Journal of the Iranian Statistical Society

On Conditional Inactivity Time of Failed Components in an (n-k+1)-out-of-n System with Nonidentical Independent Components

Authors

1 Department of Statistics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, Iran.

2 Department of Statistics, Lorestan University, Khoramabad, Iran.

3 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran.

10.29252/jirss.19.1.69

Abstract

In this paper, we study an (n-k+1)-out-of-n system by adopting their components to be statistically independent though nonidentically distributed. By assuming that at least m components at a fixed time have failed while the system is still working, we obtain the mixture representation of survival function for a quantity called the conditional inactivity time of failed components in the system. Moreover, this quantity for (n-k+1)-out-of-n system, in one sample with respect to k and m and in two samples, are stochastically compared.

Keywords

  1. 1-Eryılmaz, S. (2013), textit{On residual lifetime of coherent systems after the $r^{text{th}}$ failure}. Statistical Papers, textbf{54},243-250. 2-Gurler, S. and Bairamov, I. (2009), textit{Parallel and k-out-of-n: G systems with nonidentical components and their mean residual life functions}. Applied Mathematical Modelling, textbf{33}, 1116-1125. 3- Kochar, S., Mukerjee, H. and Samaniego, F. J. (1999) textit{The “signature” of a coherent system and its application to comparison among systems}. Naval Research Logistics, textbf{46},507-523. 4- Kochar, S. and Xu, M. (1010), textit{On residual lifetimes of k-out-of-n systems with nonidentical components}. Probability in the Engineering and Informational Sciences, textbf{24}, 109-127. 5-Khaledi, B. E. and Shaked, M. (2007), Otextit{rdering conditional lifetimes of coherent systems}. Journal of Statistical Planning and Inference, textbf{137}, 1173-1184. 6- Li, X. and Zhao, P.( 2006), textit{Some aging properties of the residual life of k-out-of-n systems}. IEEE Transactions on Reliability, textbf{55}, 535-541. 7-Li, X. and Zhao, P.( 2008), textit{Stochastic comparison on general inactivity time and general residual life of $k$-out-of-$n$ system}. Communication in Statistics- Simulation and Computation, textbf{37}, 1005-1019. 8-Li, X. and Zhang, Z. (2008), textit{Some stochastic comparisons of conditional coherent systems}. Applied Stochastic Models in Business and Industry, textbf{24}, 541-549. 9- Navarro, J., Ruiz,J. M. and Sandoval, C. J. (2005), textit{A note on comparisons among coherent systems with dependent components using signatures}. Statistics and Probability Letters, textbf{72}, 179-185. 10- Navarro, J., Balakrishnan, N. and Samaniego, F. J. (2008), textit{Mixture representations of residual lifetimes of used systems}. Journal of Applied Probability, textbf{45}, 1097-1112. 11- Navarro, J., Aguila, Y., Sordo, M. A. and Suarez-Llorens, A. (2013), textit{Stochastic ordering properties for systems with dependent identically distributed components}. Applied Stochastic Models in Business and Industry, textbf{29}, 264-278. 12- Navarro, J., Samaniego, F.J. and Balakrishnan, N. (2013), textit{Mixture representations for the joint distribution of lifetimes of two coherent systems with shared components}. Advances in Applied Probability, textbf{45}, 1011-1027. 13-Sadegh, M. K. (2008), textit{Mean past and mean residual Life of parallel system with nonidentical components}. Communications in Statistics Theory and Methods, textbf{37}, 1137-1145. 14- Salehi, E. T. , Asadi, M. and Eryılmaz, S. (2012), textit{On the mean residual lifetime of consecutive $k$-out-of-$n$ systems }. Test, textbf{21}, 93-115. 15-Salehi, E. and Tavangar M. (2019), textit{Stochastic comparisons on conditional residual lifetime and inactivity time of coherent systems with exchangeable components}, Statistics and Probability Letters, textbf{145}, 327-337. 16- Tavangar, M. (2016), textit{Conditional inactivity time of components in a coherent operating system}. IEEE Transactions on Reliability, textbf{65}, 359-369. 17- Zhang, Z. (2010), textit{Ordering conditional general coherent systems with exchangeable components}. Journal of Statistical Planning and Inference. textbf{140}, 454-460. 18-Zhao, P. Li, X. and Balakrishnan, N. (2008), textit{Conditional ordering of $k$-out-of-$n$ systems with independent but nonidentical components}. Journal of Applied Probability, textbf{45}, 1113-1125.
