In this paper we develop a new Taylor series expansion method for computing model output metrics under epistemic uncertainty in the model input parameters. Specifically, we compute the expected value and the variance of the stationary distribution associated with Markov reliability models. In the multi-parameter case, our approach allows to analyze the impact of correlation between the uncertainty on the individual parameters the model output metric. In addition, we also approximate true risk by using the Chebyshev’ inequality. Numerical results are presented and compared to the corresponding Monte Carlo simulations ones.
Keywords: Markov reliability model, epistemic uncertainty, correlation, risk analysis, fundamental matrix, Taylor series expansion, Monte Carlo simulation
@article{RO_2021__55_S1_S593_0,
author = {Bachi, Katia and Abbas, Karim and Heidergott, Bernd},
title = {Statistical {Taylor} series expansion: {An} approach for epistemic uncertainty propagation in {Markov} reliability models},
journal = {RAIRO. Operations Research},
pages = {S593--S624},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
doi = {10.1051/ro/2019091},
mrnumber = {4223091},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ro/2019091/}
}
TY - JOUR AU - Bachi, Katia AU - Abbas, Karim AU - Heidergott, Bernd TI - Statistical Taylor series expansion: An approach for epistemic uncertainty propagation in Markov reliability models JO - RAIRO. Operations Research PY - 2021 SP - S593 EP - S624 VL - 55 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ro/2019091/ DO - 10.1051/ro/2019091 LA - en ID - RO_2021__55_S1_S593_0 ER -
%0 Journal Article %A Bachi, Katia %A Abbas, Karim %A Heidergott, Bernd %T Statistical Taylor series expansion: An approach for epistemic uncertainty propagation in Markov reliability models %J RAIRO. Operations Research %D 2021 %P S593-S624 %V 55 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ro/2019091/ %R 10.1051/ro/2019091 %G en %F RO_2021__55_S1_S593_0
Bachi, Katia; Abbas, Karim; Heidergott, Bernd. Statistical Taylor series expansion: An approach for epistemic uncertainty propagation in Markov reliability models. RAIRO. Operations Research, Tome 55 (2021), pp. S593-S624. doi: 10.1051/ro/2019091
, and , A functional approximation for the queue. Discrete Event Dyn. Syst. 23 (2013) 93–104. | MR | Zbl | DOI
, Stable computation with the fundamental matrix of a markov chain. SIAM J. Matrix Anal. App. 22 (2000) 230–241. | MR | Zbl | DOI
, Multivariate quadratic forms of random vectors. J. Multivariate Anal. 87 (2003) 2–23. | MR | Zbl | DOI
, and , Sensitivity analysis of reliability and performability measures for multiprocessor systems. In: Vol. 16 of ACM SIGMETRICS Performance Evaluation Review. ACM (1988) 177–186. | DOI
and , The deviation matrix of a continuous-time markov chain. Probab. Eng. Inf. Sci. 16 (2002) 351–366. | MR | Zbl | DOI
and , A parametric uncertainty analysis method for markov reliability and reward models. IEEE Trans. Reliab. 61 (2012) 634–648. | DOI
, Importance and sensitivity analysis in assessing system reliability. IEEE Trans. Reliab. 39 (1990) 61–70. | Zbl | DOI
and , Sensitivity and uncertainty analysis in performability modelling. In: Proceedings 11th Symposium on Reliable Distributed Systems, 1992. IEEE (1992) 93–102.
and , Sensitivity and uncertainty analysis of markov-reward models. IEEE Trans. Reliab. 44 (1995) 147–154. | DOI
and , Taylor series expansions for stationary markov chains. Adv. Appl. Probab. 35 (2003) 1046–1070. | MR | Zbl | DOI
, and , Series expansions for continuous-time markov processes. Oper. Res. 58 (2010) 756–767. | MR | Zbl | DOI
, Accurate computation of the fundamental matrix of a markov chain. SIAM J. Matrix Anal. App. 16 (1995) 954–963. | MR | Zbl | DOI
and , What is fundamental for markov chains: first passage times, fundamental matrices, and group generalized inverses. In: Computations with Markov Chains. Springer (1995) 151–161. | Zbl
and , Numerical solution of linear equations arising in markov chain models. ORSA J. Comput. 1 (1989) 52–60. | Zbl | DOI
, Moments and cumulants of the multivariate normal distribution. Stochastic Anal. App. 6 (1988) 273–278. | MR | Zbl | DOI
and , A new formula for the deviation matrix, chapter 36 of Probability Statistics and Optimization. Wiley (1994). | MR | Zbl
, Generalized inverses and their application to applied probability problems. Linear Algebra App. 45 (1982) 157–198. | MR | Zbl | DOI
, On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. Biometrika 12 (1918) 134–139. | DOI
, From moments of sum to moments of product. J. Multivariate Anal. 99 (2008) 542–554. | MR | Zbl | DOI
and , Finite Markov Chains. D van Nostad Co. Inc., Princeton, NJ (1960). | MR | Zbl
and , Finite continuous time markov chains. Theory Probab. App. 6 (1961) 101–105. | Zbl | MR | DOI
, and , A divide and conquer approach to computing the mean first passage matrix for markov chains via perron complement reductions. Numer. Linear Algebra App. 8 (2001) 287–295. | MR | Zbl | DOI
and , On deviation matrices for birth–death processes. Prob. Eng. Inf. Sci. 15 (2001) 239–258. | MR | Zbl | DOI
, Iterative aggregation/disaggregation methods for computing some characteristics of markov chains. II. Fast convergence. Appl. Numer. Math. 45 (2003) 11–28. | MR | Zbl | DOI
, , and , An isserlis theorem for mixed gaussian variables: application to the auto-bispectral density. J. Stat. Phys. 136 (2009) 89–102. | MR | Zbl | DOI
, , and , A general isserlis theorem for mixed-gaussian random variables. Stat. Probab. Lett. 81 (2011) 1233–1240. | MR | Zbl | DOI
and , A non-obtrusive method for uncertainty propagation in analytic dependability models. In: Proceedings of 4th Asia–Pacific Symposium on Advanced Reliability and Maintenance Modeling (2010).
and , Development of computational algorithm for multiserver queue with renewal input and synchronous vacation. Appl. Math. Modell. 40 (2016) 1137–1156. | MR | DOI
and , Modeling for Reliability Analysis: Markov Modeling for Reliability, Maintainability, Safety, and Supportability. Wiley (1998).
, and , Performance and Reliability Analysis of Computer Systems: An Example-based Approach Using the SHARPE Software Package. Springer Science & Business Media (1996). | Zbl
, Kronecker product permutation matrices and their application to moment matrices of the normal distribution. J. Multivariate Anal. 87 (2003) 177–190. | MR | Zbl | DOI
, , and , Analysis of jackson networks with infinite supply and unreliable nodes. Queue. Syst. 87 (2017) 181–207. | MR | DOI
and , Recursive algorithm for the fundamental/group inverse matrix of a markov chain from an explicit formula. SIAM J. Matrix Anal. App. 23 (2001) 209–224. | MR | Zbl | DOI
, and , A probability bound estimation method in markov reliability analysis. IEEE Trans. Reliab. 34 (1985) 257–261. | Zbl | DOI
, A generalized isserlis theorem for location mixtures of gaussian random vectors. Stat. Probab. Lett. 82 (2012) 67–71. | MR | Zbl | DOI
, and , Uncertainty analysis in reliability modeling. In: Proceedings Annual Reliability and Maintainability Symposium, 2001. IEEE (2001) 229–234.
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