410.90077 · doi:10.1002/nav.21554 |
| [37] | Zhao, P.; Chan, P.; Li, L.; Ng, H. K. T., On allocation of redundancies in two-component series systems, Operations Research Letters, 41, 6, 690-3 (2013) · Zbl 1287.90016 · doi:10.1016/j.orl.2013.09.006 |
| [38] | Zhao, P.; Zhang, Y.; Chen, J., Optimal allocation policy of one redundancy in a n-component series system, European Journal of Operational Research, 257, 2, 656-68 (2017) · Zbl 1394.90245 · doi:10.1016/j.ejor.2016.07.055 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
