Dependent Competing-Risk Degradation Systems

  • Yaping Wang
  • Hoang Pham
Part of the Springer Series in Reliability Engineering book series (RELIABILITY)


The failure of many units or systems, such as components, parts, machines, can be generally classified into two kinds of failure modes: one is catastrophic failure in which units break down by some sudden external shocks; the other is degradation failure in which units fail to function due to the physical deterioration. There are a great number of such cases for this kind of competing failure modes in our real life.


Preventive Maintenance Lifetime Distribution Shock Model Copula Model Maintenance Policy 
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.


  1. 1.
    Esary JD, Marshall AW (1973) Shock models and wear process. Annu Prob 1(4):627–649MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Hameed MSA, Proschan F (1973) Nonstationary shock models. Stoch Process Appl 1(10):383–404MATHCrossRefGoogle Scholar
  3. 3.
    Agrafiotis GK, Tsoukalas MZ (1987) On excess-time correlated cumulative process. J Oper Res Soc 46:1269–1280Google Scholar
  4. 4.
    Gut A, Husler J (2005) Realistic variation of shock models. Stat Prob Lett 74(2):187–204MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Nakagawa T, Kijima M (1989) Replacement policies for a cumulative damage model with minimal repair at failure. IEEE Trans Reliab 28:581–584CrossRefGoogle Scholar
  6. 6.
    Qian C, Nakamura S, Nakagawa T (2003) Replacement and minimal repair polices for a cumulative damage model with maintenance. Comput Math Appl 46:1111–1118MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Wortman MA, Klutke G-A, Ayhan H (1994) A maintenance strategy for systems subjected to deterioration governed by random shocks. IEEE Trans Reliab 43(3):439–445CrossRefGoogle Scholar
  8. 8.
    Chelbi A, Ait-Kadi D (2000) Generalized inspection strategy for randomly failing systems subjected to random shocks. Int J Prod Econ 64:379–384CrossRefGoogle Scholar
  9. 9.
    Ross SM (1981) Generalized poisson models. Annu Prob 9(5):896–898MATHCrossRefGoogle Scholar
  10. 10.
    Shanthikumar JG, Sumita U (1983) General shock models associated with correlated renewal sequences. J Appl Prob 20:600–614MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Chen J, Li Z (2008) An extended extreme shock maintenance model for a deteriorating system. Reliab Eng Syst Safety 93:1123–1129CrossRefGoogle Scholar
  12. 12.
    Li Z (1984) Some probability distribution on Poisson shocks and its application in city traffic. J Lanzhou Univ 20:127–136Google Scholar
  13. 13.
    Li Z, Kong X (2007) Life behavior of δ-shock model. Stat Prob Lett 77(6):577–587MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Li Z, Zhao P (2007) Reliability analysis on the δ-shock model of complex systems. IEEE Trans Reliab 56(2):340–348CrossRefGoogle Scholar
  15. 15.
    Lam Y, Zhang YL (2004) A shock model for the maintenance problem of repairable system. Comput Oper Res 31:1807–1820MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Rangan A, Tansu A (2008) A new shock model for system subject to random threshold failure. Proc World Acad Sci Eng Technol 30:1065–1070Google Scholar
  17. 17.
    Fan J, Ghurke SG, Levine RA (2000) Multicomponent lifetime distribution in the presence of ageing. J Appl Prob 37:521–533MATHCrossRefGoogle Scholar
  18. 18.
    Igaki N, Sumita U, Kowada M (1995) Analysis of Markov renewal shock models. J Appl prob 32:821–831MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Gut A (2001) Mixed shock models. Bernoulli 7(3):541–555MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Mallor F, Omey E (2001) Shocks, runs and random sums. J Appl Prob 38:438–448MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Mallor F, Santos J (2003) Classification of shock model in system reliability. Monografias del Semin Matem Garcia de Galdeano 27:405–412MathSciNetGoogle Scholar
  22. 22.
