Optimal redundancy allocations for series systems under hierarchical dependence structures

Redundancy allocations strategies are commonly used in reliability engineering to enhance the performance of reliability systems. This paper makes use the hierarchical copula to characterize the dependence structures among the components and hot/cold standbys in series systems. Optimal allocation policies of hot and cold standbys are presented for series systems with dependent components and redundancies under certain conditions imposed on the hierarchical copula as well as the components and spares reliabilities. It is shown that, under certain mild conditions imposed on the copula function and lifetimes of components and redundancies, the optimal allocations should be following the way that the resultant lifetimes for all nodes should be balanced as much as possible to enhance the system's performance. The results extend some related ones well-known in the literature from the independent case to the dependent setting. Some numerical examples are presented as illustrations. We also present a real application in optimal allocations of redundant wires for improving the reliability of high-voltage electricity transmission network systems.