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Mathematical Foundations of Reliability Theory with Applications in Engineering and...
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Mathematical Foundations of Reliability Theory with Applications in Engineering and Applied Statistics

A Special Issue of Mathematics (ISSN 2227-7390) belonging to the section "E: Applied Mathematics".

Deadline for manuscript submissions: 31 March 2027 | Viewed by 2605

Editors

Prof. Dr. Serguei Maximov

E-Mail Website
Guest Editor
Tecnológico Nacional de México, Instituto Tecnológico de Morelia, PGIIE, Morelia 58120, Michoacán, Mexico
Interests: reliability; electromagnetic engineering
Dr. Francisco Rivas-Dávalos

E-Mail Website
Guest Editor
Tecnogógico Nacional de México, Campus Morelia, Morelia 58120, Michoacán, México
Interests: reliability engineering; failure analysis in electrical networks; cascading failures

Special Issue Information

Dear Colleagues,

Reliability theory has become a cornerstone of modern engineering, applied statistics, and risk analysis. The growing complexity of technological systems—ranging from aerospace and energy infrastructure to biomedical devices and information networks—demands mathematical tools that are both rigorous and applicable to real-world decision-making. This Special Issue aims to bring together cutting-edge contributions that deepen the mathematical foundations of reliability while highlighting their impact on engineering practice and applied statistics.

The scope of the Issue spans both theoretical advances and methodological innovations, with emphasis on problems that bridge pure mathematics and practical applications. We invite papers that address fundamental challenges such as renewal and renewal–reward processes, asymptotic methods for distribution functions in reliability, and optimal maintenance policies under short- and long-horizon planning. Contributions that integrate advanced probability, stochastic processes, asymptotic analysis, and computational methods are particularly welcome.

Applications are expected to cover a broad spectrum: maintenance optimization in complex systems, reliability assessment of renewable energy technologies, survival analysis in biomedical contexts, and reliability-inspired statistical models for emerging data-rich environments. By uniting mathematical challenges with engineering and statistical applications, this Special Issue will showcase the interdisciplinary vitality of reliability theory.

Topics of interest include, but are not limited to, the following:

  • Foundations of reliability mathematics:
    • Renewal and renewal–reward theory;
    • Stochastic processes in reliability analysis;
    • Distributional methods: survival, hazard, and cumulative failure models;
    • Asymptotic expansions, Laplace-type methods, and error bounds.
  • Maintenance and optimization policies:
    • Short- and long-horizon maintenance strategies;
    • Cost-per-unit-time and replacement policies;
    • Block and group maintenance for modular systems;
    • Age-dependent and state-dependent policies.
  • Statistical modeling and inference:
    • Parametric and nonparametric methods in reliability data analysis;
    • Goodness-of-fit and model selection for lifetime distributions;
    • Bayesian approaches to reliability and failure prediction;
    • Reliability in the presence of censored and grouped data.
  • Applications in engineering and applied sciences:
    • Reliability of energy systems and renewable technologies;
    • Reliability in electrical and electronic engineering;
    • Survival and reliability models in biostatistics;
    • Reliability under uncertainty, risk analysis, and decision theory.
  • Stochastic maintenance.
  • Renewal and renewal–reward models.
  • Bayesian reliability.
  • Degradation models.
  • Survival analysis..
  • Reliability in power systems.
  • Statistical inference under censoring.
  • Machine learning for reliability and predictive maintenance.
  • Risk and decision models.

Prof. Dr. Serguei Maximov
Dr. Francisco Rivas-Dávalos
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • reliability theory
  • renewal processes
  • maintenance policies
  • hazard functions
  • asymptotic analysis
  • applied probability
  • reliability statistics
  • engineering applications
  • survival analysis
  • risk and uncertainty

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Published Papers (3 papers)

