Integrated Departure Time and Parking Location Choices in a Morning Commute Problem under a Partially Automated Environment

Abstract

This study formulates the joint decisions of commuters on departure time and parking location choices in a morning commute problem where the commuters travel with autonomous vehicles (AVs) or human-driven vehicles (HVs). Under a mixed traffic environment, we aim to explore the impacts of parking capacity and parking pricing on the equilibrium travel pattern and the system performance. We build a dynamic equilibrium model for the morning commute problem by assuming that the parking slots can be grouped into central and peripheral clusters based on the distance between the parking location and the workplace. We first analyze the parking location preferences of commuters towards the two parking clusters under a mixed traffic environment. We then examine the equilibrium conditions and identify all the equilibrium travel patterns. We further analyze the system performance measured by the total travel cost with respect to the parking prices and the capacity of the central cluster. The optimal parking pricing scheme is also derived to minimize the total travel cost. We conduct numerical analysis to demonstrate the change in the total travel cost against the parking capacity of the central cluster and its parking price. Sensitivity analysis is performed to show the impacts of the network configuration on the total travel cost.

1. Introduction

Traffic congestion and limited parking space are the two major problems for commuters in transportation networks. In many urban megacities, travelers may experience severe traffic congestion when completing commuting trips during morning peak hours. Since the parking space may be insufficient to meet the high parking demand in the city center, commuters may also spend much time searching for parking slots, which increases their travel costs. Under such circumstances, the critical challenge is managing traffic congestion and parking provision to reduce individual travel costs and improve social welfare. From the perspective of a transportation agency, it is essential to encourage the adoption of new travel modes or resort to new transportation management instruments.

With the development of autonomous vehicle (AV) technologies, commuters may have the opportunity to purchase and utilize AVs privately in their daily lives, providing an alternative to existing travel modes. The advantages of using AVs for commuting are manifested in both individual and social (societal) aspects. From the perspective of commuters, AVs allow commuters to focus on their productive matters in vehicles with the assistance of self-driving technology. Moreover, the vehicles can self-drive to the selected parking slots without the presence of the owners. The experienced travel time cost and the cost associated with parking may be lower compared to using traditional vehicles. From the perspective of society, road traffic may be maintained in a smooth state with a certain penetration of AVs, which helps to alleviate traffic congestion and avoid traffic accidents. Traveling with AVs can also reduce vehicle emissions, especially during the congested period. Enabled by self-driving and self-parking capabilities, commuters’ decision making and travel preferences may change dramatically, which may reshape the future urban travel patterns during morning peak hours.

During the transition period, commuters can travel with AVs or HVs to complete morning commuting trips, which incurs a partially automated environment. Commuters traveling with different travel modes may have different behaviors toward departure time choices and parking location choices. And the departure time choices and the parking location choices of commuters interact with each other, which adds to the complexity of the problem. The network configuration also influences the parking location choices of commuters. Most relevant studies assume that the parking slots are distributed linearly along a corridor between the common workplace and the residence area, e.g., [1]. The parking slots could be grouped into on-street or off-street parking to analyze commuters’ parking behaviors, e.g., [2]. In this work, we consider an ideal network for the morning commute with a single corridor and two parking clusters near the workplace. We aim to establish a modeling framework to formulate the joint decisions of commuters on departure times and parking locations in the morning commute under the partially automated environment. Furthermore, we will investigate the impacts of parking management policies and network configurations on commuters’ decision making and system performance.

The morning commute problem, in which a fixed number of commuters choose departure time, is formulated using a bottleneck model by Vickrey [3]. Based on the user equilibrium (UE) principle, this classic bottleneck model has been extended to model the multidimensional decisions of commuters, such as combined departure time and route choices, e.g., [4,5]; combined departure time and travel mode choices, e.g., [6,7]; and the integrated morning and evening commuting patterns, e.g., [8,9]. These studies investigate the impacts of commuters’ decision making on congestion evolution or social welfare. The impacts of commuters’ parking decisions on congestion evolution during the morning peak period have also been extensively analyzed by modeling the combined departure time and parking location choices. The existing literature considers various network configurations and parking arrangements in its modeling and analysis. By incorporating privately owned parking slots during peak periods, Qian et al. [10] formulate the commuters’ parking location and departure time choice and investigate impacts of parking provision strategies on the total travel cost.

