Properties of some time dependent probability failure functions in reliability theory

Vali Zardasht, Majid Asadi, Panlop Zeephongsekul


We consider systems having many independent components connected in a parallel or series configuration with non-identical failure distributions. A time dependent measure is proposed which evaluate the probability that a failure of the system occurred at a specified component under the condition that the system has failed by some time. We proved several properties of this measure for parallel and serial systems.

  • R. E. Barlow and F. Proschan. Importance of System Components and Fault Tree Events. Stochastic Processes and their Applications, 3:153--173, 1975.
  • F. Beichelt and K. Fischer. General failure model applied to preventive maintenance policies. IEEE Trans. Rel., 29:39--41, 1980.
  • J. H. Cha and J. Mi. Some Probability Functions in Reliability and Their Applications. Naval Research Logistics., 54:128--135, 2007. doi:10.1002/nav.20192
  • D. R. Cox. Regression models and life-tables (with discussion). J. R. Stat. Soc., B34:187--208, 1972.
  • R. C. Gupta and R. D. Gupta. Proportional reversed hazard rate model and its applications. J. Statist. Plann. Inference, 137:3525--3536, 2007. doi:10.1016/j.jspi.2007.03.029
  • R. C. Gupta, R. D. Gupta and P. L. Gupta. Modeling failure time data by Lehman alternatives. Comm. Statist. Theory Methods, 27:887--904, 1998. doi:10.1080/03610929808832134
  • T. Nakagava. Generalized models for determining optimal number of minimal repairs before replacement. J. Operat. Res. Soc. Japan, 24:325--357, 1981.
  • T. J. O'Neill. The inverse Proportional Hazards Model. Stat. and Prob. Letters, 12:125--129, 1991. doi:10.1016/0167-7152(91)90055-V
  • M. Shaked, and J. G. Shanthikumar. Stochastic Orders., Springer, 2007.
  • S. H. Sheu, and W. S. Griffith. Optimal number of minimal repairs before replacement of a system subject to shocks. Naval Res. Logist., 43:319--333, 1996. doi:10.1002/(SICI)1520-6750(199604)43:3<319::AID-NAV1>3.0.CO;2-C


time dependent failure; functions; Reversed Hazard Rate; Mean Past Lifetime; Proportional Reversed Hazard Rates (RHRs)

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