Stochastic Constituents for the Probabilistic Metric of Random Hardware Failures in ISO 26262

SUMMARY & CONCLUSIONSThe second edition of ISO 26262, published in 2018, is an international standard for the functional safety of electrical and/or electronic (E/E) systems within road vehicles. One of the revisions is the probabilistic metric for random hardware failures (PMHF) formula. We hereafter refer to the PMHF formula in the first edition as the standard’s PMHF formula 1 (SF1) and to the PMHF formula in the second edition as the standard’s PMHF formula 2 (SF2) to compare with our PMHF formulas.The standard’s PMHF formulas, SF1 and SF2, have been widely considered to be mathematical PMHF definitions in the industry. However, we identified the unwritten mathematical PMHF definition in the first edition and demonstrated that SF1 was only a resulting equation derived from our PMHF definition for the first time in the industry.The first edition implied the repairability assumptions of SF1; the intended functionality (IF) is unrepairable, and the safety mechanism (SM) is repairable. Thus, we built an Intended Functionality Unrepairable (IFU) model with the same repairability assumptions in SF1 based on a continuous-time Markov chain (CTMC) for a general subsystem and proposed a method, which we call the CTMC-based method, to derive the PMHF formula using a stochastic integration of the unavailability density. We derived a PMHF formula, hereafter referred to as Atsushi’s PMHF formula 1 (AF1), of an IFU model corresponding to SF1. The consistency between SF1 and AF1 indicates that SF1 can be derived from the IFU model.We then built an Intended Functionality Repairable (IFR) model where both IF and SM were repairable for the purpose of validating SF2. We derived the PMHF formula, hereafter referred to as Atsushi’s PMHF formula 2 (AF2), of an IFR model using the identical methodology that proved the consistency between SF1 and AF1. Unlike the consistency of these two formulas of SF1 and AF1, AF2 is not consistent with SF2. We focus on the validity of AF2 in this paper, although the discrepancy might be attributed to SF2.Here, we decompose the verified SF1 based on the IFU model into each stochastic constituent that is IF-related and SM-related to validate AF2. We then derive the PMHF formula, hereafter referred to as Atsushi’s PMHF formula 3 (AF3), for a combined element of the IF and SM based on the stochastic constituents for the case where the IF and SM are both repairable. As a result, AF2 and AF3 are identical.In conclusion, we established a novel PMHF formula, AF3, where both elements are repairable by decomposing SF1 into stochastic constituents and applying them to a model combining the IF and SM. The fact that the same formula can be obtained with the simpler method proposed in this paper in addition to the conventional CTMC-based method reinforces the validity of AF2.In our previous papers, we identified specific numerical advantages of the methodology based on AF2 over standards with respect to design constraints for fault-tolerant systems, which is 31-fold smaller than the standard. Therefore, we hope that this paper enables engineers to accurately calculate the PMHF, thereby reducing excessive design constraints on E/E systems applying ISO 26262.