In addition to extending the classic bottleneck models, various alternative approaches can be utilized in modeling commuters’ decisions in morning commute problems. Following the UE principle, traffic assignment models are formulated to predict the flow distribution over road networks. By developing a static traffic assignment model, Kang et al. [11] study how commuters’ joint decisions on route, mode, and parking choices affect the congestion distribution under a partially autonomous environment. In a generous case, the time-of-day features of road networks are significant. In the meantime, multiple bottlenecks may be observed in a road network and the congestion may correlate spatially. In this case, dynamic traffic assignment models are established to account for the time-varying properties of road networks and spatial interaction of traffic congestion, e.g., [12,13]. The travel patterns during the morning peak period have also been investigated using data analysis methods, e.g., [14]. Sato et al. [15] explore the dynamic congestion pricing scheme design by integrating reinforcement learning techniques with dynamic traffic assignment modeling. However, AVs are not widely adopted by commuters in the real world, which incurs the lack of historical trip data of AVs. Currently, the historical trip data of AVs are limited since AVs are not widely adopted by commuters in the real world. In this case, simulated data of AVs may be used when data analysis methods are used [16].

With the emergence of autonomous vehicle technologies, the bottleneck model has been extended to study how AV technologies change equilibrium travel patterns. Van Essen et al. [17] study the influences of different penetration rates of AVs on traffic congestion. By establishing a many-to-one network, Su and Wang [18] investigate a parking sharing problem in which AV commuters have different parking options. In their framework, commuters can choose to park at home, use shared parking slots, or share their own parking slots out, and in the meantime park their own vehicles at a public parking lot. Nourinejad and Amirgholy [19] formulate the departure time and parking location decisions of commuters in a mixed traffic environment with both AVs and HVs. Zhang et al. [20] explore the impacts of different road capacity strategies on morning commuting patterns when commuters make combined departure time and parking decisions.

Various parking management policies are developed to improve system performance measured by system-wide travel costs or total traffic congestion. When ridesharing services are introduced into the morning commute, Ma and Zhang [21] design combined ridesharing compensation and dynamic parking pricing schemes to eliminate traffic congestion. Qian et al. [10] conclude that both individual welfare and the total system cost can be decreased through parking regulation strategies, including price-ceiling and quantity subsidy. Assuming that the parking slots near the workplace can be grouped into two clusters, Qian et al. [22] model the joint decisions of morning commuters on departure times and parking locations. After examining the parking profiles at equilibrium, they explore the optimal design of parking provision strategies regarding parking pricing and parking capacity setting. After AVs are introduced to road networks, parking management policies are designed to manage traffic congestion and parking patterns. Liao et al. [23] propose a tradable credit scheme to manage traffic congestion in morning commute by assuming that all commuters travel with AVs. By releasing this assumption, we further investigate the implications of parking pricing and parking capacity in alleviating the morning commute problem under a mixed environment.

In this work, we aim to examine the impacts of parking on decision-making of AV users and HV users in the morning commute problem where two parking clusters exist in the central business district (CBD) area. The bottleneck model generally admits closed-form equilibrium solutions, which allows us to formulate decisions and evaluate various policies analytically. The contributions of this work are as follows. First, we extend the classic bottleneck model to formulate the combined decisions on departure time and parking location under a partially automated traffic environment. By examining the equilibrium conditions, we can identify 14 equilibrium travel patterns in total and categorize the equilibrium travel patterns into five cases. Second, we explore the parking location preferences of both AV and HV commuters. When the difference in parking fees is very low, all commuters prefer the central parking cluster. When the difference in parking fees is very high, all commuters prefer the peripheral cluster. Otherwise, we find hybrid parking preferences for the two classes of commuters. Third, based on the modeling framework, we analyze how the parking capacity of the central cluster affects parking patterns and the total travel cost at equilibrium. The analytical results show that the total travel cost decreases when the capacity of the central cluster grows, given that the capacity of this cluster can satisfy the parking demand. The impacts of the parking fees are also investigated to understand their influences on the equilibrium travel patterns. We identify the conditions under which the total travel cost does not vary with the parking prices. For most cases, we conclude that the total travel cost increases or decreases with the parking price at the central cluster. In order to optimize social welfare, we design the optimal parking pricing scheme for the central cluster. Finally, we perform numerical analysis to demonstrate the theoretical results.

The remainder of this paper is structured as follows. In Section 2, we build the dynamic user equilibrium model for the morning commute by introducing two parking clusters, and we explore the parking preferences of commuters towards the two clusters. We derive the equilibrium conditions in Section 3. Section 4 analyzes the impacts of parking fees and the capacity of the central cluster on system-wide travel cost. Section 5 derives the optimal parking price charged at the central cluster to optimize the social welfare. We demonstrate the commuters’ preferences and the welfare effects of parking pricing policy by performing numerical experiments in Section 6. Section 7 concludes our findings.

