Optimal allocation of active redundancies in rr-out-of-nn systems

Shaked and Shanthikumar [1992. Optimal allocation of resources to nodes of series and parallel systems. Advances in Applied Probability 24, 894–914] and Singh and Singh [1997a. Optimal allocation of resources to nodes of series systems with respect to failure-rate ordering. Naval Research Logistics 44, 147–152] considered the problem of optimal allocation of KK active redundancies to nn independent and identical components of a series system with respect to the usual stochastic order and to the hazard rate order, respectively. In this note we consider the problems of optimal allocation of a rr-out-of-nn system with respect to the usual stochastic order and of optimal allocation of a series system with respect to the reversed hazard rate order. It is shown that for each rr, by balancing the allocation of active redundancies one can stochastically maximize the lifetime of the resulting rr-out-of-nn system, and that there exists an optimal allocation to optimize the reversed hazard rate function of the system for n=2n=2 while such an optimal allocation does not exist for n>2n>2.