On allocating redundancies to k -out-of- n reliability systems

For two components in series and one redundancy with their lifetimes following the proportional hazard models, we build the likelihood ratio order and the hazard rate order for lifetimes of the redundant systems. Also, for k-out-of- n system with components' lifetimes having the arrangement increasing joint density and the redundancies having identically distributed lifetimes, allocating more redundancies to weaker components is shown to help improve the system's reliability.