Abstract: The current research examines the reliability of the $$P(T<X<Z)$$ system for two stress-strength models which have Dagum distribution, a $$(1+1)$$ cascade model, an active component and a backup component, and a $$(2+1)$$ cascade model, consisting of three elements, two active components and a backup component. The mathematical formula for the reliability was reached of the cascade system $$(1+1)$$ and $$(2+1)$$, is expressed by a component of the strength $$X$$ that be between of two stresses $$T$$ and $$Z$$. It is shown that the reliability of the system increases as the reliability of the components connected increased, and sometimes reliability reached to certain limits. Through the use of the probability $$P(T<X<Z)$$, stress and strength variables follow the Dagum distribution with unknown shape parameters and known scale parameters. In the current study, the maximum likelihood estimation (MLE) and the Method of Moments were used to estimate the unknown shape parameters of the system reliabilities $$R_{11}$$ and $$R_{21}$$ for the stress-strength model. The maximum likelihood and moment methods were evaluated through a comparative study of experimental simulations using the mean square error criterion. Besides, this investigation showed through simulation results that the maximum likelihood estimation is the best in the studied experiments at different sample sizes $$n$$ except some very simple cases. The reliability of the systems that were derived and formulated in the research was compared.

Keywords: Dagum Distribution, Reliability, Estimation, Probability P(T<X<Z), Stress-Strength, Cascade

Pages: 245-251