2. Model Formulation

2.1. Model Setting

Consider a single corridor connecting a residential area and a workplace (central business district, CBD), as shown in Figure 1. A bottleneck exists at the exit of this corridor so commuters may experience queuing time to traverse the congestion segment. Following Qian et al. [22], we assume the parking slots around the workplace can be categorized into two discrete clusters according to the distance between each cluster and the workplace. Specifically, the central cluster represents a group of parking slots close to the workplace. And the peripheral cluster includes parking slots located relatively far from the workplace. Let j denote the index of the parking cluster where $j=1$ represents the central cluster and $j=2$ represents the peripheral cluster. In practice, there is a sufficient number of parking slots in the peripheral area, whereas the parking space near the CBD is usually limited. Let $K_j$ denote the parking capacity, i.e., the number of parking slots in each cluster. $K_2$ should be large enough to accommodate all commuters, but $K_1$ may not meet the parking demand in the CBD area. In other words, we assume a limited capacity for the central cluster.

During the morning peak period, there are N commuters travelling from the residential area to the workplace in the CBD. A commuter may travel by an autonomous vehicle (AV) or a human-driven vehicle (HV). Let $N^a$ and $N^h$ denote the travel demand for AVs and HVs, respectively. The two groups of commuters follow different procedures to complete their morning commuting trips. As illustrated in Figure 2, the HV commuters will travel from home to a selected parking location within a cluster and then walk back from the parking location to the workplace. Unlike HV commuters, AV commuters can drop off directly at the workplace, and their vehicles will self-drive to the parking slots. The differences in the travel process result in different cost considerations between the two groups of commuters.

The commuters decide on departure time from home (t) and the parking location choices (j). When the departure rate from home exceeds the bottleneck capacity s, a queue forms, and commuters experience a certain level of queuing time to pass the congested corridor. The travel time experienced by commuters departing at time t is composed of free-flow travel time and queuing time. Without loss of generality, the free-flow travel time on this corridor is assumed to be 0. Let $Q\left(t\right)$ denote the queuing length, i.e., the total number of vehicles waiting at the bottleneck at time t. Thus, the travel time is given by $T\left(t\right)=\frac{Q\left(t\right)}{s}$.

Next, we proceed to define the individual travel costs for both AV and HV commuters. Each commuter experiences a bottleneck cost, including travel time cost and schedule delay cost, as in Vickrey’s bottleneck model. We assume that the AV and HV commuters are heterogeneous regarding the value of time. Let ${\alpha }^a$ and ${\alpha }^h$ denote the values of time for AV commuters and HV commuters, respectively. The commuters only experience schedule delay costs for early arrivals, as late arrivals are not allowed. Based on the travel process discussed earlier, the schedule delay time is $t^{*}-t-T\left(t\right)$ for AV commuters and $t^{*}-t-T\left(t\right)-w_j$ for HV commuters.

In addition to bottleneck cost, the topology of the single-corridor network affects the individual travel costs of two groups of commuters. As illustrated in Figure 1, the closer cluster corresponds to a lower travel distance between the parking location and the workplace. In contrast, the travel distance between the parking location and the workplace will increase if commuters park at the farther cluster. Therefore, the HV and AV commuters may experience different levels of walking time costs after parking and self-driving time costs after dropping off, respectively. Let $v_j$ denote the self-driving time from the workplace to parking cluster j for AV commuters. Let $w_j$ denote the walking time from the parking cluster j to the workplace for HV commuters.

Additionally, the commuters make decisions based on the parking pricing scheme designed and implemented by the public parking manager. As the central cluster is close to the CBD area, the resulting self-driving time from the workplace to the parking location is lower for AV commuters than parking at the peripheral cluster. Similarly, the commuters bear a lower walking time by parking at the central cluster. The slots within each cluster are homogeneous for commuters, so the distance between any two slots is negligible. In this situation, an HV commuter experiences the same walking time by choosing any slot in a cluster. Similarly, the self-driving time from the workplace to any parking slot within a cluster is the same and constant. Since the central cluster is close to the workplace, it allows commuters to save schedule delay costs and provides parking convenience, which may lead to a high parking demand during peak hours. Hence, we also assume that the average parking fee at the central cluster $p_1$ is higher than that at the peripheral cluster $p_2$.