    Mallor F, Santos J (2003) Reliability of systems subject to shocks with a stochastic dependence for the damages. Test 12(2):427–444MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Mallor F, Omey E, Santos J (2006) Asymptotic results for a run and cumulative mixed shock model. J Math Sci 138(1):5410–5414MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Finkelstein MS, Zarudnij VI (2001) A shock process with a non-cumulative damage. Reliab Eng Syst Safety 71:103–107CrossRefGoogle Scholar
  25. 25.
    Bai J-M, Li Z-H, Kong X-B (2006) Generalized shock models based on a cluster point process. IEEE Trans Reliab 55(3):542–550CrossRefGoogle Scholar
  26. 26.
    Lu CJ, Meeker WQ (1993) Using degradation measures to estimate of time-to-failure distribution. Technometrics 35:161–176MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Yuan X-X, Pandey MD (2009) A nonlinear mixed-effects model for degradation data obtained from in-service inspections. Reliab Eng Syst Safety 94:509–519CrossRefGoogle Scholar
  28. 28.
    Robinson ME, Crowder MJ (2000) Bayesian methods for a growth-curve degradation model with repeated measures. Life Data Anal 6:357–374MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Wu S-J, Shao J (1999) Reliability analysis using the least squares method in nonlinear mixed-effect degradation models. Stat Sinica 9:855–877MathSciNetMATHGoogle Scholar
  30. 30.
    van Noortwijk JM, Cooke RM, Kok M (1995) A Bayesian failure model based on isotropic deterioration. Eur J Oper Res 82:270–282MATHCrossRefGoogle Scholar
  31. 31.
    van Noortwijk JM, Pandey MD (2003) A stochastic deterioration process for time-dependent reliability analysis. Proceedings of the eleventh IFIP WG 7.5 working conference on reliability and optimization of structural systems, pp 259–265Google Scholar
  32. 32.
    Nicolai RP, Dekker R, van Noortwijk JM (2007) A comparison of models for measurable deterioration: an application to coatings on steel structures. Reliab Eng Syst Safety 92:1635–1650CrossRefGoogle Scholar
  33. 33.
    Hsieh M-H, Jeng S-L, Shen P-S (2009) Assessing device reliability based on scheduled discrete degradation measurements. Prob Eng Mech 24:151–158CrossRefGoogle Scholar
  34. 34.
    Xue J, Yang K (1995) Dynamic reliability analysis of coherent multistate systems. IEEE Trans Reliab 44(4):683–688CrossRefGoogle Scholar
  35. 35.
    Kharoufeh JP (2003) Explicit results for wear processes in a Markovian environment. Oper Res Lett 31:237–244MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Saassouh B, Dieulle L, Grall A (2007) Online maintenance policy for a depredating system with random change of mode. Reliab Eng Syst Safety 92:1677–1685CrossRefGoogle Scholar
  37. 37.
    Kharoufeh JP, Cox SM (2005) Stochastic models for degradation-based reliability. IIE Trans 37(6):533–542CrossRefGoogle Scholar
  38. 38.
    Ebrahimi N (2001) A stochastic covariate failure model for assessing system reliability. J Appl Prob 38:761–767MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Huang W, Dietrich DL (2005) An alternative degradation reliability modeling approach using maximum likelihood estimation. IEEE Trans Reliab 54(2):310–317CrossRefGoogle Scholar
  40. 40.
    Bae SJ, Kuo W, Kvam PH (2007) Degradation models and implied lifetime distributions. Reliab Eng Syst Safety 92:601–608CrossRefGoogle Scholar
  41. 41.
    Zuo MJ, Jiang R, Yam RCM (1999) Approaches for reliability modeling of continuous-state devices. IEEE Trans Reliab 48(1):9–18CrossRefGoogle Scholar
  42. 42.
    Bae SJ, Kvam PH (2005) A nonlinear random coefficients model for degradation testing. Technometrics 46(4):460–469MathSciNetCrossRefGoogle Scholar
  43. 43.
    Grall A, Berenguer C, Dieulle L (2002) A condition-based maintenance policy for stochastically deteriorating systems. Reliab Eng Syst Safety 76:167–180CrossRefGoogle Scholar
  44. 44.
    Wang L, Chu J, Mao W (2009) A condition-based replacement and spare provisioning policy for deteriorating systems with uncertain deterioration to failure. Eur J Oper Res 194:184–205MATHCrossRefGoogle Scholar
  45. 45.
