## 3 Methodology
### Determination of Full Course Yellow Phases in Combination with Accidents and Failures
The first lap has to be considered separately since over 36 % of the SC phases start here, which can be explained by the small distances between the drivers shortly after the start that cause a high probability of accidents. The duration of an SC phase is chosen to be between two and eight laps with empirical probabilities \(P_{\text{sc,duration}}\) derived from data of the seasons 2014-2019. The exact values are given in Table 4. The start of an SC phase is further modified by a uniform distribution \(\mathcal{U}(0,\,1)\) to include the fact that it does not start precisely at the point laps are completed. Before finally saving the created phase, it is assured that it does not overlap an already existing phase. This would be the case if  
\[r_{\text{fcy,s,n}}\leq r_{\text{fcy,e}}+r_{\text{fcy,d}}\quad \text{and} \tag{13}\] \[r_{\text{fcy,s,e}}-r_{\text{fcy,d}}\leq r_{\text{fcy,e}} \tag{14}\]  
both hold. \(r_{\text{fcy,s/e,n}}\) is the race progress at the start and end of the new and \(r_{\text{fcy,s/e,e}}\) the race progress at the start and end of the existing FCY phase currently in comparison. \(r_{\text{fcy,d}}\) is a minimum distance which should be kept between two phases. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Probability** & **2 Laps** & **3 Laps** & **4 Laps** & **5 Laps** & **6 Laps** & **7 Laps** & **8 Laps** \\ \hline \(P_{\text{sc,duration}}\) & 0.182 & 0.25 & 0.227 & 0.193 & 0.057 & 0.068 & 0.023 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Probabilities for the duration of a safety car phase (seasons 2014–2019). Figure 8: Real and modeled cumulative distribution functions (CDF) of the start of the safety car phases in the seasons 2014–2019. Figure 7: Fraction of races in the seasons 2014–2019 with the specified number of safety car deployments. As mentioned before, we assume that every SC phase is caused by an accident. Therefore, the simulation chooses one driver who retires at the start of every SC phase. The selection happens based on the drivers' accident probabilities \(P_{\text{accident}}\) that were determined earlier. Selecting only a single driver for an accident is a simplification, since sometimes two or even more drivers are involved in reality. However, our available data is not detailed enough to be able to modeling and parameterize these cases. Furthermore, retired drivers are not crucial for race strategy determination. Determine failures and derive VSC phases  
Thereafter, the simulation determines, for those drivers not involved in an accident, whether they suffer a failure. The team-specific failure probability \(P_{\text{failure}}\) determined earlier is used in this respect. Subsequently, the simulation checks for every failure appearing if it causes a VSC phase using the conditional probability \(P(\text{vsc}|\text{failure})\). Assuming that every VSC is caused by a failure, it can be calculated using the number of VSC phases \(n_{\text{vsc}}\) and the number of failures \(n_{\text{failures}}\) (2015-2019, as the VSC was introduced in 2015) in the database:  
\[P(\text{vsc}|\text{failure})=\frac{n_{\text{vsc}}}{n_{\text{failures}}}=0.227. \tag{15}\]  
This is a simplification because there are some cases where, after an accident, VSC phases were first activated and shortly afterward replaced by an SC phase, for example. However, as before, the available data is not detailed enough to analyze these cases. The start of the failure (and probably of the phase) is sampled from a uniform distribution \(\mathcal{U}(0,\,n_{\text{laps}})\) since no outstanding race section could be identified in the data. \(n_{\text{laps}}\) stands for the number of laps in the race. The duration of a possible VSC phase is chosen in the range between one and four laps, with empirical probabilities \(P_{\text{vsc,duration}}\), and modified by a uniform distribution \(\mathcal{U}(0,\,1)\) as with the start of SC phases.