## 3 Equilibrium Conditions  
In this part, we proceed to derive the equilibrium solutions for the morning commute problem in the presence of AV commuters and HV commuters. During the morning peak period, a commuter makes joint decisions on departure time and parking location to minimize her/his generalized travel cost. The decision making of commuters and the resulting equilibrium travel patterns are mainly affected by three aspects. First, AV and HV commuters may have the same or different parking preferences. It is necessary to analyze the equilibrium conditions for each possible case of parking preference presentedin Section 2.2. Second, in addition to the parking preferences, the equilibrium travel patterns depend on the departure order of two groups of commuters. In cases where two groups of commuters prefer the same parking cluster, the commuters can park their vehicles according to a first come, first served rule. Third, the capacity of each parking cluster and its potential parking demand further impact the equilibrium travel patterns. In particular, the capacity of the central cluster is limited, so some commuters need to park at the peripheral cluster if the central one cannot accommodate the total demand. Finally, we identify 16 equilibrium travel patterns, as shown in Table 3. For each equilibrium travel pattern, we obtain the parking demand at two clusters and the equilibrium travel costs for two groups of commuters. At equilibrium, no commuters could reduce their generalized travel costs by unilaterally altering their departure time or parking location choices. Thus, we can derive the queuing rates of AV commuters and HV commuters, respectively:  
\[\frac{\mathrm{d}c_{t,j}^{a}}{\mathrm{d}t} =0\Rightarrow\frac{\mathrm{d}Q(t)}{\mathrm{d}t}=\frac{\beta s}{ \alpha^{a}-\beta}, \tag{5}\] \[\frac{\mathrm{d}c_{t,j}^{h}}{\mathrm{d}t} =0\Rightarrow\frac{\mathrm{d}Q(t)}{\mathrm{d}t}=\frac{\beta s}{ \alpha^{h}-\beta}. \tag{6}\]  
With the derived queuing rate, we can further obtain the departure rate \(r^{a}=\frac{\alpha^{a}s}{\alpha^{a}-\beta}\) for AV commuters and \(r^{h}=\frac{\alpha^{h}s}{\alpha^{h}-\beta}\) for HV commuters, respectively. Under the assumption that \(\alpha^{a}<\alpha^{h}\), the queuing rate of HV commuters is lower than that of AV commuters based on Equations (5) and (6). Similarly, the departure rate of HV commuters is lower than that of AV commuters. Thus, we obtain the departure order of the two groups of commuters in the following proposition. **Proposition 2** (The departure order at UE). : \(HV\) _commuters depart from home earlier than AV commuters._  
The departure order presented in Proposition 2 should be attributed to the commuters' tradeoffs among different cost components. According to Equations (1) and (2), the costs associated with the parking process depend on the parking location choices, which are not affected directly by the departure time choices. And we can determine the departure order based on tradeoffs between schedule delay and travel time costs. The condition \(r^{h}<r^{a}\) indicates that the impacts of schedule delay on individual travel costs are more significant than the queuing delay from the perspective of AV commuters. An HV commuter will have a shorter schedule delay time than an AV commuter if they depart from home at time \(t\). This is because the parking process for HV commuters involves parking at a cluster and then walking back to the workplace, which leads to an arrival time at the workplace close to \(t^{*}\). Thus, HV commuters can accept earlier departure time slots than AV commuters. In addition, the VOT of HV commuters is larger than that of AV commuters, so they tend to choose earlier departure time slots to reduce travel time costs. To summarize, the HV commuters will depart from home earlier than the AV commuters in the single-corridor network with two parking clusters. Next, following the parking preferences derived in Section 2.2, we examine the travel patterns by incorporating the impacts of parking demand and parking capacity.