Active redundancy allocation in systems

An effective way of improving the reliability of a system is the allocation of active redundancy. Let $X_{1}$, $X_{2}$ be independent lifetimes of the components $C_{1}$ and $C_{2}$, respectively, which form a series system. Let denote $U_{1} = \min ( \max (X_{1},X),X_{2})$ and $U_{2} = \min (X_{1},\max (X_{2},X))$, where X is the lifetime of a redundancy (say S) independent of $X_{1}$ and $X_{2}$. That is $U_{1}(U_{2})$ denote the lifetime of a system obtained by allocating S to $C_{1}(C_{2})$ as an active redundancy. Singh and Misra (1994) considered the criterion where $C_{1}$ is preferred to $C_{2}$ for redundancy allocation if $P(U_{1} > U_{2})\geq P(U_{2} > U_{1})$. In this paper we use the same criterion of Singh and Misra (1994) and we investigate the allocation of one active redundancy when it differs depending on the component with which it is to be allocated. We find sufficient conditions for the optimization which depend on the components and redundancies probability distributions. We also compare the allocation of two active redundancies (say $S_{1}$ and $S_{2}$) in two different ways, that is $S_{1}$ with $C_{1}$ and $S_{2}$ with $C_{2}$ and viceversa. For this case the hazard rate order plays an important role. We obtain results for the allocation of more than two active redundancies to a k-out-of-n systems.