On system failure probability density function

The assessment of the system unreliability is usually accomplished through well-known tools such as block diagram, fault tree, Monte Carlo and others. These methods imply the knowledge of the failure probability density function of each component 'k' (pdf pk). For this reason, possibly, the system failure probability density function (psys) has never been explicitly derived. The present paper fills this gap achieving an enlightening formulation which explicitly gives psys as the sum of (positive) terms representing the complete set of transitions leading the system from an operating to a failed configuration, due to the failure of 'a last' component. As a matter of fact, these are all the independent sequences leading the system to the failure. In our opinion, this formulation is important from both methodological and practical point of views. From the methodological one, a clear insight of the system-vs-components behaviors can be grasped and, in general, the explicit link between psys and pk seems to be a notable result. From a practical point of view, psys allows a rigorous derivation of Monte Carlo algorithms and suggests a systematic tool for investigating the system failure sequences