  2. Asadi , M. ( 2006 ), On the mean past lifetime of the components of a parallel system. Journal of Statistical Planning and Inference, 136( 4 ), 1197-1206 . [DOI:10.1016/j.jspi.2004.08.021]
  3. Eryılmaz, S. (2013), On residual lifetime of coherent systems after the r failure. Statistical Papers, 54, 243-250. [DOI:10.1007/s00362-011-0422-1]
  4. Gurler, S. and Bairamov, I. (2009), Parallel and k-out-of-n: G systems with nonidentical components and their mean residual life functions. Applied Mathematical Modelling, 33, 1116-1125. [DOI:10.1016/j.apm.2008.01.003]
  5. Kochar, S., Mukerjee, H. and Samaniego, F. J. (1999), The "signature" of a coherent system and its application to comparison among systems. Naval Research Logistics, 46, 507-523. https://doi.org/10.1002/(SICI)1520-6750(199908)46:5<507::AID-NAV4>3.0.CO;2-D [DOI:10.1002/(SICI)1520-6750(199908)46:53.0.CO;2-D]
  6. Kochar, S. and Xu, M. (2010), On residual lifetimes of k-out-of-n systems with nonidentical components. Probability in the Engineering and Informational Sciences, 24, 109-127. [DOI:10.1017/S0269964809990167]
  7. Khaledi, B. E. and Shaked, M. (2007), Ordering conditional lifetimes of coherent systems. Journal of Statistical Planning and Inference, 137, 1173-1184. [DOI:10.1016/j.jspi.2006.01.012]
  8. Li, X. and Zhao, P. ( 2006), Some aging properties of the residual life of k-out-of-n systems. IEEE Transactions on Reliability, 55, 535-541. [DOI:10.1109/TR.2006.879652]
  9. Li, X. and Zhao, P. ( 2008), Stochastic comparison on general inactivity time and general residual life of k-out-of-n system. Communication in Statistics- Simulation and Computation, 37, 1005-1019. [DOI:10.1080/03610910801943784]
  10. Li, X. and Zhang, Z. (2008), Some stochastic comparisons of conditional coherent systems. Applied Stochastic Models in Business and Industry, 24, 541-549. [DOI:10.1002/asmb.715]
  11. Navarro, J., Ruiz,J. M. and Sandoval, C. J. (2005), A note on comparisons among coherent systems with dependent components using signatures. Statistics and Probability Letters, 72, 179-185. [DOI:10.1016/j.spl.2004.12.017]
  12. Navarro, J., Aguila, Y., Sordo, M. A. and Suarez-Llorens, A. (2013), Stochastic ordering properties for systems with dependent identically distributed components. Applied Stochastic Models in Business and Industry, 29, 264-278. [DOI:10.1002/asmb.1917]
  13. Navarro, J., Samaniego, F.J. and Balakrishnan, N. (2013a), Mixture representations for the joint distribution of lifetimes of two coherent systems with shared components. Advances in Applied Probability, 45, 1011-1027. [DOI:10.1239/aap/1386857855]
  14. Sadegh, M. K. (2008), Mean past and mean residual Life of parallel system with nonidentical components. Communications in Statistics Theory and Methods, 37, 1137-1145. [DOI:10.1080/03610920701713278]
  15. Salehi, E. T. , Asadi, M. and Eryılmaz, S. (2012), On the mean residual lifetime of consecutive k-out-of-n systems. Test, 21, 93-115. [DOI:10.1007/s11749-011-0237-3]
  16. Salehi, E. and Tavangar M. (2019), Stochastic comparisons on conditional residual lifetime and inactivity time of coherent systems with exchangeable components, Statistics and Probability Letters, 145, 327-337. [DOI:10.1016/j.spl.2018.10.007]
  17. Tavangar, M. (2016), Conditional inactivity time of components in a coherent operating system. IEEE Transactions on Reliability, 65, 359-369. [DOI:10.1109/TR.2015.2422773]
  18. Tavangar, M. and Asadi, M. (2010) , A study on the mean past lifetime of the components of (n - k + 1)-out-of-n system at the system level. Metrika, 72(1), 59-73. [DOI:10.1007/s00184-009-0241-8]
  19. Zhang, Z. (2010), Ordering conditional general coherent systems with exchangeable components. Journal of Statistical Planning and Inference. 140, 454-460. [DOI:10.1016/j.jspi.2009.07.029]
  20. Zhao, P. Li, X. and Balakrishnan, N. (2008), Conditional ordering of $k$-out-of-$n$ systems with independent but nonidentical components. Journal of Applied Probability, 45, 1113-1125. [DOI:10.1239/jap/1231340237]
Journal of the Iranian Statistical Society
Volume 19, Issue 1
June 2020
Pages 69-83
  • Receive Date: 23 July 2022
  • Revise Date: 20 May 2024
  • Accept Date: 23 July 2022