Lifetime Data Analysis

, Volume 10, Issue 1, pp 83–94 | Cite as

Reliability Estimation Based on System Data with an Unknown Load Share Rule

  • Hyoungtae Kim
  • Paul H. Kvam


We consider a multicomponent load-sharing system in which the failure rate of a given component depends on the set of working components at any given time. Such systems can arise in software reliability models and in multivariate failure-time models in biostatistics, for example. A load-share rule dictates how stress or load is redistributed to the surviving components after a component fails within the system. In this paper, we assume the load share rule is unknown and derive methods for statistical inference on load-share parameters based on maximum likelihood. Components with (individual) constant failure rates are observed in two environments: (1) the system load is distributed evenly among the working components, and (2) we assume only the load for each working component increases when other components in the system fail. Tests for these special load-share models are investigated.

maximum likelihood software reliability order restricted inference system dependence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Z. Birnbaum and S. Saunders, “A statistical model for life-length of materials,” J. Am. Stat. Assoc. vol. 53 pp. 151–160, 1958.Google Scholar
  2. B. Coleman, “Time dependence of mechanical breakdown in bundles of fibers, i. Constant total load,” J. Appl. Phys. vol. 28 pp. 1058–1064, 1957a.Google Scholar
  3. B. Coleman, “Time dependence of mechanical breakdown in bundles of fibers, ii. The infinite ideal bundle under linearly increasing loads,” J. Appl. Phys. vol. 28 pp. 1065–1067, 1957b.Google Scholar
  4. D. R. Cox, Prediction Intervals and Empirical Bayes Confidence Intervals, Academic Press: Cambridge, UK, 1975.Google Scholar
  5. H. E. Daniels, “The statistical theory of the strength of bundles of threads,” Proc. R. Soc. London Ser. A vol. 183 pp. 405–435, 1945.Google Scholar
  6. B. Glanz and E. Lipton, “The height of ambition: In the epic story of how the world trade towers rose, their fall was foretold,” NY Times Mag. vol. 8, (September) pp. 32–63, 2002.Google Scholar
  7. D. G. Harlow and S. L. Phoenix, “The chain-of-bundles probability model for the strength of fibrous materials 1: Analysis and conjectures,” J. Compos. Mater. vol. 12 pp. 195–214, 1978.Google Scholar
  8. D. G. Harlow and S. L. Phoenix, “Probability distributions for the strength of fibrous materials under local load sharing 1: Two-level failure and edge effects,” Adv. Appl. Prob. vol. 14 pp. 68–94, 1982.Google Scholar
  9. M. Hollander and E. A. Pena, “Dynamic reliability models with conditional proportional hazards,” Lifetime Data Anal. vol. 1 pp. 377–401, 1995.Google Scholar
  10. S. Lee, S. Durham, and J. Lynch, “On the calculation of the reliability of general load sharing systems,” J. Appl. Prob. vol. 32 pp. 777–792, 1995.Google Scholar
  11. H. Liu, “Reliability of a load-sharing k-out-of-n: G system: Non-iid components with arbitrary distributions,” IEEE Trans. Reliab. vol. 47(3) pp. 279–284, 1998.Google Scholar
  12. W. Q. Meeker and L. A. Escobar, Statistical Methods for Reliability Data, John Wiley: New York, 1998.Google Scholar
  13. J. M. Ortega and C. R. Rheinbold, Iterative Solution of Nonlinear Equation in Several Variables, Academic Press: New York, 1970.Google Scholar
  14. S. Phoenix, “The asymptotic time to failure of a mechanical system of parallel members,” SIAM J. Appl. Math. vol. 34 pp. 227–246, 1978.Google Scholar
  15. B. W. Rosen, “Tensile failure of fibrous composites,” AIAA J. vol. 2 pp. 1985–1991, 1964.Google Scholar
  16. S. M. Ross, “A model in which component failure rates depend on the working set,” Naval Res. Logist. Q. vol. 31 pp. 297–300, 1984.Google Scholar
  17. Z. Schechner, “A load-sharing model: The linear breakdown rule,” Naval Res. Logist. Q. vol. 31 pp. 137–144, 1984.Google Scholar
  18. F. T. Wright, T. Robertson, and R. L. Dykstra, Order Restricted Statistical Inference, Wiley: New York, USA, 1988.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • Hyoungtae Kim
    • 1
  • Paul H. Kvam
    • 1
  1. 1.School of Industrial and Systems EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations