Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T02:25:34.377Z Has data issue: false hasContentIssue false
  • English
  • Français

BAYESIAN ANALYSIS OF DOUBLY STOCHASTIC MARKOV PROCESSES IN RELIABILITY

Published online by Cambridge University Press:  08 April 2020

Atilla Ay Open the ORCID record for Atilla Ay [Opens in a new window] ,
Refik Soyer ,
Joshua Landon  and
Süleyman Özekici
Atilla Ay
Affiliation:
Department of Decision Sciences, The George Washington University, Washington, DC 20052, USA E-mail: [email protected]
Refik Soyer
Affiliation:
Department of Decision Sciences, The George Washington University, Washington, DC 20052, USA E-mail: [email protected]
Joshua Landon
Affiliation:
Department of Statistics, The George Washington University, Washington, DC 20052, USA
Süleyman Özekici
Affiliation:
Department of Industrial Engineering, Koç University, 34450 İstanbul, Turkey
Get access Rights & Permissions [Opens in a new window]

Abstract

Markov processes play an important role in reliability analysis and particularly in modeling the stochastic evolution of survival/failure behavior of systems. The probability law of Markov processes is described by its generator or the transition rate matrix. In this paper, we suppose that the process is doubly stochastic in the sense that the generator is also stochastic. In our model, we suppose that the entries in the generator change with respect to the changing states of yet another Markov process. This process represents the random environment that the stochastic model operates in. In fact, we have a Markov modulated Markov process which can be modeled as a bivariate Markov process that can be analyzed probabilistically using Markovian analysis. In this setting, however, we are interested in Bayesian inference on model parameters. We present a computationally tractable approach using Gibbs sampling and demonstrate it by numerical illustrations. We also discuss cases that involve complete and partial data sets on both processes.

Keywords

Bayesian inferenceBayesian reliability analysishidden Markov modelMarkov modulated Markov process
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 3 , July 2021 , pp. 708 - 729
Copyright
© Cambridge University Press 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Ahmed, R. & Fouladirad, M. (2017). Maintenance planning for a deteriorating production process. Reliability Engineering & System Safety 159: 108–118.CrossRefGoogle Scholar
2.Arifoğlu, K. & Özekici, S. (2010). Optimal policies for inventory systems with finite capacity and partially observed Markov-modulated demand and supply processes. European Journal of Operational Research 204: 421–483.CrossRefGoogle Scholar
3.Asmussen, S. (2000). Matrix-analytic models and their analysis. Scandinavian Journal of Statistics 27: 193–226.CrossRefGoogle Scholar
4.Çanakoğlu, E. & Özekici, S. (2010). Portfolio selection in stochastic markets with HARA utility functions. European Journal of Operational Research 201: 520–536.CrossRefGoogle Scholar
5.Çekyay, B. & Özekici, S. (2010). Mean time to failure and availability of semi-Markov missions with maximal repair. European Journal of Operational Research 207: 1442–1454.CrossRefGoogle Scholar
6.Çekyay, B. & Özekici, S. (2012). Optimal maintenance of systems with Markovian mission and deterioration. European Journal of Operational Research 219: 123–133.CrossRefGoogle Scholar
7.Çekyay, B. & Özekici, S. (2012). Performance measures for systems with Markovian missions and aging. IEEE Transactions on Reliability 61: 769–778.CrossRefGoogle Scholar
8.Çekyay, B. & Özekici, S. (2015). Optimal maintenance of semi-Markov missions. Probability in the Engineering and Informational Sciences 29: 77–98.CrossRefGoogle Scholar
9.Çekyay, B. & Özekici, S. (2015). Reliability, MTTF and steady-state availability analysis of systems with exponential lifetimes. Applied Mathematical Modelling 39: 284–296.CrossRefGoogle Scholar
10.Chib, S. (1995). Marginal likelihood from the Gibbs output. Journal of the American Statistical Association 90: 1313–1321.CrossRefGoogle Scholar
11.Çınlar, E. & Özekici, S. (1987). Reliability of complex devices in random environments. Probability in the Engineering and Informational Sciences 1: 97–115.CrossRefGoogle Scholar
12.Çınlar, E., Shaked, M., & Shanthikumar, J.G. (1989). On lifetimes influenced by a common environment. Stochastic Processes and Their Applications 33: 347–359.CrossRefGoogle Scholar
13.Eisen, M. & Tainiter, M. (1963). Stochastic variations in queuing processes. Operations Research 11: 922–927.CrossRefGoogle Scholar
14.Fearnhead, P. & Sherlock, C. (2006). An exact Gibbs sampler for the Markov-modulated Poisson process. Journal of the Royal Statistical Society: Series B 68: 767–784.CrossRefGoogle Scholar
15.Kass, R.E. & Raftery, A.E. (1995). Bayes factors. Journal of the American Statistical Association 90: 773–795.CrossRefGoogle Scholar
16.Landon, J., Özekici, S., & Soyer, R. (2013). A Markov modulated Poisson model for software reliability. European Journal of Operational Research 229: 404–410.CrossRefGoogle Scholar
17.Lefèvre, C. & Milhaud, X. (1990). On the association of the lifelenghts of components subjected to a stochastic environment. Advances in Applied Probability 22: 961–964.CrossRefGoogle Scholar
18.Moler, C. & van Loan, C. (2003). Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45: 3–49.CrossRefGoogle Scholar
19.Neuts, M.F. (1974). A queue subject to extraneous phase changes. Advances in Applied Probability 3: 78–119.CrossRefGoogle Scholar
20.Neuts, M.F. (1994). Matrix-geometric solutions in stochastic models, an algorithic approach. New York: Dover Publications, Inc.Google Scholar
21.Özekici, S. & Soyer, R. (2003). Reliability of software with an operational profile. European Journal of Operational Research 149: 459–474.CrossRefGoogle Scholar
22.Özekici, S. & Soyer, R. (2006). Semi-Markov modulated Poisson process: probabilistic and statistical analysis. Mathematical Methods of Operations Research 64: 125–144.CrossRefGoogle Scholar
23.Pievatolo, A., Ruggeri, F., & Soyer, R. (2012). A Bayesian hidden Markov model for imperfect debugging. Reliability Engineering & System Safety 103: 11–21.CrossRefGoogle Scholar
24.Prabhu, N.U. & Zhu, Y. (1989). Markov-modulated queueing systems. Queueing Systems 5: 215–246.CrossRefGoogle Scholar
25.Purdue, P. (1974). The M/M/1 queue in a Markovian environment. Operations Research 22: 562–569.CrossRefGoogle Scholar
26.Ross, S. (1996). Stochastic processes, 2nd ed. New York: Wiley.Google Scholar
27.Şahinoğlu, M. (1992). Compound-poisson software reliability model. IEEE Transactions on Software Engineering 18: 624–630.CrossRefGoogle Scholar
28.Singpurwalla, N.D. (2006). The hazard potential: introduction and overview. Journal of the American Statistical Association 101: 1705–1717.CrossRefGoogle Scholar
29.van der Weide, J.A.M. & Pandley, M.D. (2011). Stochastic analysis of shock process and modeling of condition-based maintenance. Reliability Engineering & System Safety 96: 619–626.CrossRefGoogle Scholar
30.Zhu, Y. (1994). Markovian queueing networks in a random environment. Operations Research Letters 15: 11–17.CrossRefGoogle Scholar