×

zbMATH — the first resource for mathematics

Stochastic bounds and dependence properties of survival times in a multicomponent shock model. (English) Zbl 0982.60008
From the authors’ summary: Consider a system that consists of several components. Shocks arrive according to a counting process (which may be nonhomogeneous and with correlated interarrival times) and each shock may simultaneously destroy a subset of the components. Shock models of this type arise naturally in reliability modeling in dynamic environments. The purpose of this paper is to provide a general framework for studying the correlation structure of shock models in the setup of a multivariate, correlated counting process and to systematically develop upper and lower bounds for its joint component lifetime distribution and survival functions. The thrust of the approach is the interplay between a newly developed notion, majorization with respect to weighted trees, and various stochastic orders, especially orthant dependence orders of random vectors and orthant dependence orders of stochastic processes.

MSC:
60E15 Inequalities; stochastic orderings
60K10 Applications of renewal theory (reliability, demand theory, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Baccelli, F.; Makowski, A.M., Multidimensional stochastic ordering and associated random variables, Oper. res., 37, 478-487, (1989) · Zbl 0671.60013
[2] Barlow, R.E.; Proschan, F., Statistical theory of reliability and life testing, (1981), To Begin With Silver Spring
[3] Block, H.W.; Savits, T.H.; Shaked, M., Some concepts of negative dependence, Ann. probab., 10, 765-772, (1982) · Zbl 0501.62037
[4] Chang, C.S.; Yao, D., Rearrangement, majorization and stochastic scheduling, Math. oper. res., (1994)
[5] Esary, J.D.; Proschan, F., A reliability bound for systems of maintained, independent components, J. amer. statist. assoc., 65, 329-338, (1970) · Zbl 0202.17001
[6] Joag-Dev, K.; Proschan, F., Negative association of random variables with applications, Ann. statist., 11, 286-295, (1983) · Zbl 0508.62041
[7] M. Kijima, H. Li, and, M. Shaked, Stochastic processes in reliability, preprint, 1998. · Zbl 0980.60108
[8] Li, H.; Zhu, H., Stochastic equivalence of ordered random variables with applications in reliability theory, Statist. probab. lett., 20, 383-393, (1994) · Zbl 0804.62015
[9] H. Li, and, S. H. Xu, On the dependent structure and bounds of correlated parallel queues and their applications to synchronized stochastic systems, preprint, 1999.
[10] Lindqvist, H., Association of probability measures on partially ordered spaces, J. multivariate anal., 26, 111-131, (1988) · Zbl 0655.60005
[11] Marshall, A.W.; Olkin, I., A multivariate exponential distribution, J. amer. statist. assoc., 2, 84-98, (1967) · Zbl 0147.38106
[12] Marshall, A.W.; Olkin, I., Inequalities: theory of majorization and its applications, (1979), Academic Press New York · Zbl 0437.26007
[13] Marshall, A.W.; Shaked, M., Multivariate shock models for distributions with increasing hazard rate average, Ann. probab., 7, 343-358, (1979) · Zbl 0394.62033
[14] Olkin, I.; Tong, Y.L., Positive dependence of a class of multivariate exponential distributions, SIAM J. control. optim., 32, 965-974, (1994) · Zbl 0796.62085
[15] Savits, T.H.; Shaked, M., Shock models and the MIFRA property, Stochast. process. appl., 11, 273-283, (1981) · Zbl 0472.62058
[16] Shaked, M.; Shanthikumar, J.G., Multivariate imperfect repair, Oper. res., 34, 435-448, (1986) · Zbl 0616.62129
[17] Shaked, M.; Shanthikumar, J.G., Stochastic orders and their applications, (1994), Academic Press New York · Zbl 0806.62009
[18] Shaked, M.; Shanthikumar, J.G., Supermodular stochastic order and positive dependence of random vectors, J. multivariate anal., 61, 86-101, (1997) · Zbl 0883.60016
[19] Shaked, M.; Szekli, R., Comparison of replacement policies via point processes, Adv. appl. probab., 27, 1079-1103, (1995) · Zbl 0849.60081
[20] Singpurwalla, N., Survival in dynamic environments, Statist. sci., 10, 86-103, (1995) · Zbl 1148.62314
[21] Stoyan, D., Comparison methods for queues and other stochastic models, (1983), Wiley New York
[22] Szekli, R., Stochastic ordering and dependence in applied probability, (1995), Springer-Verlag New York · Zbl 0815.60017
[23] Tchen, A.H., Inequalities for distributions with given marginals, Ann. prob., 8, 814-827, (1980) · Zbl 0459.62010
[24] Tong, Y.L., Probability inequalities in multivariate distributions, (1980), Academic Press New York · Zbl 0455.60003
[25] S. H. Xu, and, H. Li, Majorization of weighted trees: A new tool to study correlated stochastic systems, preprint, 1998.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.