Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-06-06T06:03:52.806Z Has data issue: false hasContentIssue false
  • English
  • Français

Mean Residual Lifetimes of Consecutive-k-out-of-n Systems

Part of: Distribution theory - Probability Special processes

Published online by Cambridge University Press:  14 July 2016

Jorge Navarro  and
Serkan Eryilmaz
Jorge Navarro*
Affiliation:
Universidad de Murcia
Serkan Eryilmaz*
Affiliation:
Izmir University of Economics
*
∗ Postal address: Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain. Email address: jorgenav@um.es
∗∗ Postal address: Department of Mathematics, Izmir University of Economics, 35330 Izmir, Turkey.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we study reliability properties of consecutive-k-out-of-n systems with exchangeable components. For 2k ≦ n, we show that the reliability functions of these systems can be written as negative mixtures (i.e. mixtures with some negative weights) of two series (or parallel) systems. Some monotonicity and asymptotic properties for the mean residual lifetime function are obtained and some ordering properties between these systems are established. We prove that, under some assumptions, the mean residual lifetime function of the consecutive-k-out-of-n: G system (i.e. a system that functions if and only if at least k consecutive components function) is asymptotically equivalent to that of a series system with k components. When the components are independent and identically distributed, we show that consecutive-k-out-of-n systems are ordered in the likelihood ratio order and, hence, in the mean residual lifetime order, for 2k ≦ n. However, we show that this is not necessarily true when the components are dependent.

Keywords

Consecutive-k-out-of-n systemexchangeable distributionsignaturemean residual lifetimestochastic order

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 1 , March 2007 , pp. 82 - 98
Copyright
Copyright © Applied Probability Trust 2007 

References

Aki, S. and Hirano, K. (1996). Lifetime distribution and estimation problems of consecutive k-out-of-n: F systems. Ann. Inst. Statist. Math. 48, 185–199.Google Scholar
Balakrishnan, N. and Koutras, M. V. (2002). Runs and Scans with Applications. John Wiley, New York.Google Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Block, H. W., Li, Y. and Savits, T. H. (2003). Initial and final behaviour of failure rate functions for mixtures and systems. J. Appl. Prob. 40, 721–740.Google Scholar
Boland, P. J. and Papastavridis, S. (1999). Consecutive k-out-of-n: F systems with cycle k. Statist. Prob. Lett. 44, 155–160.Google Scholar
Boland, P. J. and Samaniego, F. J. (2004). Stochastic ordering results for consecutive k-out-of-n: F systems. IEEE Trans. Reliab. 53, 7–10.Google Scholar
Chen, R. W. and Hwang, F. K. (1985). Failure distributions of consecutive k-out-of-n: F systems. IEEE Trans. Reliab. 34, 338–341.CrossRefGoogle Scholar
David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd edn. Wiley, New York.Google Scholar
Derman, C., Lieberman, G. J. and Ross, S. M. (1982). On the consecutive k-out-of-n: F system. IEEE Trans. Reliab. 31, 57–63.Google Scholar
Eryilmaz, S. (2005). On the distribution and expectation of success runs in nonhomogeneous Markov dependent trials. Statist. Papers 46, 117–128.CrossRefGoogle Scholar
Fu, J. C. and Lou, W. Y. W. (2003). Distribution Theory of Runs and Patterns and Its Applications. World Scientific, River Edge, NJ.Google Scholar
Kochar, S., Mukerjee, H. and Samaniego, F. J. (1999). The ‘signature’ of a coherent system and its application to comparisons among systems. Naval Res. Logistics 46, 507–523.Google Scholar
Navarro, J. and Hernandez, P. J. (2005). Mean residual life functions of finite mixtures and systems. Submitted.Google Scholar
Navarro, J. and Rychlik, T. (2006). Reliability and expectation bounds for coherent systems with exchangeable components. J. Multivariate Anal. 98, 102–113.Google Scholar
Navarro, J. and Shaked, M. (2006). Hazard rate ordering of order statistics and systems. J. Appl. Prob. 43, 391–408.Google Scholar
Navarro, J., Belzunce, F. and Ruiz, J. M. (1997). New stochastic orders based on double truncation. Prob. Eng. Inf. Sci. 11, 395–402.Google Scholar
Navarro, J., Ruiz, J. M. and Sandoval, C. J. (2005). A note on comparisons among coherent systems with dependent components using signature. Statist. Prob. Lett. 72, 179–185.Google Scholar
Navarro, J., Ruiz, J. M. and Sandoval, C. J. (2007). Modelling coherent systems under dependence. To appear in Commun. Statist. Theory Meth.Google Scholar
Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. 34, 69–72.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, San Diego, CA.Google Scholar
Shanthikumar, J. G. (1985). Lifetime distribution of consecutive k-out-of-n: F systems with exchangeable life times. IEEE Trans. Reliab. 34, 480–483.Google Scholar
Tong, Y. L. (1985). A rearrangement inequality for the longest run with an application to network reliability. J. Appl. Prob. 22, 386–393.Google Scholar
Wondmagegnehu, E., Navarro, J. and Hernández, P. J. (2005). Bathtub shaped failure rates from mixtures: a practical point of view. IEEE Trans. Reliab. 54, 270–275.Google Scholar