The stress-strength model is a commonly utilized topic in reliability studies. In many reliability analyses involving stress-strength models, it is typically assumed that the stress and strength variables are unrelated. Nevertheless, this assumption is often impractical in real-world scenarios. This research assumes that the strength and stress variables follow the Pareto distribution, and a Gumbel copula is employed to represent their relationship. Additionally, the data is gathered through the Type-I progressively hybrid censoring scheme. The method of maximum likelihood estimation is used for point estimation, while asymptotic and Bootstrap percentile confidence intervals are employed for interval estimation of the unknown parameters and system reliability. Simulation is employed to assess the effectiveness of the suggested estimators. Subsequently, an actual dataset is examined to showcase the practicality of the stress-strength model. Simulation is employed to assess the effectiveness of the suggested estimators. Subsequently, a real dataset is examined to demonstrate the practicality of the stress-strength model.

This study focuses on estimating the reliability of a Multicomponent stress-strength model using two Bayesian approaches‎: ‎E-Bayesian and hierarchical Bayesian‎. ‎This model follows Inverse Rayleigh distributions with distinct parameters‎. ‎Additionally‎, ‎the efficiency of the proposed methods is compared by employing Monte Carlo simulation and analyzing a data set.

The lattice Boltzmann models, especially the pseudopotential models, have been developed to investigate Multicomponent multiphase fluids in presence of phase change process. However, the interparticle force between different components causes compressibility error in the non-phase-change component. This restricts the model capability in quantitative analysis of the physical foaming process, such as expansion rate and decay time. In the present study, a Multicomponent multiphase pseudopotential phase change model (the MMPPCM) is improved by introducing an effective mass form of high-pressure-difference Multicomponent model in the non-phase-change component. The improved model is compared with the MMPPCM based on simulations of the phase change process of static and moving fluids, as well as the physical foaming process. Density variation of non-phase-change component and its effect on flow field characteristics are analyzed during the phase change process. Simulation results of physical foaming process lead to about 10% ~ 20% reduction of the compressibility error for the improved model as compared with the results of MMPPCM. The improved model also enhances the computational stability of phase change simulation of the static droplets.