Let $c_{t,j}^a$ denote the generalized travel cost of AV commuters who depart from home at time t and park their vehicles at location j. Similarly, $c_{t,j}^h$ represents the generalized cost of HV commuters departing from home at time t and parking at location j. Their generalized cost functions are defined as follows.

$$\begin{array}{ccc}\hfill c_{t,j}^a& =& {\alpha }^{a}T\left(t\right)+\beta (t^{*}-t-T\left(t\right))+p_j+{\eta }^a{v}_j\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill c_{t,j}^h& =& {\alpha }^{h}T\left(t\right)+\beta (t^{*}-t-T\left(t\right)-w_j)+p_j+{\eta }^h{w}_j\hfill \end{array}$$

(2)

where ${\eta }^a$ represents the unit cost of AV self-driving time after parking and ${\eta }^h$ represents the value of walking time for HV commuters.

By making tradeoffs among travel time cost, schedule delay cost, parking fee, and the cost associated with the parking process, commuters make joint decisions on departure time t and parking location j to minimize their generalized travel costs.

Table 1. Notations.

Notation	Description
t	Departure time
$N^a$	Demand of AV commuters
$N^h$	Demand of HV commuters
$t^{*}$	Desired arrival time at destination
${\alpha }^a$	Value of travel time for AV users
${\alpha }^h$	Value of travel time for human-driven vehicle users
${\eta }^a$	Value of self-driving time for AV users
$\beta$	Unit cost of early schedule delay
${\eta }^h$	Value of walking time for HV commuters
$v_j$	Self-driving time from workplace to parking cluster j, i.e., access time to parking cluster
$w_j$	Walking time from parking cluster j to workplace
$T\left(t\right)$	Travel time (i.e., queuing time) for commuters departing from home at time t
$Q\left(t\right)$	Queuing length for commuters departing from home at time t

summarizes the notations used in this work.

The assumptions adopted in this work are listed here.

Assumption 1.

$\beta <{\alpha }^a<{\alpha }^h$. It is commonly assumed that AV commuters have a lower VOT compared with the HV commuters e.g., [24]. Following Lindsey [25], we assume ${\alpha }^a>\beta$ to ensure the existence of equilibrium.

Assumption 2.

Late arrivals are not allowed, e.g., [26].

2.2. Parking Preferences

In this part, we aim to examine the parking preferences of commuters in a morning commute under a partially autonomous environment. Each commuter always prefers a parking location that minimizes their generalized travel cost. By comparing the magnitude of individual travel costs defined in Equations (1) and (2), we can determine the parking preferences of commuters.

We examine the parking preferences of AV commuters and HV commuters separately. We define $\Delta v=v_2-v_1$ to measure the difference in self-driving time for AV users between parking at the peripheral and the central cluster. Similarly, we introduce $\Delta w=w_2-w_1$ to indicate the difference in walking time for HV users between parking at the peripheral and the central clusters. Since the peripheral cluster is more distant than the central cluster, $\Delta v$ and $\Delta w$ are both positive. The difference in parking fees charged at the two clusters is $\Delta p=p_1-p_2$. We assume $\Delta p>0$ considering that the parking demand at the central cluster is usually higher than that at the peripheral cluster.

Given a fixed departure time, we calculate the difference in generalized travel cost between commuters between using the peripheral cluster and the central cluster. Based on Equation (1), the derived cost difference for AV commuters is given by

$$\begin{array}{c}\hfill c_{t,2}^a-c_{t,1}^a={\eta }^a\Delta v-\Delta p.\end{array}$$

(3)

The parking preference of AV commuters depends on $\Delta p$ and $\Delta v$. Specifically, the AV commuters prefer the central cluster when $\Delta p<{\eta }^a\Delta v$, prefer the peripheral cluster when $\Delta p>{\eta }^a\Delta v$, and are indifferent between the two clusters otherwise. If $\Delta p<{\eta }^a\Delta v$, the parking fee or the corresponding self-driving time for choosing the central cluster is lower than the peripheral cluster, which makes the closer parking cluster more attractive to AV commuters. In this situation, AV commuters prefer parking at the central cluster in order to decrease individual generalized travel costs. In contrast, if $\Delta p>{\eta }^a\Delta v$, the parking fee or the corresponding self-driving time for choosing the peripheral cluster is lower than the central cluster. Thus, AV commuters could reduce individual generalized travel costs by parking at the peripheral cluster.

