On bivariate signatures for systems with independent modules. (English) Zbl 1317.90094
Li, Haijun (ed.) et al., Stochastic orders in reliability and risk. In honor of Professor Moshe Shaked. Based on the talks presented at the international workshop on stochastic orders in reliability and risk management, SORR2011, Xiamen, China, June 27–29, 2011. New York, NY: Springer (ISBN 978-1-4614-6891-2/hbk; 978-1-4614-6892-9/ebook). Lecture Notes in Statistics 208. Proceedings, 143-166 (2013).
Summary: I. Gertsbakh et al. [J. Appl. Probab. 49, No. 2, 416–429 (2012; Zbl 1242.90070)] proposed the concept of bivariate signature for a 3-state system. In this paper, we first give an equivalent definition for the bivariate signature in the sense of order statistics of component lifetimes and establish the formula for computing the bivariate signature of the dual of a 3-state system. A useful method for computing the bivariate signature based on the combinatorial meaning of the bivariate tail signature is presented. By this method, we derive formulas for computing the bivariate signatures of some systems consisting of independent modules. Some examples are also presented to illustrate our main results.
For the entire collection see [Zbl 1268.60007].
For the entire collection see [Zbl 1268.60007].
MSC:
| 90B25 | Reliability, availability, maintenance, inspection in operations research |
| 62H99 | Multivariate analysis |
| 62N05 | Reliability and life testing |
Citations:
Zbl 1242.90070References:
| [1] | Barlow, R. E.; Proschan, F., Statistical Theory of Reliability and Life Testing (1981), Silver Spring, Maryland: To Begin With, Silver Spring, Maryland |
| [2] | Boland, P. J., Signatures of indirect majority systems, Journal of Applied Probability, 38, 597-603 (2001) · Zbl 1042.62090 · doi:10.1239/jap/996986765 |
| [3] | Boland, P. J.; El-Newehi, E., Component redundancy vs. system redundancy in the hazard rate ordering, IEEE Transactions on Reliability, 44, 614-619 (1995) · doi:10.1109/24.475980 |
| [4] | Da, G., Xia, L. and Hu, T.: On computing signatures of k-out-of-n systems consisting of modules. Methodology and Computing in Applied Probability, DOI:10.1007/s11009-012-9308-5 (2012) · Zbl 1291.60180 |
| [5] | Da, G.; Zheng, B.; Hu, T., On computing signatures of coherent systems, Journal of Multivariate Analysis, 103, 142-150 (2012) · Zbl 1262.60085 · doi:10.1016/j.jmva.2011.06.015 |
| [6] | EI-Newehi, E.; Proschan, F.; Sethuraman, J., Multi-state coherent system, Journal of Applied Probability, 15, 675-688 (1978) · Zbl 0386.60061 |
| [7] | Eryilmaz, S.; Zuo, M., Computing and applying the signature of a system with two common failure critera, IEEE Transaction on Reliability, 59, 576-580 (2010) · doi:10.1109/TR.2010.2054170 |
| [8] | Gertsbakh, I. and Shpungin, Y.: Multidimensional Spectra of Multistate Binary Systems. Chapter 4 in Recent Advances in System Reliability, Signatures, Multistate Systems and Statistical Inference. Lisnianski A. and Frenkel I. (Eds). Springer Series in Reliability Engineering. Springer (2011) · Zbl 1230.60096 |
| [9] | Gertsbakh, I.; Shpungin, Y., Network reliabilty and Resilience (2011), Springer Briefs in Electrical and Computer Engineering: Springer, Springer Briefs in Electrical and Computer Engineering · doi:10.1007/978-3-642-22374-7 |
| [10] | Gertsbakh, I.; Shpungin, Y.; Spizzichino, F., On the signature of coherent systems obtained by connecting independent modules, Journal of Applied Probability, 48, 843-855 (2011) · Zbl 1230.60096 · doi:10.1239/jap/1316796919 |
