Generalised importance measures for multi-state elements based on performance level restrictions

In this paper we consider some commonly used importance measures in a generalised version proposed by some of the authors for application to multi-state systems constituted by multi-state elements. Physically, these measures characterize the importance for a multi-state element of achieving a given level of performance and their definitions entail evaluating the system availability and/or performance when the functioning of the element of interest is restricted in performance. With reference to a predefined threshold of element performance, two different types of restrictions are considered. The first one limits the elementsʹ reachable states to those corresponding to performances either larger or not larger than the threshold level. The second one allows the element to visit all its states but limits its performance to values larger or not larger than the performance threshold. An approach based on the universal generating function technique is proposed for the evaluation of the introduced importance measures. A numerical application is provided in order to highlight the informative content of the introduced measures.