Similarly, the difference in individual generalized travel costs for HV commuters can be derived based on Equation (2).

$$\begin{array}{c}\hfill c_{t,2}^h-c_{t,1}^h=({\eta }^h-\beta )\Delta w-\Delta p\end{array}$$

(4)

The parking preference of HV commuters relies on the difference in walking time between choosing the peripheral cluster and the central cluster, in addition to the difference in parking fee $\Delta p$. Specifically, the HV commuters prefer the central cluster when $\Delta p<({\eta }^h-\beta )\Delta w$, prefer the peripheral cluster when $\Delta p>({\eta }^h-\beta )\Delta w$, and are indifferent between the two clusters otherwise. If $\Delta p<({\eta }^h-\beta )\Delta w$, the parking fee $p_1$ or the corresponding walking time $w_1$ at the central cluster is lower than the peripheral cluster, which makes the central cluster more attractive to HV commuters. They tend to parking vehicles associated with the central cluster to decrease individual generalized travel cost. In contrast, under the condition that $\Delta p>({\eta }^h-\beta )\Delta w$, the HV commuters are able to decrease parking fee or walking time by parking at the peripheral cluster, so as to decrease individual generalized travel cost.

We make the following assumption on the relation between ${\eta }^h$ and $\beta$. The implication is that the HV commuters prefer arriving at the workplace before the desired arrival time to walking on the road.

Assumption 3.

${\eta }^h>\beta$.

Furthermore, we can describe the combined parking preferences of AV commuters and HV commuters by jointly considering the effects of $\Delta p$, $\Delta v$, and $\Delta w$. By characterizing the range of $\Delta p$ in terms of $\Delta v$ and $\Delta w$, we provide all the cases of parking preferences for the morning commute under a partially automated environment with two parking clusters in Proposition 1. Moreover, we can divide the parking preferences of commuters into five cases and nine subcases, as summarized in

Table 2. Parking location preferences.

Cases		Sub-Cases		HV Commuters	AV Commuters
1	-		$\Delta p<min\{{\eta }^a\Delta v,({\eta }^h-\beta )\Delta w\}$	1	1
2	-		$\Delta p>max\{{\eta }^a\Delta v,({\eta }^h-\beta )\Delta w\}$	2	2
3	${\eta }^a\Delta v<({\eta }^h-\beta )\Delta w$	3-1	$\Delta p={\eta }^a\Delta v$	1	Indifferent
		3-2	$\Delta p\in ({\eta }^a\Delta v,({\eta }^h-\beta )\Delta w)$	1	2
		3-3	$\Delta p=({\eta }^h-\beta )\Delta w$	Indifferent	2
4	${\eta }^a\Delta v>({\eta }^h-\beta )\Delta w$	4-1	$\Delta p=({\eta }^h-\beta )\Delta w$	Indifferent	1
		4-2	$\Delta p\in (({\eta }^h-\beta )\Delta w,{\eta }^a\Delta v)$	2	1
		4-3	$\Delta p={\eta }^a\Delta v$	2	Indifferent
5	${\eta }^a\Delta v=({\eta }^h-\beta )\Delta w$		$\Delta p={\eta }^a\Delta v$	Indifferent	Indifferent

.

Proposition 1.

(The parking preferences). For a single-route bottleneck with two parking clusters, the parking preferences of AV commuters and HV commuters depend on the relation between $\Delta p$, $\Delta v$, and $\Delta w$, specifically as follows:

• Case 1: When the difference in parking fee $\Delta p$ is very low, i.e., $\Delta p<min\{{\eta }^a\Delta v,({\eta }^h-\beta )\Delta w\}$, both AV commuters and HV commuters prefer the central (closer) parking cluster, i.e., the central cluster.
• Case 2: When the difference in parking fee $\Delta p$ is very high, i.e., $\Delta p>max\{{\eta }^a\Delta v,({\eta }^h-\beta )\Delta w\}$, both AV commuters and HV commuters prefer the peripheral (farther) parking cluster, i.e., the peripheral cluster.
• Otherwise, the cases of hybrid parking preferences are obtained. Moreover, we can further decompose the cases of hybrid parking preferences into three cases (Cases 3–5) and seven subcases, as shown in Table 2.

