Optimal active redundancy allocation in k-out-of-n system

A k-out-of-n system consisting of n components is one that works if and only if at least k of the n components work. Suppose there are n + r (1 less than or equal to r less than or equal to n) components available of which r will be used for active redundancy. From the given n + r components, r components are chosen to be used as active redundancies, and another r components receive active redundancies (i.e. these r components are bolstered). Consider a k-out-of-n system consisting of the r bolstered and the other n - r original components. The problem of which r components should be used for active redundancy, and where to allocate them in order to maximize the lifetime of the resulting k-out-of-n system is studied. It is shown that under the usual stochastic ordering 'less than or equal to(st)' the first r weakest components should be used for active redundancy and allocated in reverse order to the next r weakest components.