Measures of importance with applications to inspection policies

A new measure of importance for a component (module) in a general coherent system, that quantifies the role of that component in the failure of the system, is proposed. A general expression for that measure is derived and its properties both for general model and for some special structures and/or lifetimes are studied. The relationship of the new measure of importance to a similar measure proposed by (Barlow and Proschan, 1975) is studied. Also some new properties of Barlow and Proschan's measure of component importance are derived. Also a computer algorithm for the computation of measure of importance is given for practical convenience.; As a natural extension, measures of importance of a group of components (modules) based on the number of failed components among a group G by the time of the system failure, denoted by NG, are studied. A useful expression of P(NG ≥ l) (1 ≤ l ≤ r) when the reliability function of the system is symmetric in the arguments corresponding to all components in G is given and then several monotonicity properties of P(NG ≥ l) are presented. Then some properties of NG when the reliability function of the system is not necessarily symmetric in the arguments corresponding to all components in G are investigated. An expression for expected value and variance of NG is given and interesting results for a k-out-of-n system are derived.; Finally, the application of the new measures of importance of component (group of components) in determining "optimal" inspection and repair policies is given, where inspection is performed at the time of system failure. The systems that are considered are 2-out-of-n system, bridge system and r-out-of-n system.