Point Unavailability and its Density Processes with Periodic Inspections in ISO 26262

ISO 26262 is an international functional safety standard for electrical and electronic systems in automotive vehicles. Part 5 of the standard covers the hardware design and verification phase of the safety lifecycle, which requires the probabilistic metric for random hardware failures (PMHF) as a quantitative safety benchmark. However, this standard provides only the resultant formula for calculating the PMHF and does not offer a clear definition or process for the derivation of the PMHF. In our prior studies, we clarified the mathematical definition and derivation of the PMHF formula. We introduced the concept of the stochastic process of the point unavailability density (PUD) and proposed a definition of the PMHF as the average PUD over the lifetime of a vehicle. This led to the development of our original PMHF formula. We also developed the deterministic point unavailability (PUA) formula and applied to derive our PMHF formula. Our approach not only identified potential inaccuracies within the standard's PMHF formula but also demonstrated improved efficiency in the design time required for fault-tolerant subsystems. In this paper, we derive the probabilistic formulas for the PUA and PUD in periodic inspection and repair (PIR) maintenance strategies. These formulas have recursive properties and are revealed to be implicit functions. The deterministic PUA and PUD formulas for the derivation of the PMHF were employed in our previous papers. In contrast, this paper evaluates the errors between those formulas and the probabilistic ones. It demonstrates that the deterministic formulas in the earlier paper are accurate, with no practical issues in the calculation of the PMHF. To understand the PMHF, it is crucial to have clear insight into the PUA and PUD, from which the PMHF is derived, under the PIR strategy. Therefore, this study details the behaviors of these four stochastic processes —the probabilistic and deterministic formulas of the PUA and PUD— aiming to improve automobile safety design through the proposed PMHF formula.