Optimal allocation of active redundancies in -out-of- 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 K active redundancies to n 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 r -out-of- n 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 r , by balancing the allocation of active redundancies one can stochastically maximize the lifetime of the resulting r -out-of- n system, and that there exists an optimal allocation to optimize the reversed hazard rate function of the system for n = 2 while such an optimal allocation does not exist for n > 2 .