In this paper, the n-cascade with P(X<Y<Z) is used to find the reliability of stress-strength system, where X and Z are the strengths of a component subjected to stress Y. The n-cascade stress-strength system reliability expression is obtained for four different distributions which are [Exponential Pareto, Inverted Exponential, Exponentiated Invers Rayleigh, Frechet]. For some specific values of the parameters of the distribution, the numerical values of the four different reliabilities have also been computed, provided, and discussed.

In this article we derive two reliability mathematical expressions of two kinds of s-out of -k stress-strength model systems; and . Both stress and strength are assumed to have an Inverse Lomax distribution with unknown shape parameters and a common known scale parameter. The increase and decrease in the real values of the two reliabilities are studied according to the increase and decrease in the distribution parameters. Two estimation methods are used to estimate the distribution parameters and the reliabilities, which are Maximum Likelihood and Regression. A comparison is made between the estimators based on a simulation study by the mean squared error criteria, which revealed that the maximum likelihood estimator works the best.

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