## 3 Methodology
### Modeling of Accidents, Failures, and Full Course Yellow Phases  
The modeling of accidents and failures is implemented by simply taking the concerned driver out of the race as soon as his race time exceeds the defined time of retirement. Modeling of the virtual safety car  
The VSC is modeled by increasing the lap times of the drivers to \(t_{\text{lap,vsc}}=1.4\cdot t_{\text{base}}\), cp. Section 2.5. However, since FCY phases can start and end at any point during a lap, we have to calculate the lap \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Probability** & **1 Lap** & **2 Laps** & **3 Laps** & **4 Laps** \\ \hline \(P_{\text{vsc,duration}}\) & 0.479 & 0.396 & 0.021 & 0.104 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Probabilities for the duration of a virtual safety car phase (seasons 2015–2019). fractions that are driven normally \(f_{\rm n}\) and affected by the VSC phase \(f_{\rm vsc}\) to obtain the correct lap time. If, for example, the phase starts within the current lap and ends in a later lap, the resulting lap time \(t_{\rm lap}\) can be calculated by  
\[f_{\rm n} =\frac{t_{\rm race,vsc,s}-t_{\rm race,l-1}}{t_{\rm lap,n}}, \tag{16}\] \[f_{\rm vsc} =1.0-f_{\rm n}\text{, and}\] (17) \[t_{\rm lap} =f_{\rm n}\cdot t_{\rm lap,n}+f_{\rm vsc}\cdot t_{\rm lap,vsc}, \tag{18}\]  
where \(t_{\rm race,vsc,s}\) is the start race time of the VSC phase, \(t_{\rm race,l-1}\) the race time of the driver at the end of the previous lap and \(t_{\rm lap,n}\) the unaffected lap time of the driver in the current lap. Similar calculations are performed when the phase ends. Overtaking is forbidden in the simulation if a VSC affects at least 50% of a lap. Due to the limited speed, tire degradation and fuel consumption are reduced to 50% during the phase. This is an estimation since the exact values cannot be derived from the publicly available data. In reality, the saved fuel is, of course, consumed after the phase, for example, by increasing the engine power. In the simulation, the average consumption per lap after an FCY phase is therefore automatically adjusted so that the saved fuel is used up by the race finish. For the realistic modeling of SCs, we use driver-individual \"safety car ghosts\" (SCGs). The concept is illustrated in Figure 9. An SCG can be imagined as a virtual car that is only visible to its corresponding driver and does not affect any other driver. Since it is a safety car, it cannot be overtaken. The driver-individual handling is necessary, since the drivers may be affected by the same SC in different laps due to lap-wise discretization. An SC deployment is modeled in two stages, a run-up stage and a following stage. If a driver reaches the start race time of an SC phase, we assume that his SCG starts driving on the finish line exactly at this time. Equally to the VSC, the lap time of the respective driver is then increased up to \(t_{\rm lap,vsc}\) for the remaining part of the lap and following laps to simulate the run-up stage under full course yellow condition. Every driver catches up with his SCG within several laps, since it drives at 160% of the base lap time, cp. Section 2.5. The first SCG lap time is even slower to modeling the real behavior where the SC waits for the leading driver at the pit exit. If a driver's calculated race time at the end of a lap is below that of his SCG, he would have overtaken it. Thus his lap time is artificially increased to stay behind. Keeping a minimum temporal spacing \(t_{\rm gap,sc}\) between the drivers is hereby assured by adding \(p\cdot t_{\rm gap,sc}\) to the individual SCG race times, where \(p\) stands for the drivers' rank positions. Tire degradation and fuel consumption are reduced to 25% (again an estimation) while driving behind an SC. The value is smaller than that of the VSC phase because the speed is even lower. The SCGs remain active until the end of the lap in which the SC phase ends, even if the originally determined end time is reached before the end of the lap.