    Kiessler PC, Klutke G-A, Yang Y (2002) Availability of periodically inspected systems subject to Markovian degradation. J Appl Prob 39:700–711MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Yang Y, Klutke G-A (2000) Lifetime-characteristics and inspection-schemes for levy degradation process. IEEE Trans Reliab 49(4):377–382MathSciNetCrossRefGoogle Scholar
  47. 47.
    Zhao YX (2003) On preventive maintenance policy of a critical reliability level for system subject to degradation. Reliab Eng Syst Safety 79:301–308CrossRefGoogle Scholar
  48. 48.
    Pandey MD, Yuan XX, van Noortwijk JM (2005) Gamma process model for reliability analysis and replacement of aging structural components, Safety and Reliability of Engineering Systems and Structures. In: Proceedings of the Ninth International Conference on Structural Safety and Reliability (ICOSSAR), Rome, pp 2439–2444Google Scholar
  49. 49.
    van Noortwijk JM, Frangopol DM (2004) Two probabilistic life-cycle maintenance models for deteriorating civil infrastructures. Prob Eng Mech 19:345–359CrossRefGoogle Scholar
  50. 50.
    Delia M-C, Rafael P-O (2006) A deteriorating two-system with two repair modes and sojourn times phase-type distributed. Reliab Eng Syst Safety 91:1–9CrossRefGoogle Scholar
  51. 51.
    Delia M-C, Rafael P-O (2008) A maintenance model with failures and inspection following Markovian arrival processes and two repair modes. Eur J Oper Res 186:694–707MathSciNetMATHCrossRefGoogle Scholar
  52. 52.
    Sim SH, Endrenyi J (1993) A failure-repair model with minimal & major maintenance. IEEE Trans Reliab 42(1):134–140MATHCrossRefGoogle Scholar
  53. 53.
    Ciampoli M (1998) Time dependent reliability of structural systems subject to deterioration. Comput Struct 67:29–35MATHCrossRefGoogle Scholar
  54. 54.
    Hosseini MM, Kerr RM, Randall RB (2000) An inspection model with minimal and major maintenance for a system with deterioration and poisson failures. IEEE Trans Reliab 49(1):88–987CrossRefGoogle Scholar
  55. 55.
    Zhu Y, Elsayed EA, Liao H, Chan LY (2010) Availability optimization of systems subject to competing risk. Eur J Oper Res 202(3):781–788MATHCrossRefGoogle Scholar
  56. 56.
    van der Weide JAM, Pandey MD, van Noortwijk JM (2010) Discounted cost model for condition-based maintenance optimization. Reliab Eng Syst Safety 95:236–246CrossRefGoogle Scholar
  57. 57.
    Chiang JH, Yuan J (2001) Optimal maintenance policy for a Markovian system under periodic inspection. Reliab Eng Syst Safety 71:165–172CrossRefGoogle Scholar
  58. 58.
    Delia M-C, Rafael P-O (2006) Replacement times and costs in a degrading system with several types of failure: the case of phase-type holding times. Eur J Oper Res 175:1193–1209MATHCrossRefGoogle Scholar
  59. 59.
    Kharoufeh JP, Finkelstein DE, Mixon DG (2007) Availability of periodically inspected systems with Markovian wear and shocks. J Appl Prob 43:303–317MathSciNetCrossRefGoogle Scholar
  60. 60.
    Tang Y-Y, Lam Y (2006) A δ-shock maintenance model for a deteriorating system. Eur J Oper Res 168:541–556MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Frostig E, Kenzin M (2009) Availability of inspected systems subject to shocks—a matrix algorithmic approach. Eur J Oper Res 193:168–183MathSciNetMATHCrossRefGoogle Scholar
  62. 62.
    Mori Y, Ellingwood BR (1994) Maintaining: reliability of concrete structures. I: role of inspection/repair. Struct Eng 120(3):824–845CrossRefGoogle Scholar
  63. 63.
    van Noortwijk JM, van der Weide JAM, Kallen MJ, Pandey MD (2007) Gamma processes and peaks-over-threshold distribution for time-dependent reliability. Reliab Eng Syst Safety 92:1651–1658CrossRefGoogle Scholar
  64. 64.
