## 6 Numerical Analysis
### System Performance Analysis  
In the following, we aim to demonstrate the impacts of the parking capacity of the central cluster and the corresponding parking price on the total travel cost. According to Proposition 1, the parking location preferences of commuters depend on the relationship between \(\Delta p\), \(\Delta w\), and \(\Delta v\), which further affects the travel patterns and the resulted system performance at equilibrium. Specifically, we find that the equilibrium cases changes following this order: Case 1, Case 3, and Case 2, when \(\eta^{a}\Delta v<(\eta^{h}-\beta)\Delta w\) and \(\Delta p\) increases from 0 to \(+\infty\). In contrast, when \(\eta^{a}\Delta v>(\eta^{h}-\beta)\Delta w\), we can observe the following equilibrium cases: Case 1, Case 4, and Case 2, when the level of \(\Delta p\) increases. In addition to \(\Delta p\), \(\Delta w\), and \(\Delta v\), the capacity of the central cluster \(K_{1}\) also affects the equilibrium patterns. For simplification, we choose to vary the parking price at the central cluster \begin{table}
\begin{tabular}{c c c} \hline \hline Parameter & Value & Unit \\ \hline \(\alpha^{a}\) & 15 & USD/h \\ \(\alpha^{a}\) & 20 & USD/h \\ \(\eta^{a}\) & 14 & USD/h \\ \(\eta^{a}\) & 25 & USD/h \\ \(\beta\) & 10 & USD/h \\ \(s\) & 1000 & vehicles/h \\ \(N^{a}\) & 1000 & vehicles \\ \(N^{a}\) & 1500 & vehicles \\ \(p_{2}\) & 0 & USD \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Input. while keeping the price at the peripheral one \(p_{2}\) fixed. Therefore, we focus on the joint impacts of \(K_{1}\) and \(p_{1}\) on the total travel cost \(\pi\) hereafter. Under the condition \(\eta^{a}\Delta v<(\eta^{h}-\beta)\Delta w\), we analyze the total travel cost \(\pi\) by varying the parking price at the central cluster in the range of \([0,16]\) and varying the parking capacity \(K_{1}\) in the range of \([0,3000]\). The remaining parameters are fixed and set according to Table 6. Before investigating the system performance measured by the total travel cost, we illustrate the impacts of \(K_{1}\) and \(p_{1}\) on the parking patterns at the central cluster. Recalling the equilibrium results derived in Section 3, the potential parking demand at the central cluster depends on the parking preferences of the AV commuters and the HV commuters. Additionally, not all commuters who prefer to park at the central cluster could finally find parking slots in some cases, as stated in Proposition 3. Here, we present the actual number of vehicles parking at the central cluster and the unfulfilled demand in Figure 3(a) and Figure 3(b), respectively. Figure 4(a) depicts the contours of the total travel cost \(\pi\) with respect to the parking price and the capacity charged at the central cluster. To illustrate the transition of equilibrium patterns, we mark the equilibrium patterns in Cases 1, 2, and 3, as shown in Figure 4(b). In Case 1, as shown in this figure, the total travel cost decreases with the parking capacity first. Meanwhile, the unfulfilled demand, defined as the difference between the potential parking demand and the physical capacity of the central cluster in Figure 3(b), is positive. Later, when \(K_{1}\) increases, the total travel cost in Figure 4(b) remains constant. The corresponding unfulfilled demand is zero, as shown in Figure 3(b). A similar tendency is also observed in Case 3. And in Case 2, the total travel cost does not change with \(K_{1}\). The above attributes of the total travel cost concerning the parking capacity of the central cluster are consistent with the analytical results obtained in Proposition 3. Figure 4: _Cont._  
Figure 4(b) depicts the contours of the total travel cost \(\pi\) with respect to the parking price and the capacity charged at the central cluster. To illustrate the transition of equilibrium patterns, we mark the equilibrium patterns in Cases 1, 2, and 3, as shown in Figure 4(b). In Case 1, as shown in this figure, the total travel cost decreases with the parking capacity first.