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Advances in Degradation Modeling pp 195–208Cite as

Deterioration Processes With Increasing Thresholds

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Abstract

The present chapter derives the reliability functions and hazard functions, when the threshold for deterioration is an increasing function of time. Four cases are considered. Case I: The threshold is a step function with K jumps at known points. Case II: The threshold is a step function with K jumps, where the location of jumps are random, following a renewal process. Case III: The controlled case. The jumps occur after the first crossing of a control limit. Case IV: The threshold increases linearly. The theory is illustrated in all four cases with a Markovian deterioration process, i.e., a compound Poisson process with i.i.d. jumps following an exponential distribution.

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References

  1. Zacks, S. (2004). Distribution of failure times associated with non-homogeneous compound Poisson damage process. A Festschrift for Herman Rubin, Institute of Mathematical Statistics, Lecture Notes-Monograph Series 45, pp. 396–407.

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  2. Zacks, S. (2005). Some recent results on the distributions of stopping times of compound Poisson processes with linear boundaries. Journal of Statistical Planning and Inference, 130, 95–109.

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  3. Zacks, S. (2006). Failure distributions associated with general renewal damage processes. Probability, Statistics and Modeling in Public Health, N. Nikulin, D. Commenges and C. Huber (eds.), pp. 466–475, Springer, New York.

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  4. Zacks, S. (2010). The availability and hazard of a system under a cumulative damage process with replacements. Methodology and Computing in Applied Probability, 12, 1 (to appear).

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Binghamton University, Binghamton, NY, USA

    S. Zacks

Authors
  1. S. Zacks
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Editor information

Editors and Affiliations

  1. Université Victor Segalen Bordeaux 2, I’lnstitut de Mathématiques de Bordeaux, Bordeaux Cedex, 33076, France

    M.S. Nikulin

  2. Département Génie Informatique, Labo. Mathématiques Appliquées Compiègne, Université de Technologie de Compiègne, Compiègne, France

    Nikolaos Limnios

  3. McMaster University, Department of Mathematics and Statistics, 1280 Main Street West, Hamilton, L8S 4K1, Canada

    N. Balakrishnan

  4. Fak. Mathematik, Inst. Mathematische Stochastik, Otto-von-Guericke-Universität, Universitätsplatz 2, Magdeburg, 39106, Germany

    Waltraud Kahle

  5. Laboratoire de Statistique Medicale, Université René Descartes, Paris 5, Paris, 75006, France

    Catherine Huber-Carol

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© 2010 Birkhaüser Boston, a part of Springer Science+Business Media, LLC

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Zacks, S. (2010). Deterioration Processes With Increasing Thresholds. In: Nikulin, M., Limnios, N., Balakrishnan, N., Kahle, W., Huber-Carol, C. (eds) Advances in Degradation Modeling. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4924-1_13

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