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Open AccessArticle

Combining Generalized Renewal Processes with Non-Extensive Entropy-Based q-Distributions for Reliability Applications

1
Center for Risk Analysis and Environmental Modeling—CEERMA, Universidade Federal de Pernambuco, Recife PE 50740-550, Brazil
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Department of Production Engineering, Universidade Federal de Pernambuco, Recife PE 50740-550, Brazil
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Department of Mechanical Engineering, University of Chile, Beauchef 850, Santiago, Chile
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Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA
*
Author to whom correspondence should be addressed.
Entropy 2018, 20(4), 223; https://doi.org/10.3390/e20040223
Received: 22 January 2018 / Revised: 12 March 2018 / Accepted: 24 March 2018 / Published: 25 March 2018
(This article belongs to the Special Issue Entropy for Characterization of Uncertainty in Risk and Reliability)
The Generalized Renewal Process (GRP) is a probabilistic model for repairable systems that can represent the usual states of a system after a repair: as new, as old, or in a condition between new and old. It is often coupled with the Weibull distribution, widely used in the reliability context. In this paper, we develop novel GRP models based on probability distributions that stem from the Tsallis’ non-extensive entropy, namely the q-Exponential and the q-Weibull distributions. The q-Exponential and Weibull distributions can model decreasing, constant or increasing failure intensity functions. However, the power law behavior of the q-Exponential probability density function for specific parameter values is an advantage over the Weibull distribution when adjusting data containing extreme values. The q-Weibull probability distribution, in turn, can also fit data with bathtub-shaped or unimodal failure intensities in addition to the behaviors already mentioned. Therefore, the q-Exponential-GRP is an alternative for the Weibull-GRP model and the q-Weibull-GRP generalizes both. The method of maximum likelihood is used for their parameters’ estimation by means of a particle swarm optimization algorithm, and Monte Carlo simulations are performed for the sake of validation. The proposed models and algorithms are applied to examples involving reliability-related data of complex systems and the obtained results suggest GRP plus q-distributions are promising techniques for the analyses of repairable systems. View Full-Text
Keywords: generalized renewal process; q-Exponential distribution; q-Weibull distribution; particle swarm optimization; reliability analysis generalized renewal process; q-Exponential distribution; q-Weibull distribution; particle swarm optimization; reliability analysis
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Lins, I.D.; Moura, M.D.C.; Droguett, E.L.; Corrêa, T.L. Combining Generalized Renewal Processes with Non-Extensive Entropy-Based q-Distributions for Reliability Applications. Entropy 2018, 20, 223.

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