    Elio C, Giovanni M (2006) Indirect reliability estimation for electric devices via a dynamic ‘stress-strength’ model, SPEEDAM international symposium on power electronics, electrical drives, automation and motionGoogle Scholar
  65. 65.
    Lehmann A (2009) Joint modeling of degradation and failure time data. J Stat Plan Infer 139(5):1693–1706MathSciNetMATHCrossRefGoogle Scholar
  66. 66.
    Fan J, Ghurye SG, Levine RA (2000) Multi-component lifetime distributions in the presence of ageing. J Appl Prob 37:521–533MathSciNetMATHCrossRefGoogle Scholar
  67. 67.
    Cha JH, Finkelstein M (2009) On a terminating shock process with independent wear increments. J Appl Prob 46:353–362MathSciNetMATHCrossRefGoogle Scholar
  68. 68.
    Finkelstein M (2009) On damage accumulation and biological aging. J Stat Plan Infer 139(5):1643–1648MathSciNetMATHCrossRefGoogle Scholar
  69. 69.
    Satow T, Teramoto K, Nakagawa T (2000) Optimal replacement policy for a cumulative damage model with time deterioration. Math Comput Model 31:313–319MathSciNetMATHCrossRefGoogle Scholar
  70. 70.
    Deloux E, Castanier B, Berenguer C (2009) Predictive maintenance policy for a gradually deteriorating system subject to stress. Reliab Eng Syst Safety 94(2):418–431CrossRefGoogle Scholar
  71. 71.
    Klutke G-A, Yang Y (2002) The availability of inspected systems subject to shocks and graceful degradation. IEEE Trans Reliab 51(3):371–374CrossRefGoogle Scholar
  72. 72.
    Wenjian L, Pham H (2005) Reliability modeling of multi-state degraded systems with multi-competing failures and random shocks. IEEE Trans Reliab 54(2):297–303CrossRefGoogle Scholar
  73. 73.
    Wenjian L, Pham H (2005) An inspection-maintenance model for systems with multiple competing processes. IEEE Trans Reliab 54(2):318–327CrossRefGoogle Scholar
  74. 74.
    Peng W, Coit DW (2004) Reliability prediction based on degradation modeling for systems with multiple degradation measures, Reliability and maintainability 2004 annual symposium-RAMS, pp 302–307Google Scholar
  75. 75.
    Yifan Z, Lin M, Rodney C, H-E Kim (2009) Asset life prediction using multiple degradation indicators and lifetime data: a gamma-based state space model approach. The 8th international conference on reliability, maintainability and safetyGoogle Scholar
  76. 76.
    Embrechts P, Puccetti G (2010) Bounds for the sum of dependent risks having overlapping marginals. J Multivar Anal 101:177–190MathSciNetMATHCrossRefGoogle Scholar
  77. 77.
    Kojadinovic I, Yan J (2010) Comparison of three semiparametric methods for estimating dependence parameters in copula models. Insur Math Econ 47(1):52–63MathSciNetCrossRefGoogle Scholar
  78. 78.
    Rodriguez-Lallena JA (2003) Manuel Ubeda-Flores, distribution functions of multivariate copulas. Stat Prob Lett 64:41–50MathSciNetMATHCrossRefGoogle Scholar
  79. 79.
    Chen X, Fan Y (2006) Estimation of copula-based semiparametric time series models. J Econ 130:307–335MathSciNetGoogle Scholar
  80. 80.
    Hurlimann W (2004) Fitting bivariate cumulative returns with copulas. Comput Stat Data Anal 45:355–372MathSciNetCrossRefGoogle Scholar
  81. 81.
    Abegaz F, Naik-Nimbalkar UV (2008) Modeling statistical dependence of Markov chains via copula models. J Stat Plan Infer 138:1131–1146MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    Zezula I (2009) On multivariate Gaussian copulas. J Stat Plan Infer 139:3942–3946MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    Van den Goorbergh RWJ, Genest C, Werker BJM (2005) Bivariate option pricing using dynamic copulae models. Insur Math Econ 37:101–114MATHCrossRefGoogle Scholar
  84. 84.
