Advertisement

Methodology and Computing in Applied Probability

, Volume 12, Issue 3, pp 431–449 | Cite as

Numerical Approach for Assessing System Dynamic Availability Via Continuous Time Homogeneous Semi-Markov Processes

  • Márcio das Chagas Moura
  • Enrique López Droguett
Article

Abstract

Continuous-time homogeneous semi-Markov processes (CTHSMP) are important stochastic tools to model reliability measures for systems whose future behavior is dependent on the current and next states occupied by the process as well as on sojourn times in these states. A method to solve the interval transition probabilities of CTHSMP consists of directly applying any general quadrature method to the N 2 coupled integral equations which describe the future behavior of a CTHSMP, where N is the number of states. However, the major drawback of this approach is its considerable computational effort. In this work, it is proposed a new more efficient numerical approach for CTHSMPs described through either transition probabilities or transition rates. Rather than N 2 coupled integral equations, the approach consists of solving only N coupled integral equations and N straightforward integrations. Two examples in the context of availability assessment are presented in order to validate the effectiveness of this method against the comparison with the results provided by the classical and Monte Carlo approaches. From these examples, it is shown that the proposed approach is significantly less time-consuming and has accuracy comparable to the method of N 2 computational effort.

Keywords

Homogeneous semi-markov processes Integral equations Quadrature methods Availability assessment Reliability 

AMS 2000 Subject Classification

60K10 60K15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Baker CTH (1977) The numerical treatment of integral equations. Clarendon, New YorkzbMATHGoogle Scholar
  2. Becker G, Camarinopoulos L, Zioutas G (2000) A semi-Markovian model allowing for inhomogenities with respect to process time. Reliab Eng Syst Saf 70:41–48. doi: 10.1016/S0951-8320(00)00044-2 CrossRefGoogle Scholar
  3. Bellman RE, Roth RS (1984) The laplace transform. World Scientific Publishing Co Pte Ltd., SingaporeGoogle Scholar
  4. Corradi G, Janssen J, Manca R (2004) Numerical treatment of homogeneous semi Markov processes in transient case—a straightforward approach. Methodol Comput Appl Probab 6:233–246. doi: 10.1023/B:MCAP.0000017715.28371.85 zbMATHCrossRefMathSciNetGoogle Scholar
  5. Droguett EL, Moura MC, Jacinto CMC, Silva Jr. MF (2008) A semi-Markov model with Bayesian belief network based human error probability for availability assessment of downhole optical monitoring systems. Simul Model Pract Theory 16:1713–1727. doi: 10.1016/j.simpat.2008.08.011
  6. Howard RA (2007) Dynamic probabilistic systems v.II: Semi-Markov and decision processes. Dover Publications, INC., Mineola, New YorkGoogle Scholar
  7. Janssen J, Limnios N (1999) Semi-Markov models and applications. SpringerGoogle Scholar
  8. Janssen J, Manca R (2006) Applied semi-Markov processes. Springer Science+Business Media, LLC, New YorkzbMATHGoogle Scholar
  9. Janssen J, Manca R (2007) Semi-Markov risk models for finance, insurance and reliability. Springer Science+Business Media, LLC, New YorkzbMATHGoogle Scholar
  10. Korb KB, Nicholson AE (2003) Bayesian artificial intelligence. Chapman & Hall/CRC, FloridaCrossRefGoogle Scholar
  11. Langseth H, Portinale L (2007) Bayesian networks in reliability. Reliab Eng Syst Saf 92:92–108. doi: 10.1016/j.ress.2005.11.037 CrossRefGoogle Scholar
  12. Limnios N (1997) Dependability analysis of semi-Markov systems. Reliab Eng Syst Saf 55:203–207. doi: 10.1016/S0951-8320(96)00121-4 CrossRefGoogle Scholar
  13. Limnios N, Oprisan G (2001) Semi-Markov processes and reliability. Birkhauser, BostonzbMATHGoogle Scholar
  14. Ouhbi B, Limnios N (1997) Reliability estimation of semi-Markov systems: a case study. Reliab Eng Syst Saf 58:201–204. doi: 10.1016/S0951-8320(97)00070-7 CrossRefGoogle Scholar
  15. Ouhbi B, Limnios N (1999) “Non-parametric estimation for semi-Markov processes based on its hazard rate”. Statist. Infer. Stoch. Processes 2(2):151–173zbMATHCrossRefMathSciNetGoogle Scholar
  16. Ouhbi B, Limnios N (2002) The rate of occurrence of failures for semi-Markov processes and estimation. Stat Probab Lett 59:245–255. doi: 10.1016/S0167-7152(02)00139-6 zbMATHCrossRefMathSciNetGoogle Scholar
  17. Ouhbi B, Limnios N (2003) Nonparametric reliability estimation of semi-Markov processes. J Statist Plann Inference 109:155–165. doi: 10.1016/S0378-3758(02)00308-7 zbMATHCrossRefMathSciNetGoogle Scholar
  18. Pearl J (2000) Causality, reasoning, and inference. Cambridge University Press, New YorkGoogle Scholar
  19. Perman M, Senegacnik A, Tuma M (1997) Semi-Markov models with an application to power-plant reliability analysis. IEEE Trans Reliab 46(4):526–532. doi: 10.1109/24.693787 CrossRefGoogle Scholar
  20. Pievatolo A, Valadè I (2003) UPS reliability analysis with non-exponential duration distribution. Reliab Eng Syst Saf 81:183–189. doi: 10.1016/S0951-8320(03)00087-5 CrossRefGoogle Scholar
  21. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2002) Numerical recipes in C++, 2nd ed. Cambridge University Press, CambridgeGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Márcio das Chagas Moura
    • 1
  • Enrique López Droguett
    • 1
  1. 1.Department of Production EngineeringFederal University of PernambucoRecifeBrazil

Personalised recommendations