## 1 Introduction  
Motorsport races are competitions held to determine a ranking among the participants. In circuit races, the result depends not only on the driver and car performance but also on race strategy. Since the participants of such races drive a certain number of laps on a closed circuit, they can drive into their pits at the end of every lap. Pit stops are mostly taken in order to obtain a fresh set of tires, allowing the driver to drive a faster lap time than with an old set. However, since pit stops take some time, one must find an effective compromise between benefit and expanse. These aspects are determined by race strategy. Race simulations are used to simulate and compare the effects of different race strategies. The common approach to building such simulations is to discretize the race lap-wise, as we presented in an earlier paper [1]. Thus, the first simulation step in every lap \(l\) is to calculate the expected lap times \(t_{\text{lap}}\) of the drivers. To obtain them, the simulation adds up a number of different time parts in a lap time model, as shown in Equation (1) [1]. \(t_{\text{base}}\) is the lap time that the fastest car-driver combination can theoretically achieve in the race, that is, when the tires are fresh and the car has almost no fuel on board. It therefore takes into account the characteristics of the race track. To this basis are added: \(t_{\text{tire}}\) for the effect of tire degradation (dependent on tire age \(a_{\text{tire}}\) and compound \(c_{\text{tire}}\)), \(t_{\text{fuel}}\) for the time lost due to the fuel mass that is carried in the car, \(t_{\text{car}}\) and \(t_{\text{driver}}\) for the car and driver abilities, \(t_{\text{grid}}\) for the time that is lost at the race start (dependent on grid position \(p_{\text{g}}\)) and \(t_{\text{pit,in-lap/out-lap}}\) for the time that is lost in pit stops [1]. \[t_{\text{lap}}(l)=t_{\text{base}}+t_{\text{tire}}(a_{\text{tire}},c_{\text{tire }})+t_{\text{fuel}}(l)+t_{\text{car}}+t_{\text{driver}}+t_{\text{grid}}(l,p_{ \text{g}})+t_{\text{pit,in-lap/out-lap}}(l) \tag{1}\]  
Consecutive lap times of a driver are summed up to obtain his race times \(t_{\text{race}}\) at the end of every lap as given by [1]  
\[t_{\text{race}}(l)=\sum_{i=1}^{l}t_{\text{lap}}(i). \tag{2}\]  
The race times are the central element of the simulation. For example, they can be compared in order to determine whether an overtaking maneuver would have taken place in reality. This will be the case if the calculated race time of the pursuer is sufficiently smaller (i.e., faster) than that of the car in front. Figure 1 visualizes the simulation flow chart. The strategy engineer specifies the participants' strategies and obtains their race durations (and therefore also their final rank positions). The lap-wise discretization provides a good compromise between computational effort, parameterization effort, and accurate results. Although it causes some modeling difficulties, as will be discussed later, it is preferred over other approaches because the fast computing times are essential for the intended application. They enable the strategy engineer to quickly compare the results of many different strategies before and during a race. Moreover, they make the application of the Monte Carlo method, and thus the ideas presented in this paper, possible in the first place. For comparison: A time-discrete simulation or a simulation of the individual driving lines of every car would be orders of magnitude slower and would be hard to parameterize. In the presented state, however, the simulation misses one crucial aspect: real races are often decided by random events. If, for example, a driver causes an accident, race control usually sends a safety car on the track. Among other things, it significantly reduces the race speed while the crashed car is removed from the track, which leads to increased lap times. Consequently, probabilistic events such as this should be modeled within the race simulation to allow the strategy engineer to determine a stable race strategy.