    Zhang L, Singh VP (2007) Bivariate rainfall frequency distributions using Archimedean copulas. J Hydrol 332:93–109CrossRefGoogle Scholar
  85. 85.
    Fernandez V (2008) Copula-based measures of dependence structure in assets returns. Physica A 387:3615–3628MathSciNetGoogle Scholar
  86. 86.
    Al-Harthy M, Begg S, Bratvold RB (2007) Copulas: a new technique to model dependence in petroleum decision making. J Petrol Sci Eng 57:195–208CrossRefGoogle Scholar
  87. 87.
    Valle LD (2009) Bayesian copulae distributions, with application to operational risk management. Methodol Comput Appl Probab 11:95–115MathSciNetCrossRefGoogle Scholar
  88. 88.
    Dakovic R, Czado C (2009) Comparing point and interval estimates in the bivariate t-copula model with application to financial data. Statistical Papers, 1613-9798 (online)Google Scholar
  89. 89.
    Ning C (2010) Dependence structure between the equity market and the foreign exchange market—a copula approach. J Int Money Finance 29(5):743–759CrossRefGoogle Scholar
  90. 90.
    Roch O, Alegre A (2006) Testing the bivariate distribution of daily equity returns using copulas. An application to the Spanish stock market. Comput Stat Data Anal 51:1312–1329MathSciNetMATHCrossRefGoogle Scholar
  91. 91.
    Concepcion Ausin M, Lopes HF (2010) Time-varying joint distribution through copulas. Comput Stat Data Anal 54(11):2383–2399CrossRefGoogle Scholar
  92. 92.
    Renard B, Lang M (2007) Use of a Gaussian copula for multivariate extreme value analysis: some case studies in hydrology. Adv Water Resour 30:897–912CrossRefGoogle Scholar
  93. 93.
    Kaishev VK, Dimitrina DS, Haberman S (2007) Modeling the joint distribution of competing risks survival times using copula functions. Insur Math Econ 41(3):339–361MATHCrossRefGoogle Scholar
  94. 94.
    Bedford T (2006) Copulas, degenerate distributions and quantile tests in competing risk problems. J Stat Plan Infer 136(5):1572–1587MathSciNetMATHCrossRefGoogle Scholar
  95. 95.
    Lo SMS, Wilke RA (2010) A copula model for dependent competing risks. J Roy Stat Soc Ser C (Appl Stat) 59(2):359–376MathSciNetCrossRefGoogle Scholar
  96. 96.
    Cossette H, Marceau E, Marri F (2008) On the compound Poisson risk model with dependence based on a generalized Farlie-Gumbel-Morgenstern. Insur Math Econ 43:444–455MathSciNetMATHCrossRefGoogle Scholar
  97. 97.
    Cossette H, Gaillardetz P, Marceau E, Rioux J (2002) On two dependent individual risk models. Insur Math Econ 30:153–166MathSciNetMATHCrossRefGoogle Scholar
  98. 98.
    Embrechts P, Hoing A, Juri A (2003) Using copulae to bound the value-at-risk for functions of dependent risks. Finance Stoch 7:145–167MathSciNetMATHCrossRefGoogle Scholar
  99. 99.
    Wang Y, Pham H (2009) The imperfect preventive maintenance policies for two-process cumulative damage model (submitted to Int J Syst Sci)Google Scholar
  100. 100.
    Wang Y, Pham H (2011) Modeling the dependent competing risks with multiple degradation processes and random shock using time-varying copulas. IEEE Trans Reliab, vol 6(4) to appearGoogle Scholar
  101. 101.
    Wang Y, Pham H (2010) Dependent competing risk model with multiple-degradation and random shock using time-varying copulas. 16th ISSAT international conference on reliability and quality in design, Washington D.C., pp 100–104Google Scholar
  102. 102.
    Wang Y, Pham H (2011) A multi-objective optimization of imperfect preventive maintenance policy for dependent competing risk system with hidden failure. IEEE Trans Reliab, vol 6(3) to appearGoogle Scholar

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© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Department of Industrial & Systems EngineeringRutgers UniversityPiscatawayUSA

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