Some Useful Approaches in Estimation of Reliability of Stress – Strength Model

Abstract

The stress-strength system is one of the systems used to calculate reliability. In this thesis, we derive five mathematical formulas of the reliability system in the stress-strength model when the strength Y and the stress X are independent and fellow Rayleigh-Pareto distribution RP(λ,β_1) and RP(λ,β_2),where (β_1,β_2) represents the shape of unknown parameters, and λ the scale known common parameter. The models that we have investigated are as follows: 1-The first model is the system contains single component which has strength Y subject to a stress X , R_1=P(Y<X). 2- The second model is the system contains single component which has strength Y subject to two bounded stresses, namely lower X1 and X2, upper bound stresses, respectively, R_2=P(X_1<Y<X_2 ). 3- The third model includes a multi-component system (s-out-k). Y works when at least s of k is subjected to a common stress X, R_((s,k))=P (at least s of Y_1, Y_2, …, Y_k exceed X). 4- The fourth model in which the system contains Kth series which have strength (Y_1,Y_2,Y_3,…,Y_k ) subject to common on stress, R_S=P(X<min⁡(Y_1,Y_2,Y_3,…,Y_k ). 5-The fifth model in which the system contains components third, parallel which have strength Y_1,Y_2 and〖 Y〗_3 subject to common on stress, R_K=P(X<max⁡〖(Y_1,Y_2,Y_3 ))〗. We used some estimation methods such as, the Maximum Likelihood Estimation(MLE), Least Square (LS), Percentile Estimation (PE) and five Shrinkage Sh methods. To estimate the parameter and system reliability of the five proposal types of stress-strength models including have assumed a new method from shrinkage methods. The estimating of reliability system obtained from all of the above approaches are compared using the Mean Square Error (MSE) and Mean Absolute Percentage Error (MAPE) criteria based on Monte-Carlo simulation experiment with different sample sizes. Significantly, the Shrinkage Function estimator fSh estimators have shown better performance compared with other methods and works well in the field of reliability of the stress-strength system