Estimating Reliability for Frechet (3+1) Cascade Model

In this paper, the mathematical formula of the reliability function of a special (3+1) cascade model is found, where this model can work with three active components if it is able to cope with the stresses to which it is subjected, while the cascade 3+1 model consists of four components, where the components (A1,A2 and A3) are the basic components in the model and component (A4) is a spare component in an active state of readiness, so when any of the three core components (A1,A2 and A3) fails to copy with the stress they are under and stops working, they are replaced by the standby component (A4) readiness to keep the model running, it was assumed that the random variables of stress and strength follow one of the statistical life distributions, which is the Frechet distribution. The unknown parameters of the Ferchet distribution were estimated by three different estimation methods (maximum likelihood, least square, and regression), and then the reliability function of the model was estimated by these different methods. A Monte Carlo simulation was performed using MATLAB software to compare the results of different estimation methods and find out which methods are the best for estimating the reliability of the model using two statistical criteria: MSE and MAPE and using different sample sizes. After completing the comparison of the simulation results, it was found that the maximum likelihood estimator is the best for estimating the reliability function of the model among the three different estimation methods