Two-Dimensional Failure Modeling

  • D.N. Murthy
  • Jaiwook Baik
  • Richard Wilson
  • Michael Bulmer


For many products (for example, automobiles), failures depend on age and usage and, in this case, failures are random points in a two-dimensional plane with the two axes representing age and usage. In contrast to the one-dimensional case (where failures are random points along the time axis) the modeling of two-dimensional failures has received very little attention. In this chapter we discuss various issues (such as modeling process, parameter estimation, model analysis) for the two-dimensional case and compare it with the one-dimensional case.


Hazard Function Usage Rate Failure Time Weibull Model Subsequent Failure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Weibull probability plot


  1. 5.1.
    W. R. Blischke, D. N. P. Murthy: Warranty Cost Analysis (Marcel Dekker, New York 1994)Google Scholar
  2. 5.2.
    W. R. Blischke, D. N. P. Murthy: Reliability (Wiley, New York 2000)zbMATHGoogle Scholar
  3. 5.3.
    J. F. Lawless: Statistical Models and Methods for Lifetime Data (Wiley, New York 1982)zbMATHGoogle Scholar
  4. 5.4.
    W. Q. Meeker, L. A. Escobar: Statistical Methods for Reliability Data (Wiley, New York 1998)zbMATHGoogle Scholar
  5. 5.5.
    C. D. Lai, D. N. P. Murthy, M. Xie: Weibull Distributions and Their Applications. In: Springer Handbook of Engineering Statistics, ed. by Pham (Springer, Berlin 2004)Google Scholar
  6. 5.6.
    D. N. P. Murthy, M. Xie, R. Jiang: Weibull Models (Wiley, New York 2003)CrossRefGoogle Scholar
  7. 5.7.
    W. Nelson: Applied Life Data Analysis (Wiley, New York 1982)CrossRefzbMATHGoogle Scholar
  8. 5.8.
    J. D. Kalbfleisch, R. L. Prentice: The Statistical Analysis of Failure Time Data (Wiley, New York 1980)zbMATHGoogle Scholar
  9. 5.9.
    N. L. Johnson, S. Kotz: Distributions in Statistics: Continuous Univariate Distributions I (Wiley, New York 1970)zbMATHGoogle Scholar
  10. 5.10.
    N. L. Johnson, S. Kotz: Distributions in Statistics: Continuous Univariate Distributions II (Wiley, New York 1970)zbMATHGoogle Scholar
  11. 5.11.
    T. Nakagawa, M. Kowada: Analysis of a system with minimal repair and its application to a replacement policy, Eur. J. Oper. Res. 12, 176–182 (1983)CrossRefzbMATHGoogle Scholar
  12. 5.12.
    D. N. P. Murthy: A note on minimal repair, IEEE Trans. Reliab. 40, 245–246 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 5.13.
    D. R. Cox: Renewal Theory (Methuen, London 1967)zbMATHGoogle Scholar
  14. 5.14.
    D. R. Cox, V. Isham: Point Processes (Chapman-Hall, New York 1980)zbMATHGoogle Scholar
  15. 5.15.
    S. M. Ross: Stochastic Processes (Wiley, New York 1983)zbMATHGoogle Scholar
  16. 5.16.
    H. Pham, H. Wang: Imperfect Maintenance, Eur. J. Oper. Res. 94, 438–452 (1996)Google Scholar
  17. 5.17.
    L. Doyen, O. Gaudoin: Classes of imperfect repair models based on reduction of failure intensity function or virtual age, Reliab. Eng. Syst. Saf. 84, 45–56 (2004)CrossRefGoogle Scholar
  18. 5.18.
    B. W. Silverman: Density Estimation for Statistics and Data Analysis (Chapman Hall, London 1986)zbMATHGoogle Scholar
  19. 5.19.
    J. T. Duane: Learning curve approach to reliability monitoring, IEEE Trans. Aerosp. 40, 563–566 (1964)CrossRefGoogle Scholar
  20. 5.20.
    S. E. Rigdon, A. P. Basu: Statistical Methods for the Reliability of Repairable Systems (Wiley, New York 2000)zbMATHGoogle Scholar
  21. 5.21.
    R. Jiang, D. N. P. Murthy: Modeling failure data by mixture of two Weibull distributions, IEEE Trans. Reliab. 44, 478–488 (1995)CrossRefGoogle Scholar
  22. 5.22.
