Stochastic Comparisons of Series and Parallel Systems with Generalized Exponential Components

This paper examines the problem of the stochastic comparison of series and parallel systems with $s$-independent heterogeneous generalized exponential components. The results established here are developed in three directions. First, we consider a system with possibly different shape and scale parameters, and obtain some ordering results when its matrix of parameters changes to another matrix, in the certain mathematical sense. Next, by using the concept of vector majorization and related orders, we establish various ordering results for the comparisons of series and parallel systems, when their component's lifetimes have either the same shape parameters with possibly different scale parameters, or the same scale parameters with possibly different shape parameters. Finally, some of the known results on various stochastic orderings between parallel systems in the exponential case are extended to the case when the lifetimes of components follow the generalized exponential distributions. The results of this paper can be used in practical situations to replace components of series and parallel systems by new components, or to find various bounds for the important aging characteristics of these systems.