Active-redundancy allocation in systems

An effective way of improving the reliability of a system is the allocation of active redundancies. Let X₁, X₂ be s-independent lifetimes of the components C₁ and C₂, respectively, which form a series system. Let us denote U₁ = min(max(X₁,X),X₂) and U₂ = min(X₁, max(X₂, X)), where X is the lifetime of a redundancy (say R) s-independent of X₁ and X₂. That is, U₁(U₂) denote the lifetime of a system obtained by allocating R to C₁(C₂) as an active redundancy. Singh and Misra (1994) considered the criterion where C₁ is preferred to C₂ for the allocation of R as active redundancy if P(U₁ > U₂) ≥ P(U₂ > U₁). In this paper, we use the same criterion of Singh and Misra (1994). We investigate the allocation of one active redundancy when it differs depending on the component with which it is to be allocated. We also compare the allocation of two active redundancies (say R₁ and R₂) in two different ways; that is, R₁ with C₁ & R₂ with C₂, and viceversa. For this case, the hazard rate order plays an important role. We furthermore consider the allocation of active redundancy to k-out-of-n: G systems.