Calculations for Functional Safety
Quantities, Formulas and Methods

B Supplements for system failure rate calculation with fault trees

B.1 Improvement for very high unavailability

Formula (65) from section 6 provides somewhat overly conservative results for large unavailabilities. Since large unavailabilities are always a sign of improperly designed systems, this is not relevant in practice. Nevertheless, one more formula should be mentioned here which provides less conservative results.

If instead of the failure rates one uses the failure frequencies \(w(t)\) which are conditional with respect to the recovery, then one can first calculate the system failure frequency \(w_{\mathrm {sys}}(t)\) using the following formula:

\begin{equation} \label {eq:h_sys_w} \begin{split} h_{\mathrm {sys}}(t) &= \sum _{i=1}^{n_{\mathrm {MCS}}} \frac {w_{\mathrm {MCS_i}}(t)}{1-Q_{\mathrm {MCS_i}}(t)} \\ &= \sum _{i=1}^{n_{\mathrm {MCS}}} \frac {\sum \limits _{j=1}^{n_{\mathrm {Lit,MCS_i}}} \left ( w_j(t) \cdot \prod \limits _{k=1,k\neq j}^{n_{\mathrm {Lit,MCS_i}}} q_{i,k}(t) \right )} {1-\prod \limits _{k=1}^{n_{\mathrm {Lit,MCS_i}}} q_{i,k}(t)} \end {split} \end{equation}