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45 pages, 25937 KB  
Article
New Power Reliability Modeling via Randomized Progressive First-Failure Beta–Binomial Censoring: Theory, Optimization, and Engineering Applications to Fiber Strengths
by Maysaa Elmahi Abd Elwahab, Osama E. Abo-Kasem, Shuhrah Alghamdi and Ahmed Elshahhat
Mathematics 2026, 14(11), 1803; https://doi.org/10.3390/math14111803 - 23 May 2026
Viewed by 534
Abstract
In modern reliability engineering, modeling bounded lifetime data under realistic experimental conditions is still challenging, especially when censoring schemes and unit removals are random. This study proposes a new and unified reliability framework by combining the flexible powering new power (PNP) distribution with [...] Read more.
In modern reliability engineering, modeling bounded lifetime data under realistic experimental conditions is still challenging, especially when censoring schemes and unit removals are random. This study proposes a new and unified reliability framework by combining the flexible powering new power (PNP) distribution with a grouping-based progressive first-failure mechanism using a beta-binomial random design. The proposed approach explicitly accounts for the randomness in group removals, providing a more realistic description of practical life-testing experiments. Classical estimation is carried out using maximum likelihood methods with the Newton-Raphson algorithm, along with confidence intervals constructed under both standard and log-transformed parameterizations. To increase flexibility in inference, a Bayesian approach is developed based on a joint gamma and shifted log-normal prior, which respects parameter constraints and incorporates prior uncertainty. Since the posterior distributions cannot be obtained in closed form, a Metropolis-Hastings Markov chain Monte Carlo algorithm is used to generate reliable posterior estimates and credible intervals. Additionally, beyond sensitivity analysis, multiple prior robustness diagnostics are incorporated to ensure reliable hyperparameter calibration and to safeguard against prior misspecification. The performance of the proposed estimators is carefully examined through extensive Monte Carlo simulations under different censoring schemes and parameter settings. The simulation results indicate that the proposed Bayesian procedures often provide more stable estimation and shorter interval estimates with competitive coverage probabilities compared with the corresponding classical methods, particularly under moderate-to-heavy censoring settings. To demonstrate its practical usefulness, the proposed model is applied to two real datasets on tensile strength of carbon and polyester fibers, where it provides a good fit and useful insights into material reliability and failure behavior. In the same applications, the practical relevance and superior performance of the proposed distribution are demonstrated, where it outperforms existing bounded versions of several well-known models, including the gamma, Weibull, and Birnbaum-Saunders distributions. Overall, this work contributes to reliability analysis by offering a flexible and computationally efficient framework that accounts for both random censoring and complex lifetime patterns, with potential applications in engineering, materials science, and applied reliability studies. Full article
(This article belongs to the Special Issue Mathematical Foundations of Reliability Theory with Applications in Engineering and Applied Statistics)
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36 pages, 6439 KB  
Article
Modelling Workload and Injury Risk in Elite Touch Rugby with Clustering Effect: A Time-Scaled Shared Frailty Approach
by Tom Huang, Shu Su, Nuttanan Wichitaksorn and Kirsten Spencer
Mathematics 2026, 14(9), 1550; https://doi.org/10.3390/math14091550 - 3 May 2026
Viewed by 511
Abstract
In this study, we propose a general mathematical modelling framework based on the characteristics of elite athletes’ movements in the touch rugby matches to investigate the dynamic relationship between physical workload and injury risk over time. Our framework extends the Cox-based model in [...] Read more.
In this study, we propose a general mathematical modelling framework based on the characteristics of elite athletes’ movements in the touch rugby matches to investigate the dynamic relationship between physical workload and injury risk over time. Our framework extends the Cox-based model in the context of touch rugby by incorporating a time-scaling component and cluster-specific heterogeneity simultaneously. In addition, we allow for the inclusion of covariates (e.g., velocity variation) to capture their effects. We applied our model to high-frequency wearable sensor data collected from 27 elite athletes (15 men and 12 women). The empirical study results show that our model, time-scaled frailty model (TSFM), demonstrates better goodness-of-fit than traditional frailty and Andersen–Gill models. The results reveal that higher velocity variation, particularly during high-intensity phases, and longer time of continuous exposure to the workload spike state significantly increased overload risk, ultimately resulting in injury. It also highlights the importance of individual differences, even under the same exercise intensity. These insights provide coaches with an evidence-based framework for athlete monitoring, allowing for more personalized training loads, tactical deployment, and injury prevention strategies in elite touch rugby environments. Full article
(This article belongs to the Special Issue Mathematical Foundations of Reliability Theory with Applications in Engineering and Applied Statistics)
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48 pages, 3619 KB  
Article
Comparative Assessment of the Reliability of Non-Recoverable Subsystems of Mining Electronic Equipment Using Various Computational Methods
by Nikita V. Martyushev, Boris V. Malozyomov, Anton Y. Demin, Alexander V. Pogrebnoy, Georgy E. Kurdyumov, Viktor V. Kondratiev and Antonina I. Karlina
Mathematics 2026, 14(4), 723; https://doi.org/10.3390/math14040723 - 19 Feb 2026
Cited by 21 | Viewed by 891
Abstract
The assessment of reliability in non-repairable subsystems of mining electronic equipment represents a computationally challenging problem, particularly for complex and highly connected structures. This study presents a systematic comparative analysis of several deterministic approaches for reliability estimation, focusing on their computational efficiency, accuracy, [...] Read more.
The assessment of reliability in non-repairable subsystems of mining electronic equipment represents a computationally challenging problem, particularly for complex and highly connected structures. This study presents a systematic comparative analysis of several deterministic approaches for reliability estimation, focusing on their computational efficiency, accuracy, and applicability. The investigated methods include classical boundary techniques (minimal paths and cuts), analytical decomposition based on the Bayes theorem, the logic–probabilistic method (LPM) employing triangle–star transformations, and the algorithmic Structure Convolution Method (SCM), which is based on matrix reduction of the system’s connectivity graph. The reliability problem is formally represented using graph theory, where each element is modeled as a binary variable with independent failures, which is a standard and practically justified assumption for power electronic subsystems operating without common-cause coupling. Numerical experiments were carried out on canonical benchmark topologies—bridge, tree, grid, and random connected graphs—representing different levels of structural complexity. The results demonstrate that the SCM achieves exact reliability values with up to six orders of magnitude acceleration compared to the LPM for systems containing more than 20 elements, while maintaining polynomial computational complexity. Qualitatively, the compared approaches differ in the nature of the output and practical applicability: boundary methods provide fast interval estimates suitable for preliminary screening, whereas decomposition may exhibit a systematic bias for highly connected (non-series–parallel) topologies. In contrast, the SCM consistently preserves exactness while remaining computationally tractable for medium and large sparse-to-moderately dense graphs, making it preferable for repeated recalculations in design and optimization workflows. The methods were implemented in Python 3.7 using NumPy and NetworkX, ensuring transparency and reproducibility. The findings confirm that the SCM is an efficient, scalable, and mathematically rigorous tool for reliability assessment and structural optimization of large-scale non-repairable systems. The presented methodology provides practical guidelines for selecting appropriate reliability evaluation techniques based on system complexity and computational resource constraints. Full article
(This article belongs to the Special Issue Mathematical Foundations of Reliability Theory with Applications in Engineering and Applied Statistics)
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