On relevation transform involved with statistical dependence between two component lifetimes

Krakowski (Rev Fr Autom Inform Rech Opèr. 1973;7:107–120.) introduced the relevation transform for component and active redundancy with independent lifetimes, and except for Johnson and Kotz (Am J Math Manag Sci. 1981;1:155–165; Nav Res Logist. 1983;30:163–169.) most subsequent researches were conducted under this framework. However, it is not uncommon that a component and its active redundancy bear some common stresses due to the environment and thus they have dependent lifetimes. In this note, we equip the involved lifetimes with a survival copula and then clarify the potential difference between the new and classical versions through making stochastic comparison. Moreover, by ordering the lifetime of system with relevation redundancy we also study the way of allocating a relevation redundancy at component level to ultimately improve the system reliability. The present results on series and parallel systems serve as a generalization of the corresponding ones of Belzunce et al. (Appl Stoch Models Bus Ind. 2019;35:492–503.). Several numerical examples are presented to illustrate these findings as well.