, Volume 78, Issue 7, pp 807–827 | Cite as

A dynamic stress–strength model with stochastically decreasing strength

  • Ji Hwan Cha
  • Maxim Finkelstein


We consider a dynamic stress–strength model under external shocks. The strength of the system decreases with time and the failure occurs when the strength finally vanishes. Furthermore, there is another cause of the system failure induced by an external shock process. Each shock is characterized by the corresponding stress. If the magnitude of the stress exceeds the current strength, then the system also fails. We assume that the initial strength of the system and its decreasing drift pattern are random. We derive the survival function of the system and interpret the time-dependent dynamic changes of the random quantities which govern the reliability performance of the system.


Stress–strength model External shock process Nonhomogeneous Poisson process Multivariate stochastic order 



The authors would like to thank the editor and the referee for very careful and helpful comments, which have improved and clarified the presentation of our paper. The work of the first author was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (No. 2011-0017338). The work of the first author was also supported by Priority Research Centers Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0093827). The work of the second author was supported by the NRF (National Research Foundation of South Africa) Grant FA2006040700002.


  1. Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing. Holt Rinehart and Winston, New YorkGoogle Scholar
  2. Beichelt FE, Fischer K (1980) General failure model applied to preventive maintenance policies. IEEE Trans Reliab 29:39–41CrossRefGoogle Scholar
  3. Block HW, Borges W, Savits TH (1985) Age-dependent minimal repair. J Appl Probab 22:370–386MathSciNetCrossRefGoogle Scholar
  4. Cha JH, Finkelstein M (2009) On a terminating shock process with independent wear increments. J Appl Probab 46:353–362MathSciNetCrossRefGoogle Scholar
  5. Cha JH, Finkelstein M (2011) On new classes of extreme shock models and some generalizations. J Appl Probab 48:258–270MathSciNetCrossRefGoogle Scholar
  6. Cha JH, Finkelstein M (2012) Information-based thinning of point processes and its application to shocks models. J Stat Plan Inference 142:2345–2350MathSciNetCrossRefGoogle Scholar
  7. Finkelstein M (2008) Failure rate modelling for reliability and risk. Springer, LondonGoogle Scholar
  8. Finkelstein M, Cha JH (2013) Stochastic modeling for reliability: shocks, burn-in and heterogeneous population. Springer, LondonCrossRefGoogle Scholar
  9. Fleming TR, Harrington DP (1991) Counting processes and survival analysis. Wiley, New YorkGoogle Scholar
  10. Li T, Anderson JJ (2013) Shaping human mortality patterns through intrinsic and extrinsic vitality processes. Demogr Res 28:342–368CrossRefGoogle Scholar
  11. Marshall AW, Olkin I (1979) Inequalities: theory of majorization and its applications. Academic Press, New YorkGoogle Scholar
  12. Marshall AW, Shaked M (1979) Multivariate shock models for distributions with increasing hazard rate average. Ann Probab 7:343–358MathSciNetCrossRefGoogle Scholar
  13. Savits TH, Shaked M (1981) Shock models and the MIFRA property. Stat Process Appl 11:273–283MathSciNetCrossRefGoogle Scholar
  14. Mori Y, Ellingwood BR (1994) Maintaining: reliability of concrete structures. I: role of inspection/repair. J Struct Eng 120:824–845CrossRefGoogle Scholar
  15. Shaked M, Shanthikumar J (2007) Stochastic orders. Springer, New YorkCrossRefGoogle Scholar
  16. Strehler BL, Mildvan AS (1960) General theory of mortality and aging. Science 132:14–21CrossRefGoogle Scholar
  17. van Noortwijk JM, van der Weide JAM, Kallen MJ, Pandey MD (2007) Gamma processes and peaks-over-threshold distributions for time-dependent reliability. Reliab Eng Syst Saf 92:1651–1658CrossRefGoogle Scholar
  18. van Noortwijk JM (2009) A survey of the application of gamma processes in maintenance. Reliab Eng Syst Saf 94:2–21CrossRefGoogle Scholar
  19. Wenocur ML (1989) A reliability model based on the gamma process and its analytic theory. Adv Appl Probab 21:899–918MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of StatisticsEwha Womans UniversitySeoulRepublic of Korea
  2. 2.Department of Mathematical StatisticsUniversity of the Free StateBloemfonteinSouth Africa
  3. 3.ITMO UniversitySt. PetersburgRussia

Personalised recommendations