## 4 MTBF Evaluation  
The MTBF of any ADS parts is derived statistically based on [44] or, when available, from part manufacturers. Reference [44] provides failure rate data for a wide variety of component types including mechanical, electromechanical, and electronic assemblies. Statistical data are useful to derive MTBF of the single sub-part in order to highlight the MTBF genesis of the Pitot probe, TAT probe, and the ADC. The results from [45] are collected in Table 4. Considering that metallic ducts, pneumatic connections, power, and data connections have a low failure rate (FR), they are grouped as a single sub-part in the present analysis. The failure rate is calculated as the ratio between the observed fails and one million operating hours. Therefore, the MTBF can be calculated if not available from the manufacturer by multiplying \(1\times 10^{6}\) times the probability of failure occurrence (PF) from [44]. On the other hand, the probability of a failure occurrence per flight hour, or the FR, can be calculated as the inverse of the MTBF as  
\[\text{MTBF}=\frac{\text{total time}}{\text{total fails}}=\frac{1\times 10^{6}}{ \text{PF}}\times 10^{6}=\frac{1}{\text{FR}} \tag{8}\]  
As an example, the TAT integrated heater has \(\text{PF}=30.1\). The corresponding MTBF is 33,223 flight hours, whereas the FR is about \(3.01\times 10^{-5}\) per flight hour. As Equation (8) refers to a single ADS sub-part, considering that the sub-part failure events are independent (i.e., in OR conditions), the ADS part's MTBF is calculated as  
\[\text{MTBF}_{\text{ADS part}}=\frac{1}{\sum\limits_{j}\text{FR}_{j}} \tag{9}\]  
where \(\text{FR}_{j}\) is the FR of \(j\)th sub-part of the ADS part calculated in Equation (8). From Table 4, it can be noticed that both estimated MTBF for the Pitot and TAT are comparable with the manufactures data. The subdivision of ADS parts into sub-parts, with the consequent MTBF evaluation, is crucial to single out and evaluate the sub-parts that are the most critical with respect to safety. In fact, for the Pitot probe, the leading sub-part is the heater element (used for anti-icing purposes) with a FR of the order of magnitude of \(1\times 10^{-4}\), whereas for the TAT probe, both the heater and the dual sensing element are the most critical sub-parts. On the other hand, both the metallic tubes and the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**ADS Part**} & \multirow{2}{*}{
\begin{tabular}{c} **MTBF from** \\ **Manufacturer** \\ \end{tabular} } & \multirow{2}{*}{**ADS Sub-Part**} & \multirow{2}{*}{**MTBF from [44]**} & \multirow{2}{*}{**FR from [44]**} & \multirow{2}{*}{**Estimated MTBF**} \\ \cline{5-6}  & & & & & \\ \hline \multirow{3}{*}{Pitot probe} & \multirow{3}{*}{13,000 h} & Tube and strut & 1,428,571 h & \(7.00\times 10^{-7}\) & \multirow{3}{*}{14,492 h} \\  & & Heater & 14,689 h & \(6.81\times 10^{-5}\) & & \\  & & Duct, piping and power cable & 4,464,286 h & \(2.2\times 10^{-7}\) & & \\ \hline \multirow{3}{*}{TAT probe} & \multirow{3}{*}{20,000 h} & Tube and strut & 2,857,143 h & \(3.50\times 10^{-7}\) & \multirow{3}{*}{17,989 h} \\  & & (single) Sensing element & 80,645 h & \(1.24\times 10^{-5}\) & & \\  & & Heater & 33,222 h & \(3.01\times 10^{-5}\) & & \\  & & data and power cable & 2,941,176 h & \(3.4\times 10^{-7}\) & & \\ \hline \multirow{3}{*}{ADC} & \multirow{3}{*}{N/A} & single board & 48,609 h & \(2.06\times 10^{-5}\) & \multirow{3}{*}{48,134 h} \\  & & Abs. press. transl. & 9,900,000 h & \(1.01\times 10^{-7}\) & & \\ \cline{1-1}  & & Diff. press. transl. & 9,900,000 h & \(1.01\times 10^{-7}\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: ADS parts and related mean time between failure (MTBF) declared by the part manufacturer, whereas the sub-part MTBF is retrieved from [44] and the “Estimated MTBF” is calculated using Equation (9). pneumatic/data/power connections have an FR of the order of magnitude of \(1\times 10^{-7}\), i.e., at least two orders of magnitude higher.