Reliability, the number of spare parts and repair time have a great impact on system availability. In this paper, we observed a repairable system comprised of several components. The aim was to determine the repair rate by emphasizing its stochastic nature. A model for the statistical analysis of the component repair rate in function of the desired level of availability is presented. Furthermore, based on the presented model, the approach for the calculation of probability density functions of maximal and minimal repair times for a system comprised of observed components was developed as an important measure that unambiguously defines the total annual repair time. The obtained generalized analytical expressions that can be used to predict the total repair time for an observed entity are the main contributions of the manuscript. The outputs of the model can be useful for making decisions in which time interval repair or replacement should be done to maintain the system and component availability. In addition to planning maintenance activities, the presented models could be used for service capacity planning and the dynamic forecasting of system characteristics.

As reliance on wind energy intensifies globally, optimizing the efficiency and reliability of wind turbines is becoming vital. This paper explores sophisticated maintenance strategies, crucial for enhancing the operational sustainability of wind turbines. It introduces an innovative approach to maintenance scheduling that utilizes a mathematical model incorporating an alternating renewal process for accurately determining repair rate thresholds. These thresholds are important for identifying optimal maintenance timings, thereby averting failures and minimizing downtime. Central to this study are the obtained generalized analytical expressions that can be used to predict the total repair time for an observed entity. Four key lemmas are developed to establish formal proofs for the probability density function (PDF) and cumulative distribution function (CDF) of repair rates, both above and below critical repair rate thresholds. The core innovation of this study lies in the methodological application of PDFs and CDFs to set repair time thresholds that refine maintenance schedules. The model’s effectiveness is illustrated using simulated data based on typical wind turbine components such as gearboxes, generators, and converters, validating its potential for improving system availability and operational readiness. By establishing measurable repair rate thresholds, the model effectively prioritizes maintenance tasks, extending the life of crucial turbine components and ensuring consistent energy output. Beyond enhancing theoretical understanding, this research provides practical insights that could inform broader maintenance strategies across various renewable energy systems, marking a significant advancement in the field of maintenance engineering

This paper discusses incidence structures and their rank. The aim of this paper is to prove that there exists a regular decomposable incidence structure J = (P, B) of maximum degree depending on the size of the set and a predetermined rank. Furthermore, an algorithm for construction of this structures is given. In the proof of the main result, the points of the set P are shown by Euler's formula of complex number. Two examples of construction the described incidence structures of maximum degree 6 and maximum degree 30 are given.

The research focuses on the optimization of numerical solutions of neutral stochastic differential equations with time delay. Analyzing approaches such as Euler-Maruyama, backward Euler, and θ-Euler-Maruyama methods, the goal is to investigate the characteristics of approximate solutions, especially stability and boundedness. This study contributes to the understanding of the complexity of stochastic processes, offering a perspective for further mathematical modeling and optimization. The study of the characteristics of approximate solutions includes a detailed analysis of their stability and limitations, providing insight into the system’s behavior in dynamic conditions. This analysis lays the foundations for the improvement of numerical methods and more precise modeling of stochastic processes with time delay. The aforementioned approaches, such as the Euler-Maruyama, backward Euler, and θ-Euler-Maruyama methods, provide tools for understanding and solving complex mathematical challenges. Through an interdisciplinary approach, this study sheds light on the field of optimization of numerical solutions, encouraging further development of theoretical and practical aspects of stochastic differential equations.