    S. B. Vardeman: Statistics for Engineering Problem Solving (PWS, Boston 1993)Google Scholar
  23. 5.23.
    R. B. DʼAgostino, M. A. Stephens: Goodness-of-Fit Techniques (Marcel Dekker, New York 1986)zbMATHGoogle Scholar
  24. 5.24.
    J. F. Lawless, J. Hu, J. Cao: Methods for the estimation of failure distributions and rates from automobile warranty data, Lifetime Data Anal. 1, 227–240 (1995)CrossRefzbMATHGoogle Scholar
  25. 5.25.
    I. B. Gertsbakh, H. B. Kordonsky: Parallel time scales and two-dimensional manufacturer and individual customer warranties, IEE Trans. 30, 1181–1189 (1998)Google Scholar
  26. 5.26.
    B. P. Iskandar, W. R. Blischke: Reliability and Warranty Analysis of a Motorcycle Based on Claims Data. In: Case Studies in Reliability and Maintenance, ed. by W. R. Blischke, D. N. P. Murthy (Wiley, New York 2003) pp. 623–656Google Scholar
  27. 5.27.
    G. Yang, Z. Zaghati: Two-dimensional reliability modelling from warranty data, Ann. Reliab. Maintainab. Symp. Proc. 272-278 (IEEE, New York 2002)Google Scholar
  28. 5.28.
    A. W. Marshall, I. Olkin: A multivariate exponential distribution, J. Am. Stat. Assoc. 62, 30–44 (1967)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 5.29.
    L. Lee: Multivariate distributions having Weibull properties, J. Multivariate Anal. 9, 267–277 (1979)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 5.30.
    J. C. Lu, G. K. Bhattacharyya: Some new constructions of bivariate Weibull Models, Ann. Inst. Stat. Math. 42, 543–559 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  31. 5.31.
    S. K. Sarkar: A continuous bivariate exponential distribution, J. Am. Stat. Assoc. 82, 667–675 (1987)CrossRefzbMATHGoogle Scholar
  32. 5.32.
    N. L. Johnson, S. Kotz: Distributions in Statistics: Continuous Multivairate Distributions (Wiley, New York 1972)Google Scholar
  33. 5.33.
    T. P. Hutchinson, C. D. Lai: Continuous Bivariate Distributions, Emphasising Applications (Rumsby Scientific, Adelaide 1990)zbMATHGoogle Scholar
  34. 5.34.
    H. G. Kim, B. M. Rao: Expected warranty cost of a two-attribute free-replacement warranties based on a bi-variate exponential distribution, Comput. Ind. Eng. 38, 425–434 (2000)CrossRefGoogle Scholar
  35. 5.35.
    D. N. P. Murthy, B. P. Iskandar, R. J. Wilson: Two-dimensional failure free warranties: Two-dimensional point process models, Oper. Res. 43, 356–366 (1995)CrossRefzbMATHGoogle Scholar
  36. 5.36.
    N. D. Singpurwalla, S. P. Wilson: Failure models indexed by two scales, Adv. Appl. Probab. 30, 1058–1072 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  37. 5.37.
    S.-C. Yang, J. A. Nachlas: Bivariate reliability and availability modeling, IEEE Trans. Reliab. 50, 26–35 (2001)CrossRefGoogle Scholar
  38. 5.38.
    J. J. Hunter: Renewal theory in two dimensions: Basic results, Adv. Appl. Probab. 6, 376–391 (1974)CrossRefMathSciNetzbMATHGoogle Scholar
  39. 5.39.
    B. P. Iskandar: Two-Dimensional Renewal Function Solver, Res. Rep. No. 4/91 (Dept. of Mechanical Engineering, Univ. Queensland, Queensland 1991)Google Scholar
  40. 5.40.
    M. S. Finkelstein: Alternative time scales for systems with random usage, IEEE Trans. Reliab. 50, 261–264 (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Division of Mechanical EngineeringThe University of QueenslandBrisbaneAustralia
  2. 2.Department of Information StatisticsKorea National Open UniversitySeoulSouth Korea
  3. 3.Department of MathematicsThe University of QueenslandBrisbaneAustralia
  4. 4.Department of MathematicsUniversity of QueenslandBrisbaneAustralia

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