The very flexible Weibull (VF-W) distribution is formulated by expressing its cumulative risk function as a logarithmic composite of auxiliary cumulative risks, making the model particularly well-suited for modeling heterogeneous life behaviors. This model admits a remarkably flexible hazard structure, capable of generating monotone increasing, unimodal (increase-then-decrease), and multi-turning-point shapes, thereby capturing complex failure behaviors far beyond those allowed by the classical Weibull distribution. This paper presents a comprehensive inferential study of the VF-W model through the unified progressive hybrid (UPH) censoring framework for modeling shock-type lifetime data. The UPH scheme integrates the advantages of Type-II, generalized hybrid, and progressive hybrid censoring mechanisms into a unified structure that ensures efficiency and adaptability in reliability testing. Classical inference is developed through maximum likelihood estimation with asymptotic interval construction, while Bayesian inference is performed using independent gamma priors and a Markov iterative algorithm. Extensive Monte Carlo experiments are conducted to evaluate the finite-sample performance of both approaches under various censoring intensities, revealing that the Bayesian MCMC-based estimators and their highest posterior density intervals provide superior precision, coverage, and robustness. The proposed VF-W model using UPH-based strategy is further validated through the analysis of a real shocks dataset, where it demonstrates a comparative performance improvement over existing models. The VF-W model exhibits stable parameter estimation under diverse censoring levels, indicating robustness in incomplete-data scenarios. Furthermore, the model maintains analytical tractability, offering closed-form expressions for key reliability measures, which facilitates practical implementation in different scenarios. The results confirm the VFW model’s strong potential as a unifying and computationally stable tool for reliability modeling, particularly in complex engineering and physical systems operating under stochastic shock environments.

In recent years, fog computing has become an important environment for dealing with the Internet of Things. Fog computing was developed to handle large-scale big data by scheduling tasks via cloud computing. Task scheduling is crucial for efficiently handling IoT user requests, thereby improving system performance, cost, and energy consumption across nodes in cloud computing. With the large amount of data and user requests, achieving the optimal solution to the task scheduling problem is challenging, particularly in terms of cost and energy efficiency. In this paper, we develop novel strategies to save energy consumption across nodes in fog computing when users execute tasks through the least-cost paths. Task scheduling is developed using modified artificial ecosystem optimization (AEO), combined with negative swarm operators, Salp Swarm Algorithm (SSA), in order to competitively optimize their capabilities during the exploitation phase of the optimal search process. In addition, the proposed strategy, Enhancement Artificial Ecosystem Optimization Salp Swarm Algorithm (EAEOSSA), attempts to find the most suitable solution. The optimization that combines cost and energy for multi-objective task scheduling optimization problems. The backpack problem is also added to improve both cost and energy in the iFogSim implementation as well. A comparison was made between the proposed strategy and other strategies in terms of time, cost, energy, and productivity. Experimental results showed that the proposed strategy improved energy consumption, cost, and time over other algorithms. Simulation results demonstrate that the proposed algorithm increases the average cost, average energy consumption, and mean service time in most scenarios, with average reductions of up to 21.15% in cost and 25.8% in energy consumption.

This paper proposes a flexible probability mass function for modeling count data, particularly over-dispersed and asymmetric observations. A novel two-parameter discrete distribution, the discrete power inverted Topp–Leone distribution, is presented using a survival discretization technique. The following statistical characteristics are examined: factorial moments, probability-generating function, quantiles, mean, variance, mean residual life, and entropy measures. The best estimators of the unknown parameters were obtained using various techniques, such as maximum likelihood, moments, least squares, Anderson–Darling, and Cramér-von Mises. A simulation study showed that the accuracy of the estimates improves with larger samples, although higher parameter values may affect precision. The findings indicate that the efficiency of these estimation methods varies under different conditions. Applications to liver lesions, chromatid aberrations, and criminal sociology datasets confirm the model's usefulness for discrete count data in fields such as social sciences, pharmacology, and environmental health. Finally, for modeling count data, the new probabilistic model can be employed as a competitive alternative to other existing distributions.

Healthcare cost acceleration and resource allocation issues have worsened across European health systems, where a small group of patients drives excessive healthcare spending. The prediction of high-cost utilization patterns is important for the sustainable management of healthcare and focused intervention measures. The aim of our study was to derive and validate machine learning algorithms for high-cost healthcare utilization prediction based on detailed administrative data and by comparing three algorithmic methods for the best risk stratification performance. The research analyzed extensive insurance beneficiary records which compile data from health group collective funds operated by non-life insurers across EU countries, across multiple service classes. The definition of high utilization was equivalent to the upper quintile of overall health expenditure using a moderate cost threshold. The research applied three machine learning algorithms, namely logistic regression using elastic net regularization, the random forest, and support vector machines. The models used a comprehensive set of predictor variables including demographics, policy profiles, and patterns of service utilization across multiple domains of healthcare. The performance of the models was evaluated using the standard train–test methodology and rigorous cross-validation procedures. All three models demonstrated outstanding discriminative ability by achieving area under the curve values at near-perfect levels. The random forest achieved the best test performance with exceptional metrics, closely followed by logistic regression with comparable exceptional performance. Service diversity proved to be the strongest predictor across all models, while dentistry services produced an extraordinarily high odds ratio with robust confidence intervals. The group of high utilizers comprised approximately one-fifth of the sample but demonstrated significantly higher utilization across all service classes. Machine learning algorithms are capable of classifying patients eligible for the high utilization of healthcare services with nearly perfect discriminative ability. The findings justify the application of predictive analytics for proactive case management, resource planning, and focused intervention measures across private group health insurance providers in EU countries.