| [11] | Gertsbakh, I.; Shpungin, Y.; Spizzichino, F., On two-dimensional signatures, Journal of Applied Probability, 49, 416-429 (2012) · Zbl 1242.90070 · doi:10.1239/jap/1339878795 |
| [12] | Kochar, S. C.; Mukerjee, H.; Samaniego, F. J., The “signature” of a coherent system and its application to comparison among systems, Naval Research Logistics, 46, 507-523 (1999) · Zbl 0948.90067 · doi:10.1002/(SICI)1520-6750(199908)46:5<507::AID-NAV4>3.0.CO;2-D |
| [13] | Levitin, G.; Gertsbakh, I.; Shpungin, Y., Evaluating the damage associate with intentional network disintegration, Reliability Engineering and System Safety, 96, 433-439 (2011) · doi:10.1016/j.ress.2010.12.022 |
| [14] | Lisnianski, A.; Levitin, G., Multistate system reliability, 6 in Quality (2003), Reliability and Engineering Statistics: World Scientific, Reliability and Engineering Statistics · Zbl 1030.90017 |
| [15] | Marichal, J-L.; Mathonet, P., Extensions of system signatures to dependent lifetimes: Explicit expressions and interpretations, Journal of Multivariate Analysis, 102, 931-936 (2011) · Zbl 1215.62108 · doi:10.1016/j.jmva.2011.01.010 |
| [16] | Marichal, J-L; Mathonet, P.; Waldhauser, T., On signature-based expressions of system reliability, Journal of Multivariate Analysis, 102, 1410-1416 (2011) · Zbl 1219.62154 · doi:10.1016/j.jmva.2011.05.007 |
| [17] | Navarro, J.; Eryilmaz, S., Mean residual lifetimes of consecutive k-out-of-n systems, Journal of Applied Probability, 44, 82-98 (2007) · Zbl 1135.62084 · doi:10.1239/jap/1175267165 |
| [18] | Navarro, J.; Ruiz, J. M.; Sandoval, C. J., A note on comparisons among coherent systems with dependent components using signatures, Statistics and Probability Letters, 72, 179-185 (2005) · Zbl 1068.60026 · doi:10.1016/j.spl.2004.12.017 |
| [19] | Navarro, J.; Ruiz, J. M.; Sandoval, C. J., Properties of coherent systems with dependent components, Communications in Statistics — Theory and Methods, 36, 175-191 (2007) · Zbl 1121.60015 |
| [20] | Navarro, J.; Rychlik, T., Reliability and expectation bounds for coherent systems with exchangeable components, Journal of Multivariate Analysis, 98, 102-113 (2007) · Zbl 1102.62111 · doi:10.1016/j.jmva.2005.09.003 |
| [21] | Navarro, J.; Rychlik, T., Comparisons and bounds for expected lifetimes of reliability systems, European Journal of Operational Research, 207, 309-317 (2010) · Zbl 1217.62155 · doi:10.1016/j.ejor.2010.05.001 |
| [22] | Navarro, J.; Samaniego, F. J.; Balakrishnan, N., The joint signature of coherent systems with shared components, Journal of Applied Probability, 47, 235-253 (2010) · Zbl 1187.60012 · doi:10.1239/jap/1269610828 |
| [23] | Navarro, J.; Samaniego, F. J.; Balakrishnan, N., Signature-based representations for the reliability of systems with heterogeneous components, Journal of Applied Probability, 48, 856-867 (2011) · Zbl 1250.62052 · doi:10.1239/jap/1316796920 |
| [24] | Navarro, J.; Samaniego, F. J.; Balakrishnan, N.; Bhattacharya, D., On the application and extension of system signatures in engineering reliability, Naval Research Logistics, 55, 313-327 (2008) · Zbl 1153.90386 · doi:10.1002/nav.20285 |
| [25] | Samaniego, F. J., On closure of the IFR class under formation of coherent systems, IEEE Transactions on Reliability, 34, 69-72 (1985) · Zbl 0585.62169 · doi:10.1109/TR.1985.5221935 |
| [26] | Samaniego, F. J., System Signatures and Their Applications in Engineering Reliability (2007), New York: Springer, New York · Zbl 1154.62075 · doi:10.1007/978-0-387-71797-5 |
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.
