Reliability; Multistate system; Dependent lifetimes; Semicoherent system; System joint signature
Abstract :
[en] The structure signature of a system made up of n components having continuous and i.i.d. lifetimes was defined in the eighties by Samaniego as the n-tuple whose k-th coordinate is the probability that the k-th component failure causes the system to fail. More recently, a bivariate version of this concept was considered as follows. The joint structure signature of a pair of systems built on a common set of components having continuous and i.i.d. lifetimes is a square matrix of order n whose (k,l)-entry is the probability that the k-th failure causes the first system to fail and the l-th failure causes the second system to fail. This concept was successfully used to derive a signature-based decomposition of the joint reliability of the two systems. In the first part of this paper we provide an explicit formula to compute the joint structure signature of two or more systems and extend this formula to the general non-i.i.d. case, assuming only that the distribution of the component lifetimes has no ties. We also provide and discuss a necessary and sufficient condition on this distribution for the joint reliability of the systems to have a signature-based decomposition. In the second part of this paper we show how our results can be efficiently applied to the investigation of the reliability and signature of multistate systems made up of two-state components. The key observation is that the structure function of such a multistate system can always be additively decomposed into a sum of classical structure functions. Considering a multistate system then reduces to considering simultaneously several two-state systems.
Disciplines :
Mathematics
Author, co-author :
Marichal, J.-L.; Mathematics Research Unit, University of Luxembourg, Maison du Nombre, 6, avenue de la Fonte, Esch-sur-Alzette, Luxembourg
Mathonet, Pierre ; Université de Liège - ULiège > Département de mathématique > Géométrie différentielle
Navarro, J.; Facultad de Matemáticas, Universidad de Murcia, Murcia, Spain
Paroissin, C.; Laboratoire de Mathématiques et de leurs Applications de Pau – Fédération IPRA, CNRS / Univ Pau & Pays Adour, UMR 5142, Pau, France
Language :
English
Title :
Joint signature of two or more systems with applications to multistate systems made up of two-state components
Barlow, R.E., Wu, A., Coherent systems with multi-state components. Mathematics of Operations Research 3:4 (1978), 275–281.
Block, H.W., Savits, T.H., A decomposition for multistate monotone systems. Journal of Applied Probability 19 (1982), 391–402.
Boland, P.J., Signatures of indirect majority systems. Journal of Applied Probability 38 (2001), 597–603.
Da, G., Hu, T., On bivariate signatures for systems with independent modules. Stochastic orders in reliability and risk, 2013, Springer, London, 143–166 Chapter 7.
Dukhovny, A., Marichal, J.-L., System reliability and weighted lattice polynomials. Probability in the Engineering and Informational Sciences 22:3 (2008), 373–388.
El-Neweihi, E., Proschan, F., Sethuraman, J., Multistate coherent system. Journal of Applied Probability 15 (1978), 675–688.
Eryilmaz, S., Lifetime of multistate k-out-of-n systems. Quality and Reliability Engineering International 30 (2014), 1015–1022.
Eryilmaz, S., Mixture representations for three-state systems with three-state components. IEEE Transactions on Reliability Theory 64:2 (2015), 829–834.
Gertsbakh, I., Shpungin, Y., Multidimensional spectra of multistate systems with binary components. Recent advances in reliability: signatures, multi-state systems and statistical inference, 2011, Springer, London., 49–61 Chapter 4.
Gertsbakh, I., Shpungin, Y., Spizzichino, F., Signatures of coherent systems built with separate modules. Journal of Applied Probability 48:3 (2011), 843–855.
Gertsbakh, I., Shpungin, Y., Spizzichino, F., Two-dimensional signatures. Journal of Applied Probability 49 (2012), 416–429.
Hatoyama, Y., Reliability analysis of 3-state systems. IEEE Transactions on Reliability Theory R-28:5 (1979), 386–393.
Janan, X., On multistate system analysis. IEEE Transactions on Reliability Theory R-34:4 (1985), 329–337.
Levitin, G., Multi-state vector-k-out-of-n systems. IEEE Transactions on Reliability Theory 62:3 (2013), 648–657.
Levitin, G., Gertsbakh, I., Shpungin, Y., Evaluating the damage associate with intentional network disintegration. Reliability Engineering and System Safety 96 (2011), 433–439.
Li, J.-A., Wu, Y., Lai, K.K., Liu, K., Reliability estimation and prediction of multi-state components and coherent systems. Reliability Engineering and System Safety 88 (2005), 93–98.
Marichal, J.-L., Mathonet, P., Extensions of system signatures to dependent lifetimes: Explicit expressions and interpretations. Journal of Multivariate Analysis 102:5 (2011), 931–936.
Marichal, J.-L., Mathonet, P., Spizzichino, F., On modular decompositions of system signatures. Journal of Multivariate Analysis 134 (2015), 19–32.
Marichal, J.-L., Mathonet, P., Waldhauser, T., On signature-based expressions of system reliability. Journal of Multivariate Analysis 102:10 (2011), 1410–1416.
Natvig, B., Two suggestions of how to define a multistate coherent system. Advances in Applied Probability 14 (1982), 434–455.
Natvig, B., Multistate systems reliability theory with applications. 2011, Wiley, Chichester, UK.
Navarro, J., Rychlik, T., Reliability and expectation bounds for coherent systems with exchangeable components. Journal of Multivariate Analysis 98:1 (2007), 102–113.
Navarro, J., Samaniego, F.J., Balakrishnan, N., The joint signature of coherent systems with shared components. Journal of Applied Probability 47 (2010), 235–253.
Navarro, J., Samaniego, F.J., Balakrishnan, N., Mixture representations for the joint distribution of lifetimes of two coherent systems with shared components. Advances in Applied Probability 45:4 (2013), 1011–1027.
Navarro, J., Samaniego, F.J., Balakrishnan, N., Bhattacharya, D., On the application and extension of system signatures in engineering reliability. Naval Research Logistics 55 (2008), 313–327.
Navarro, J., Spizzichino, F., Balakrishnan, N., Applications of average and projected systems to the study of coherent systems. Journal of Multivariate Analysis 101:6 (2010), 1471–1482.
Samaniego, F.J., On closure of the IFR class under formation of coherent systems. IEEE Transactions on Reliability Theory 34 (1985), 69–72.
Samaniego, F.J., System signatures and their applications in engineering reliability. International series in operations research & management science, 110, 2007, Springer, New York.
Spizzichino, F. (2008). The role of signature and symmetrization for systems with non-exchangeable components. In Advances in mathematical modeling for reliability, (pp. 138–148). Amsterdam: IOS.
