Ordering properties concerned with three dimensional random vectors coupled by Archimedean copulas and applications

For a three dimensional nonnegative random vector with an Archimedean copula, one can construct different random variables by consecutively applying bivariate maxima, minima or addition to its marginals. This paper studies ordering properties of such random variables. The present results successfully improve or extend the results on mutually independent random variables in the related literature. Applications in allocating one redundancy to series systems and merging channels in traffic flows regulating are presented as well.