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The failure rate dynamics in heterogeneous populations

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  • Cha, Ji Hwan
  • Finkelstein, Maxim

Abstract

Most populations encountered in real world are heterogeneous. In reliability applications, the mixture (observed) failure rate, obviously, can be considered as a measure of ‘average’ quality in these populations. However, in addition to this average measure, some variability characteristics for failure rates can be very helpful in describing the time-dependent changes in quality of heterogeneous populations. In this paper, we discuss variance and the coefficient of variation of the corresponding random failure rate as variability measures for items in heterogeneous populations. Furthermore, there is often a risk that items of poor quality are selected for important missions. Therefore, along with the ‘average quality’ of a population, more ‘conservative’ quality measures should be also defined and studied. For this purpose, we propose the percentile and the tail-mixture of the failure rates as the corresponding conservative measures. Some illustrative examples are given.

Suggested Citation

  • Cha, Ji Hwan & Finkelstein, Maxim, 2013. "The failure rate dynamics in heterogeneous populations," Reliability Engineering and System Safety, Elsevier, vol. 112(C), pages 120-128.
    • Handle: RePEc:eee:reensy:v:112:y:2013:i:c:p:120-128
    DOI: 10.1016/j.ress.2012.11.012
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    References listed on IDEAS

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    1. Maxim S. Finkelstein, 2009. "Understanding the shape of the mixture failure rate (with engineering and demographic applications)," MPIDR Working Papers WP-2009-031, Max Planck Institute for Demographic Research, Rostock, Germany.
    2. Maxim Finkelstein, 2008. "Failure Rate Modelling for Reliability and Risk," Springer Series in Reliability Engineering, Springer, number 978-1-84800-986-8, January.
    3. Bebbington, Mark & Lai, Chin-Diew & Zitikis, RiÄ ardas, 2009. "Balancing burn-in and mission times in environments with catastrophic and repairable failures," Reliability Engineering and System Safety, Elsevier, vol. 94(8), pages 1314-1321.
    4. Cha, Ji Hwan & Finkelstein, Maxim, 2010. "Burn-in by environmental shocks for two ordered subpopulations," European Journal of Operational Research, Elsevier, vol. 206(1), pages 111-117, October.
    5. Maxim Finkelstein, 2009. "Understanding the shape of the mixture failure rate (with engineering and demographic applications)," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 25(6), pages 643-663, November.
    6. Finkelstein, Maxim, 2007. "Shocks in homogeneous and heterogeneous populations," Reliability Engineering and System Safety, Elsevier, vol. 92(5), pages 569-574.
    7. Y. Kebir, 1991. "On hazard rate processes," Naval Research Logistics (NRL), John Wiley & Sons, vol. 38(6), pages 865-876, December.
    8. Cha, Ji Hwan & Finkelstein, Maxim, 2011. "Burn-in and the performance quality measures in heterogeneous populations," European Journal of Operational Research, Elsevier, vol. 210(2), pages 273-280, April.
    9. James Vaupel & Kenneth Manton & Eric Stallard, 1979. "The impact of heterogeneity in individual frailty on the dynamics of mortality," Demography, Springer;Population Association of America (PAA), vol. 16(3), pages 439-454, August.
    10. Jie Mi, 1996. "Minimizing Some Cost Functions Related to Both Burn-In and Field Use," Operations Research, INFORMS, vol. 44(3), pages 497-500, June.
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    Cited by:

    1. Hal Caswell, 2014. "A matrix approach to the statistics of longevity in heterogeneous frailty models," Demographic Research, Max Planck Institute for Demographic Research, Rostock, Germany, vol. 31(19), pages 553-592.
    2. Lin, Kunsong & Chen, Yunxia & Xu, Dan, 2017. "Reliability assessment model considering heterogeneous population in a multiple stresses accelerated test," Reliability Engineering and System Safety, Elsevier, vol. 165(C), pages 134-143.
    3. Raju Bhakta & Pradip Kundu & Suchandan Kayal & Morad Alizadeh, 2024. "Stochastic Orderings between Two Finite Mixtures with Inverted-Kumaraswamy Distributed Components," Mathematics, MDPI, vol. 12(6), pages 1-20, March.
    4. Omid Shojaee & Manoochehr Babanezhad, 2023. "On some stochastic comparisons of arithmetic and geometric mixture models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(5), pages 499-515, July.
    5. Slimacek, Vaclav & Lindqvist, Bo Henry, 2016. "Nonhomogeneous Poisson process with nonparametric frailty," Reliability Engineering and System Safety, Elsevier, vol. 149(C), pages 14-23.
    6. Nil Kamal Hazra & Maxim Finkelstein, 2018. "On stochastic comparisons of finite mixtures for some semiparametric families of distributions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 27(4), pages 988-1006, December.
    7. XiaoFei, Lu & Min, Liu, 2014. "Hazard rate function in dynamic environment," Reliability Engineering and System Safety, Elsevier, vol. 130(C), pages 50-60.
    8. Cha, Ji Hwan & Finkelstein, Maxim, 2014. "Some notes on unobserved parameters (frailties) in reliability modeling," Reliability Engineering and System Safety, Elsevier, vol. 123(C), pages 99-103.
    9. J. Jarrahiferiz & M. Kayid & S. Izadkhah, 2019. "Stochastic properties of a weighted frailty model," Statistical Papers, Springer, vol. 60(1), pages 53-72, February.
    10. Barmalzan, Ghobad & Kosari, Sajad & Zhang, Yiying, 2021. "On stochastic comparisons of finite α-mixture models," Statistics & Probability Letters, Elsevier, vol. 173(C).

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