Comparisons of coherent systems with two types of heterogeneous components having proportional reversed hazard rates

The comparison of coherent systems in terms of stochastic orders is vital in reliability theory. While there is a considerable amount of literature devoted to comparing systems with homogeneous and independent components, real-world systems often consist of heterogeneous components. Hence, this article aims to investigate systems with heterogeneous and independent components, as well as, those with heterogeneous and dependent components. For this purpose, we consider systems comprise of three components, which are of two different types of components, namely two components of type A and one component of type B. The system's lifetime distribution is represented using the failure signature when the components are independent, which is a function of the component's life distribution. However, when the components are dependent, the system's lifetime distribution is represented using copula and diagonal sections. Additionally, distorted distributions are utilized to enable distribution-free stochastic comparisons. Using these representations, we compare systems with components having proportional reversed hazard rates, in three scenarios: (i) when components are heterogeneous and independent; (ii) when components are heterogeneous and dependent; and finally, (iii) comparing systems with homogeneous and independent components with those that have heterogeneous components. To illustrate the applicability of these results, we provide some examples and applications.