Skip to main content
SpringerLink
Log in

On Some Shock Models with Poisson and Generalized Poisson Shock Processes

  • Chapter
  • First Online:
Statistical Modeling for Degradation Data

Part of the book series: ICSA Book Series in Statistics ((ICSABSS))

Abstract

We consider systems subject to shocks that follow the generalized Polya process (GPP), which has been recently introduced and characterized in the literature. Distinct from the nonhomogeneous Poisson process that has been widely used in applications, the important feature of this process is the dependence of its future behaviour on the number of previous events (shocks). We consider the delayed events model and the corresponding shot noise process governed by the GPP. We also present some results on the preventive maintenance for systems with failure/repair times following the GPP.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 119.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Asfaw ZG, Linqvist B (2015) Extending minimal repair models for repairable systems: a comparison of dynamic and heterogeneous extensions of a nonhomogeneous Poisson process. Reliab Eng Syst Saf 140:153–158

    Article  Google Scholar 

  2. Aven T, Jensen U (1999) Stochastic models in reliability. Springer, New York

    Book  MATH  Google Scholar 

  3. Babykina G, Couallier V (2014) Modelling pipe failures in water distribution systems: accounting for harmful repairs and a time-dependent covariate. Int J Perform Eng 10:31–42

    Google Scholar 

  4. Block HW, Borges WS, Savits TH (1985) Age-dependent minimal repair. J Appl Probab 22:370–385

    Article  MathSciNet  MATH  Google Scholar 

  5. Caballé NC, Castro IT, Pérez CJ, Lanza-Gutiérrez JM (2015) A condition-based maintenance of a dependent degradation-threshold-shock model in a system with multiple degradation processes. Reliab Eng Syst Saf 134:98–109

    Article  Google Scholar 

  6. Cha JH (2014) Characterization of the generalized Polya process and its applications. Adv Appl Probab 46:1148–1171

    Article  MATH  Google Scholar 

  7. Cha JH, Finkelstein M (2009) On a terminating shock process with independent wear increments. J Appl Probab 46:353–362

    Article  MathSciNet  MATH  Google Scholar 

  8. Cha JH, Finkelstein M (2011) On new classes of extreme shock models and some generalizations. J Appl Probab 48:258–270

    Article  MathSciNet  MATH  Google Scholar 

  9. Cha JH, Finkelstein M (2015) On some mortality rate processes and mortality deceleration with age. J Math Biol 72:331–342

    Article  MathSciNet  MATH  Google Scholar 

  10. Cha JH, Finkelstein M (2016) New shock models based on the generalized Polya process. Eur J Oper Res 251:135–141

    Article  MathSciNet  MATH  Google Scholar 

  11. Cha JH, Finkelstein M (2016) Justifying the Gompertz curve of mortality via the generalized Polya process of shocks. Theor Popul Biol 109:54–62

    Article  MATH  Google Scholar 

  12. Cox DR, Isham V (1980) Point processes. University Press, Cambridge

    MATH  Google Scholar 

  13. Finkelstein M (2008) Failure rate modelling for reliability and risk. Springer, London

    MATH  Google Scholar 

  14. Finkelstein M, Cha JH (2013) Stochastic modelling for reliability. Shocks, burn-in and heterogeneous populations. Springer, London

    Google Scholar 

  15. Kebir Y (1991) On hazard rate processes. Nav Res Logist 38:865–877

    Article  MathSciNet  MATH  Google Scholar 

  16. Kuniewski SP, van der Weide JA, van Noortwijk JM (2009) Sampling inspection for the evaluation of time-dependent reliability of deteriorating systems under imperfect defect detection. Reliab Eng Syst Saf 94:1480–1490

    Article  Google Scholar 

  17. Le Gat Y (2014) Extending the Yule process to model recurrent pipe failures in water supply networks. Urban Water J 11:617–30

    Article  Google Scholar 

  18. Lemoine AJ, Wenocur ML (1986) A note on shot-noise and reliability modeling. Oper Res 34:320–323

    Article  MATH  Google Scholar 

  19. Lund R, McCormic W, Xiao U (2004) Limiting properties of poisson shot noise processes. J Appl Probab 41:911–918

    Article  MathSciNet  MATH  Google Scholar 

  20. Nakagawa T (2007) Shock and damage models in reliability theory. Springer, London

    MATH  Google Scholar 

  21. Rice J (1977) On generalized shot noise. Adv Appl Probab 9:553–565

    Article  MathSciNet  MATH  Google Scholar 

  22. Ross SM (1996) Stochastic processes. Wiley, New York

    MATH  Google Scholar 

  23. Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Statistics, Ewha Womans University, Seoul, 120-750, Republic of Korea

    Ji Hwan Cha

  2. Department of Mathematical Statistics, University of the Free State, 339, Bloemfontein, 9300, South Africa

    Maxim Finkelstein

Authors
  1. Ji Hwan Cha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maxim Finkelstein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maxim Finkelstein .

Editor information

Editors and Affiliations

  1. School of Social Work & Department of Biostatistics, Department of Statistics, University of North Carolina, University of Pretoria, Chapel Hill, North Carolina, USA

    Ding-Geng (Din) Chen

  2. Department of Mathematical Sciences, University of South Dakota, Vermillion, South Dakota, USA

    Yuhlong Lio

  3. Department of Statistical Science, Southern Methodist University, Dallas, Texas, USA

    Hon Keung Tony Ng

  4. Department of Statistics, Tamkang University, New Taipei City, Taiwan

    Tzong-Ru Tsai

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Nature Singapore Pte Ltd.

About this chapter

Cite this chapter

Cha, J.H., Finkelstein, M. (2017). On Some Shock Models with Poisson and Generalized Poisson Shock Processes. In: Chen, DG., Lio, Y., Ng, H., Tsai, TR. (eds) Statistical Modeling for Degradation Data. ICSA Book Series in Statistics. Springer, Singapore. https://doi.org/10.1007/978-981-10-5194-4_4

Download citation

Publish with us

Policies and ethics

Search

Navigation