Generic Equations for a Probabilistic Metric for Random Hardware Failures According to ISO 26262

The International organization for Standardization (ISO) standard 26262:2018 “Road vehicles- Functional safety,” also known as the second edition of ISO 26262:2011, was issued in December 2018. The resulting equation for the probabilistic metric for random hardware failures (PMHF), which describes an average probability per hour over the operational lifetime of an item, was revised in Part 10 of the second edition from that in the first edition, although only a minimal mathematical definition of the PMHF is given. In this paper, we assume that the mathematical definition of the PMHF is the average of the point unavailability density (PUD). We also assume conditions that are not clearly described in the standards, including the repairability of the elements of the item in question. We propose a generic PMHF formula with continuous -time Markov chains (CTMC) and the point unavailability (PUA) function of an item subjected to periodic inspections, which we have developed for the first time in the industry. To validate the mathematical definition of the PMHF and our assumptions, we derive PMHF equations identical to those in Part 10 of the first edition by fixing the intended functionality (IF) as unrepairable. However, we fi nd potential problems with the resulting PMHF formula in the second edition. With regard to the dual point failure (DPF) calculations in the formula, it seems that either the IF or the safety mechanism 1 (SM1) is fixed as unrepairable. Since all DPF cases need to be counted, it is necessary to perform calculations such that both the IF and SM1 are repairable in the initial state to comply with the generic subsystem model described in the second edition. We conclude that the PMHF equation in the second edition may overestimate the PMHF metric because of its overly strict constraints, which could significantly impact the automotive industry, especially with regard to the emergency operation tolerance time intervals (EOTTIs) of fault -tolerant items. Finally, we propose a practical solution to avoid this underestimation.