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Mathematical Problems in Engineering
Volume 2010, Article ID 242567, 16 pages
http://dx.doi.org/10.1155/2010/242567
Research Article

An Expectation Maximization Algorithm to Model Failure Times by Continuous-Time Markov Chains

1Department of Mathematics, Faculty of Science, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China
2School of Electrical Engineering, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China

Received 5 June 2010; Revised 28 July 2010; Accepted 29 July 2010

Academic Editor: Ming Li

Copyright © 2010 Qihong Duan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. C. K. Pil, M. Rausand, and J. Vatn, “Reliability assessment of reliquefaction systems on LNG carriers,” Reliability Engineering and System Safety, vol. 93, pp. 1345–1353, 2008. View at Google Scholar
  2. S. Y. Chen, C. Y. Yao, G. Xiao, Y. S. Ying, and W. L. Wang, “Fault detection and prediction of clocks and timers based on computer audition and probabilistic neural networks,” in Proceedings of the 8th International Workshop on Artificial Neural Networks (IWANN '05), vol. 3512 of Lecture Notes on Computer Science, pp. 952–959, 2005. View at Scopus
  3. S. Y. Chen, Y. F. Li, and J. W. Zhang, “Vision processing for realtime 3-D data acquisition based on coded structured light,” IEEE Transactions on Image Processing, vol. 17, no. 2, pp. 167–176, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  4. X. Zhao and L. Cui, “On the accelerated scan finite Markov chain imbedding approach,” IEEE Transactions on Reliability, vol. 58, no. 2, pp. 383–388, 2009. View at Publisher · View at Google Scholar · View at Scopus
  5. S. Y. Chen, Y. F. Li, Q. Guan, and G. Xiao, “Real-time three-dimensional surface measurement by color encoded light projection,” Applied Physics Letters, vol. 89, no. 11, article no. 111108, 2006. View at Publisher · View at Google Scholar · View at Scopus
  6. “IEC61511. Functional safety—safety instrumented systems for the process industry sector,” Geneva, Switzerland, IEC, 2004.
  7. M. Xie, Y. Tang, and T. N. Goh, “A modified Weibull extension with bathtubshaped failure rate function,” Reliability Engineering and System Safety, vol. 76, pp. 279–285, 2002. View at Google Scholar
  8. A. Pievatolo, E. Tironi, and I. Valadé, “Semi-Markov processes for power system reliability assessment with application to uninterruptible power supply,” IEEE Transactions on Power Systems, vol. 19, no. 3, pp. 1326–1333, 2004. View at Publisher · View at Google Scholar · View at Scopus
  9. H. R. Guo, H. Liao, W. Zhao, and A. Mettas, “A new stochastic model for systems under general repairs,” IEEE Transactions on Reliability, vol. 56, no. 1, pp. 40–49, 2007. View at Publisher · View at Google Scholar · View at Scopus
  10. M. Li, “Fractal time series—a tutorial review,” Mathematical Problems in Engineering, vol. 2010, Article ID 157264, 26 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. Endrenyi, G. J. Anders, and A. M. Leite Da Suva, “Probabilistic evaluation of the effect of maintenance on reliability—an application,” IEEE Transactions on Power Systems, vol. 13, no. 2, pp. 576–583, 1998. View at Google Scholar · View at Scopus
  12. M. Li and W. Zhao, “Representation of a stochastic traffic bound,” IEEE Transactions on Parallel and Distributed Systems, vol. 21, no. 9, pp. 1368–1372, 2010. View at Google Scholar
  13. M. Li and S. C. Lim, “Modeling network traffic using generalized Cauchy process,” Physica A, vol. 387, no. 11, pp. 2584–2594, 2008. View at Publisher · View at Google Scholar · View at Scopus
  14. D. R. Cox, “The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 51, pp. 433–441, 1955. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. C. Singh, R. Billinton, and S. Y. Lee, “The method of stages for non-Markov models,” IEEE Transactions on Reliability, vol. 26, no. 2, pp. 135–137, 1977. View at Google Scholar · View at Scopus
  16. Y. Lam, “Calculating the rate of occurrence of failures for continuous-time markov chains with application to a two-component parallel system,” Journal of the Operational Research Society, vol. 46, pp. 528–536, 1995. View at Google Scholar
  17. E. A. Oliveira, A. C.M. Alvim, and P. Melo, “Unavailability analysis of safety systems under aging by supplementary variables with imperfect repair,” Annals of Nuclear Energy, vol. 32, no. 2, pp. 241–252, 2005. View at Publisher · View at Google Scholar · View at Scopus
  18. A. Bobbio, A. Horváth, M. Scarpa, and M. Telek, “Acyclic discrete phase type distributions: properties and a parameter estimation algorithm,” Performance Evaluation, vol. 54, no. 1, pp. 1–32, 2003. View at Publisher · View at Google Scholar · View at Scopus
  19. R. J. Elliott, L. Aggoun, and J. B. Moore, Hidden Markov Models, vol. 29 of Applications of Mathematics, Springer, New York, NY, USA, 1995. View at MathSciNet
  20. R. J. Elliott, Z. P. Chen, and Q. H. Duan, “Insurance claims modulated by a hidden Brownian marked point process,” Insurance: Mathematics & Economics, vol. 45, no. 2, pp. 163–172, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  21. R. N. Bhattacharya and E. C. Waymire, Stochastic Processes with Applications, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, NY, USA, 1990. View at MathSciNet
  22. C. F. Van Loan, “Computing integrals involving the matrix exponential,” IEEE Transactions on Automatic Control, vol. 23, no. 3, pp. 395–404, 1978. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. F. Carbonell, J. C. Jímenez, and L. M. Pedroso, “Computing multiple integrals involving matrix exponentials,” Journal of Computational and Applied Mathematics, vol. 213, no. 1, pp. 300–305, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. R. J. Elliott, “New finite-dimensional filters and smoothers for noisily observed Markov chains,” IEEE Transactions on Information Theory, vol. 39, no. 1, pp. 265–271, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. R. J. Elliott and W. P. Malcolm, “Discrete-time expectation maximization algorithms for Markov-modulated Poisson processes,” IEEE Transactions on Automatic Control, vol. 53, no. 1, pp. 247–256, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  26. M. V. Aarset, “How to identify bathtub hazard rate,” IEEE Transactions on Reliability, vol. 36, no. 1, pp. 106–108, 1987. View at Google Scholar · View at Scopus
  27. M. Xie and C. D. Lai, “Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function,” Reliability Engineering and System Safety, vol. 52, no. 1, pp. 87–93, 1996. View at Publisher · View at Google Scholar · View at Scopus