Poisson Processes

  • Toshio Nakagawa
Part of the Springer Series in Reliability Engineering book series (RELIABILITY)


It is well-known that most units operating in a useful life period, and complex systems that consist of many kinds of components, fail normally due to random causes independently over the time interval. Then, it is said in technical terms of stochastic processes, that failures occur in a Poisson process that counts the number of failures through time. This is a natural modeling tool in reliability problems. Some reliability measures such as MTTF (Mean Time To Failure), availability, and failure rate are estimated statistically from life data and are in practical use under such modelings without much theoretical arguments. Furthermore, because a Poisson process has stationary and independent properties, it is much convenient for formulating stochastic models in mathematical reliability theory.


Failure Rate Exponential Distribution Poisson Process Failure Time Interarrival Time 
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  1. 1.
    Kingman JFC (1993) Poisson processes. Oxford University Press, OxfordMATHGoogle Scholar
  2. 2.
    Barlow RE, Proschan F (1965) Mathematical theory of reliability. Wiley, New YorkMATHGoogle Scholar
  3. 3.
    Nakagawa T (2005) Maintenance theory of reliability. Springer, LondonGoogle Scholar
  4. 4.
    Osaki S (1992) Applied stochastic system modeling. Springer, BerlinMATHGoogle Scholar
  5. 5.
    Çinlar E (1975) Introduction to stochastic processes. Prentice-Hall, Englewood CliffsMATHGoogle Scholar
  6. 6.
    Nakagawa T (2008) Advanced reliability models and maintenance policies. Springer, LondonGoogle Scholar
  7. 7.
    Havil J (2003) GAMMA: exploring Euler’s constant. Princeton University Press, PrincetonMATHGoogle Scholar
  8. 8.
    Ross SM (1983) Stochastic processes. Wiley, New YorkMATHGoogle Scholar
  9. 9.
    Beichelt FE, Fatti LP (2002) Stochastic processes and their applications. CRC Press, Boca RatonMATHGoogle Scholar
  10. 10.
    Tijms HC (2003) A first course in stochastic models. Wiley, ChichesterMATHCrossRefGoogle Scholar
  11. 11.
    Yamada S (1994) Software reliability models: fundamentals and applications. JUSE Press, TokyoGoogle Scholar
  12. 12.
    Finkelstein M (2008) Failure rate modelling for reliability and risk. Springer, LondonGoogle Scholar
  13. 13.
    Murthy DNP, Xie M, Jiang R (2004) Weibull models. Wiley, HobokenMATHGoogle Scholar
  14. 14.
    Goel AL, Okumoto K (1979) Time-dependent error-detection rate model for software reliability and other performance measures. IEEE Trans Reliab R-28:206–211CrossRefGoogle Scholar
  15. 15.
    Pham H (2006) System software reliability. Springer, LondonGoogle Scholar
  16. 16.
    Pham, H (ed) (2008) Recent advances in reliability and quality in design. Springer, LondonMATHGoogle Scholar
  17. 17.
    Nakagawa T (2007) Shock and damage models in reliability theory. Springer, LondonMATHGoogle Scholar
  18. 18.
    Cox DR (1962) Renewal theory. Methuen, LondonMATHGoogle Scholar
  19. 19.
    Esary JD, Marshall AW, Proschan F (1973) Shock models and wear processes. Ann Probab 1:627–649MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Yun WY, Nakagawa T (2010) Note on MTTF of a parallel system. In: 16th ISSAT International Conference on Reliability and Quality in Design, pp 235–237Google Scholar
  21. 21.
    Chen M, Mizutani S, Nakagawa T (2010) Optimal backward and backup policies in reliability theory. J Oper Res Soc Jpn 53:101–118MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Department of Business AdministrationAichi Institute of TechnologyToyotaJapan

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