From differential to difference importance measures for Markov reliability models

This paper presents the development of the differential importance measures (DIM), proposed recently for the use in risk-informed decision-making, in the context of Markov reliability models. The proposed DIM are essentially based on directional derivatives. They can be used to quantify the relative contribution of a component (or a group of components, a state or a group of states) of the system on the total variation of system performance provoked by the changes in system parameters values. The estimation of DIM at steady state using only a single sample path of a Markov process is also investigated. A numerical example of a dynamic system is finally introduced to illustrate the use of DIM, as well as the advantages of proposed evaluation approaches.

[1]  J. Meyer.  The Role of the Group Generalized Inverse in the Theory of Finite Markov Chains , 1975 .

[2]  Emanuele Borgonovo,et al.  Comparison of global sensitivity analysis techniques and importance measures in PSA , 2003, Reliab. Eng. Syst. Saf..

[3]  A. Barros,et al.  New Results on the Differential Importance Measures of Markovian Systems , 2008 .

[4]  Xi-Ren Cao,et al.  Algorithms for sensitivity analysis of Markov systems through potentials and perturbation realization , 1998, IEEE Trans. Control. Syst. Technol..

[5]  Martin Jacobsen.  Statistical Analysis of Counting Processes , 1982 .

[6]  Qing Zhang,et al.  Continuous-Time Markov Chains and Applications , 1998 .

[7]  Joanne Bechta Dugan,et al.  Approximate sensitivity analysis for acyclic Markov reliability models , 2003, IEEE Trans. Reliab..

[8]  Aarnout Brombacher,et al.  Probability... , 2009, Qual. Reliab. Eng. Int..

[9]  Luca Podofillini,et al.  First-order differential sensitivity analysis of a nuclear safety system by Monte Carlo simulation , 2005, Reliab. Eng. Syst. Saf..

[10]  C. D. Meyer.  Sensitivity of the Stationary Distribution of a Markov Chain , 1994, SIAM J. Matrix Anal. Appl..

[11]  Christophe Bérenguer,et al.  Importance measure on finite time horizon and application to Markovian multistate production systems , 2008 .

[12]  Enrico Zio,et al.  Monte Carlo estimation of the differential importance measure: application to the protection system of a nuclear reactor , 2004, Reliab. Eng. Syst. Saf..

[13]  Xi-Ren Cao,et al.  Perturbation realization, potentials, and sensitivity analysis of Markov processes , 1997, IEEE Trans. Autom. Control..

[14]  Ajit Srividya,et al.  Importance measures in ranking piping components for risk informed in-service inspection , 2003, Reliab. Eng. Syst. Saf..

[15]  HU Qing-lin.  Application of Mathematics , 2004 .

[16]  Luca Podofillini,et al.  Accounting for components interactions in the differential importance measure , 2006, Reliab. Eng. Syst. Saf..

[17]  Emanuele Borgonovo,et al.  A new importance measure for risk-informed decision making , 2001, Reliab. Eng. Syst. Saf..

[18]  Christophe Bérenguer,et al.  Reliability importance analysis of Markovian systems at steady state using perturbation analysis , 2008, Reliab. Eng. Syst. Saf..

[19]  LIYI DAI,et al.  Rate of Convergence for Derivative Estimation of Discrete-Time Markov Chains via Finite-Difference Approximation with Common Random Numbers , 1997, SIAM J. Appl. Math..

[20]  Emanuele Borgonovo,et al.  The reliability importance of components and prime implicants in coherent and non-coherent systems including total-order interactions , 2010, Eur. J. Oper. Res..

[21]  CARL D. MEYER,et al.  The Condition of a Finite Markov Chain and Perturbation Bounds for the Limiting Probabilities , 1980, SIAM J. Algebraic Discret. Methods.

[22]  Xi-Ren Cao.  The Maclaurin series for performance functions of Markov chains , 1998, Advances in Applied Probability.

[23]  Paul S. Dwyer,et al.  On Errors in Matrix Inversion , 1953 .

[24]  M. Sørensen,et al.  Exponential Families of Stochastic Processes , 1997 .

[25]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.