An importance measure for multi-component systems with Semi-Markov dynamics

We consider semi-Markov reliability models of multi-component systems with a discrete state space, general enough to include systems with maintenance or repair. We assume that for all system states the functioning or failure of each component is specified. In this setup we propose a component importance measure which is close in spirit to the classical steady state Barlow–Proschan importance measure for repairable binary coherent systems. We discuss our importance measure to some extent, highlighting the relation to the classical Barlow–Proschan measure, and present formulas expressing it in terms of quantities easily obtained from the building blocks of the semi-Markov process. Finally an example of a two-component cold standby system with maintenance and repair is presented which illustrates how our importance measure can be used in practical applications.