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

Area of Science:

• Reliability Engineering
• Statistical Modeling
• Non-extensive Statistical Mechanics

Background:

• The Generalized Renewal Process (GRP) is a key probabilistic model for repairable systems, often combined with the Weibull distribution.
• Existing models may have limitations in capturing complex failure intensity behaviors and handling extreme data values.

Purpose of the Study:

• To develop novel GRP models utilizing Tsallis' non-extensive entropy-based q-Exponential and q-Weibull distributions.
• To enhance the modeling capabilities for repairable systems, particularly for data with extreme values and diverse failure patterns.
• To provide advanced alternatives to the traditional Weibull-GRP model.

Main Methods:

• Development of Generalized Renewal Process (GRP) models incorporating q-Exponential and q-Weibull probability distributions.
• Parameter estimation using the maximum likelihood method.
• Optimization via a particle swarm algorithm and validation through Monte Carlo simulations.

Main Results:

• The q-Exponential-GRP serves as an alternative to the Weibull-GRP, adept at handling extreme values due to its power-law behavior.
• The q-Weibull-GRP generalizes existing models, capable of fitting decreasing, constant, increasing, bathtub-shaped, and unimodal failure intensity functions.
• Application to complex system reliability data demonstrated the efficacy of the proposed models.

Conclusions:

• The integration of q-distributions with GRP offers a powerful and flexible framework for analyzing repairable systems.
• These novel models provide superior performance in fitting diverse and complex reliability data compared to traditional methods.
• The developed q-distribution-based GRP models are promising for advanced reliability analysis of complex systems.