## 4 The Impacts of Parking Capacity and Parking Fees
### Parking Capacity  
In this part, we examine the impacts of parking capacity on the total travel cost. It is assumed that only the closer parking cluster is capacity-limited and the capacity of the farther one is sufficient to accommodate all commuters. In this situation, the parking capacity of the closer cluster affects the total travel cost. Thus, we focus on how the capacity of the central cluster impacts the travel patterns and the resulting total travel cost at equilibrium. As the parking slots are publicly owned, the revenue collected from commuters can be invested for developing urban transportation infrastructures and maintenance. So, the revenue from providing public parking services can be excluded from the total commuting costs of commuters. The resulting total travel cost denoted by \(\pi\) is composed of travel time cost, schedule delay cost, the walking time cost of HV commuters, and the self-driving cost of AV commuters. That is,  
\[\pi=N^{h}c^{h}+N^{a}c^{a}-n_{1}p_{1}-n_{2}p_{2}, \tag{10}\]  
where the term \(N^{h}c^{h}+N^{a}c^{a}\), indicating the total commuting cost, is independent of the capacity of the central cluster \(K_{1}\). The parking patterns described by \((n_{1},n_{2})\) depend on the capacity setting of the closer cluster, as shown in Table 3. We can further obtain the  
\begin{table}
\begin{tabular}{l l l l} \hline \hline
**Cases** & **Pattern Based on K\({}_{1}\)** & \((n_{1},n_{2})\) & \(e^{a}\) & \(e^{a}\) \\ \hline \multirow{3}{*}{1} & (1) \(K_{1}<N^{h}\) & \((K_{1},N-K_{1})\) & \(\beta\frac{N^{h}}{x}+\frac{e^{a}}{N}(\frac{e^{h}}{x}+(\eta^{h}-\beta)\Delta w- \Delta p)+p_{2}+\eta^{a}p_{2}\) & \\  & (2) \(N^{h}\leq K_{i}<N\) & \((K_{1},N-K_{1})\) & \(\beta\frac{N^{h}}{x}+\frac{e^{a}}{N}(\frac{e^{h}}{x}+(\eta^{h}-\beta)\Delta w- \Delta p)+p_{2}+\eta^{a}p_{2}\) & \\  & (3) \(K_{1}\geq N\) & \((N,0)\) & \(\beta\frac{N^{h}}{x}+\frac{e^{a}}{N}(\frac{e^{h}}{x}+p_{1}+\eta^{a}p_{1})\) & \\ \hline \multirow{3}{*}{2} & (4) \(K_{1}\) free & \((0,N)\) & \(\beta\frac{N^{h}}{x}+\frac{e^{a}}{N}(\frac{e^{h}}{x}+p_{2}+\eta^{a}p_{2}\) & \(\beta\frac{N}{x}+p_{2}+(\eta^{a}-\beta)\Delta w-\Delta p)\) \\ \cline{2-4}  & (1) \(K_{1}<N^{h}\) & \((K_{1},N-K_{1})\) & \(\beta\frac{N^{h}}{x}+\frac{e^{a}}{N}(\frac{e^{h}}{x}+(\eta^{h}-\beta)\Delta w- \Delta p)+p_{2}+\eta^{a}p_{2}\) & \\  & (1) \(N^{h}\leq K_{1}<N^{h}+y\) & \((K_{1},N-K_{1})\) & \(\beta\frac{N^{h}}{x}+\frac{e^{a}}{N}(\frac{e^{h}}{x}+(\eta^{h}-\beta)\Delta w- \Delta p)+p_{2}+\eta^{a}p_{2}\) & \\  & (5) \(N^{h}+y\leq K_{1}<N\) & \((N^{h}+y,N^{h}-y)\) & \(\beta\frac{N^{h}}{x}+\frac{e^{a}}{N}(\frac{e^{h}}{x}+p_{1}+\eta^{derivative of total travel cost \(\pi\) with respect to the capacity of the closer cluster \(K_{1}\), i.e., \(\frac{\partial\pi}{\partial K_{1}}=-p_{1}\frac{dn_{1}}{dK_{1}}-p_{2}\frac{dn_{2} }{dK_{1}}\). The corresponding results of the derivative \(\frac{\partial\pi}{\partial K_{1}}\) are provided in Table 1 in Appendix B. Given fixed parking fees and the topology of the single-corridor network, we summarize the impacts of the parking capacity of the central cluster on the total travel cost in Proposition 3. **Proposition 3**. : _At equilibrium, the total travel cost \(\pi\) decreases with the capacity of the central cluster \(K_{1}\) if it is insufficient to meet its potential parking demand._  
Proposition 3 indicates that the system performance measured by total travel cost depends on the relative magnitude of the parking capacity of the closer cluster and the potential parking demand for this cluster. The potential parking demand at the closer cluster is determined by the demand of two groups of commuters and their parking preferences stated in Table 2. For example, the parking preferences of HV and AV commuters in Case 3-2 are the central and the peripheral clusters, respectively. Thus, the potential parking demand for the closer cluster is the total demand of HV commuters \(N^{h}\).