Optimal allocation policy of one redundancy in a n-component series system

It is of great importance to optimize system performance by allocating redundancies in a coherent system in reliability engineering and system security. In this paper, we focus on the problem of how to optimally allocate one active [standby] redundancy in a n-component series system in the sense of stochastic ordering. For the active case, it is showed that allocating the redundancy to the relatively weaker component leads to longer system's lifetime in the likelihood ratio and reversed hazard rate orders, respectively. For the standby case, we show that the redundancy should be allocated to the weakest component of the series system in the likelihood ratio order. Based on these results, two optimal allocation policies are proposed. Also, some numerical examples are presented to explicate the theoretic results established here.