# Safety-Oriented System Hardware Architecture Exploration in Compliance with ISO 26262
## 4 Safety-Oriented System Hardware Architecture Exploration Framework
### Problem Formulation  
Before the problem can be formalized, the following notations are defined first:  
* \(n\): The number of safety-related hardware elements in the system;
* \(C(i)\): The \(i\)th safety-related component, where \(1\leq i\leq n\);
* \(\lambda_{C(i)}\): Failure rate of the \(i\)th safety-related component;
* \(\lambda_{S}\), \(\lambda_{SPF}\), \(\lambda_{RF}\), and \(\lambda_{DPFL}\): The failure rates associated with a safe fault, single-point fault (SPF), residual fault (RF), and latent dual-point fault (DPF), respectively;
* \(SPFM_{tar}\), \(LFM_{tar}\), and \(PMHF_{tar}\): The target values for SPFM, LFM, and PMHF in accordance with the target ASIL;
* \(PF_{SPFM_{tar}}\left(t\right),PF_{LFM_{tar}}\left(t\right)\), and \(PF_{PMHF_{tar}}\left(t\right)\) : The target failure probability for achieving \(SPFM_{tar}\), \(LFM_{tar}\), and \(PMHF_{tar}\) safety metrics at mission time \(t\);
* \(SPFM_{it_{e}d}\), \(LFM_{it_{e}d}\) and \(PMHF_{it_{e}d}\): The values of SPFM, LFM, and PMHF in the \(d\)th design iteration, where \(d\geq 0\).  
With the notations defined above, the main goal of the safety-oriented system hardware architecture exploration framework can be formalized as:  
1. Explore the system hardware design space to determine a hardware architecture such that the following requirements can be fulfilled: \(\blacksquare\)\(PMHF_{ite\_d}<PMHF_{tar}\). \(\blacksquare\)\(SPFM_{it_{e}d}\geq SPFM_{tar}\). \(\blacksquare\)\(LFM_{it_{e}d}\geq LFM_{tar}\).  
Based on Table 1, the target values of \(PMHF_{tar}\), \(SPFM_{tar}\) and \(LFM_{tar}\) can be determined according to the target ASIL. For example, if the target ASIL B is selected, then th \(PMHF_{tar}\), \(SPFM_{tar}\), and \(LFM_{tar}\) are specified as \(10^{-7}\) h\({}^{-1}\), 90%, and 60%, respectively. On the other hand, \(PMHF_{ite\_d}\), \(SPFM_{ite\_d}\), and \(LFM_{ite\_d}\) can be derived from the following formulae provided by ISO-26262 for the hardware architecture metrics calculation.  
\(PMHF_{ite\_d}\) can be calculated by the following expression (3)  
\[PMHF_{ite\_d}=\sum\
olimits_{Safety\_Related\ hardware(HW)\ elements}(\lambda_{SPF}+\lambda_{RF}+\lambda_{DPF,L}) \tag{3}\]  
Next, \(SPFM_{ite\_d}\) can be computed by the following expression (4)  
\[SPFM_{ite\_d}=1-\frac{\sum\
olimits_{Safety\_Related\ HW\ elements}( \lambda_{SPF}+\lambda_{RF})}{\sum\
olimits_{Safety\_Related\ HW\ elements}\lambda} \tag{4}\]  
where \(\sum\
olimits_{Safety\_Related\ HW\ elements}\lambda\) represents the total failure rates of all safety-related hardware elements and can be calculated by \(\sum\
olimits_{Safety\_Related\ HW\ elements}\lambda=\sum\
olimits_{Safety\_Related\ HW\ elements}\lambda_{SPF}+\lambda_{RF}+\lambda_{DPF}+\lambda_{S}\) (assuming all failures are independent and follow the exponential distribution).  
Moreover, the following expression can be derived from the safety requirement \(SPFM_{ite\_d}\geq SPFM_{tar}\).  
\[\sum\
olimits_{Safety\_Related\ HW\ elements}(\lambda_{SPF}+\lambda_{RF}) \leq(1-SPFM_{tar})\times\sum\
olimits_{Safety\_Related\ HW\ elements}\lambda \tag{5}\]  
Lastly, the calculation of \(LFM_{ite\_d}\) is based on the following expression (6)  
\[LFM_{ite\_d}=1-\frac{\sum\
olimits_{Safety\_Related\ HW\ elements}( \lambda_{DPF,L})}{\sum\
olimits_{Safety\_Related\ HW\ elements}( \lambda-\lambda_{SPF}-\lambda_{RF})} \tag{6}\]  
Similarly, the following expression can be derived from the safety requirement \(LFM_{ite\_d}\geq LFM_{tar}\).  
\[\sum\
olimits_{Safety\_Related\ HW\ elements}(\lambda_{DPF,L}) \leq(1-LFM_{tar})\times\sum\
olimits_{Safety\_Related\ HW\ elements}(\lambda-\lambda_{SPF}-\lambda_{RF}) \tag{7}\]  
The failure rates \(\lambda_{S}\), \(\lambda_{SPF}\), \(\lambda_{RF}\), and \(\lambda_{DPFL}\) in expressions (3)-(7) can be derived from the process as exhibited in Figure 1 and expressions (1) and (2) as shown in Section 3.