Proposition 1 indicates that the parking location preferences of commuters under a partially automated environment depend on the joint impacts of $\Delta p$, $\Delta w$, and $\Delta v$. Specifically, Proposition 1 reveals that (1) the closer parking cluster (the central cluster ) is more attractive to all commuters when the difference in parking fees charged at the two clusters is significantly low; (2) the farther parking cluster (the peripheral cluster) is more attractive to all commuters when the difference in parking fees charged at two clusters is significantly high; (3) when the difference in parking fees charged at two clusters varies in an intermediate range, the parking preferences are determined by the range of $\Delta p$ and the relation between ${\eta }^a\Delta v$ and $({\eta }^h-\beta )\Delta w$.

As stated in Proposition 1, when $\Delta p$ takes very low or very high values, the two groups of commuters have the same preference towards parking location, and they prefer only one of the two parking clusters. When $\Delta p$ takes very low values, i.e., $\Delta p<min\{{\eta }^a\Delta v,({\eta }^h-\beta )\Delta w\}$ such that $c_{t,2}^a>c_{t,1}^a$ and $c_{t,2}^h>c_{t,1}^h$, both AV commuters and HV commuters parking at the central cluster experience lower individual generalized travel costs compared with parking at the peripheral cluster. Thus, all commuters prefer the central cluster. Similarly, when $\Delta p$ takes very high values, i.e., $\Delta p>max\{{\eta }^a\Delta v,({\eta }^h-\beta )\Delta w\}$ such that $c_{t,2}^a<c_{t,1}^a$ and $c_{t,2}^h<c_{t,1}^h$, both AV commuters and HV commuters parking at the peripheral cluster experience lower individual generalized travel costs compared with parking at the central cluster. Thus, all commuters prefer the peripheral cluster when the difference in parking fees charged at the central and the peripheral clusters is very high.

Moreover, when $\Delta p$ varies in an intermediate range, i.e., $\Delta p\in [min\{{\eta }^a\Delta v,({\eta }^h-\beta )\Delta w\},max\{{\eta }^a\Delta v,({\eta }^h-\beta )\Delta w\}]$, we can characterize the hybrid parking preferences of AV commuters and HV commuters, as described in Cases 3–5. The hybrid case of parking preference describes that the two groups of commuters may have different preferences towards two parking clusters. Given ${\eta }^a\Delta v<({\eta }^h-\beta )\Delta w$ in Case 3, the impacts of walking time on the generalized travel costs of HV commuters are more significant than the impacts of self-driving time on the individual generalized travel costs of AV commuters. In this situation, we can further identify three subcases of parking preferences based on the specific levels of $\Delta p$, specifically as follows:

a.

When $\Delta p\in ({\eta }^a\Delta v,({\eta }^h-\beta )\Delta w)$, the AV commuters prefer the central cluster and HV commuters prefer the peripheral cluster, which corresponds to Case 3-2;

b.

When $\Delta p={\eta }^a\Delta v$ in Case 3-1, the AV commuters are indifferent between the two parking clusters and the HV commuters prefer the central cluster;

c.

When $\Delta p=({\eta }^h-\beta )\Delta w$ in Case 3-3, the AV commuters prefer the peripheral cluster and the HV commuters are indifferent between two parking clusters.

We can analyze the parking preferences of two groups of commuters for Case 4 in a similar way. In Case 5, where $\Delta p={\eta }^a\Delta v$ and ${\eta }^a\Delta v=({\eta }^h-\beta )\Delta w$, both AV and HV commuters are indifferent with parking at the central cluster and the peripheral cluster, as they respectively experience the same level of individual generalized travel cost, i.e., $c_{t,2}^a=c_{t,1}^a$ and $c_{t,2}^h=c_{t,1}^h$. And the equilibrium in Case 5 reduces to the classic morning commute problem with a single parking cluster.

Based on the results in Table 1, we can observe different transition patterns concerning equilibrium cases. Given fixed $\Delta v$ and $\Delta w$ that ensures ${\eta }^a\Delta v<({\eta }^h-\beta )\Delta w$, we can find that the equilibrium cases evolve in the order of Case 1, Case 3 and Case 2 when we increase $\Delta p$. Given fixed $\Delta v$ and $\Delta w$ that ensures ${\eta }^a\Delta v>({\eta }^h-\beta )\Delta w$, the equilibrium cases evolve in the order of Case 1, Case 4, and Case 2 when we increase $\Delta p$. Finally, we obtain Case 5 when ${\eta }^a\Delta v=({\eta }^h-\beta )\Delta w$ and $\Delta p={\eta }^a\Delta v$ are satisfied. We will further illustrate the transition among different equilibrium cases with respect to the range of $\Delta p$ in Section 6.2.

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 presented in 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. Parking patterns and travel costs at equilibrium.

Cases	Pattern Based on ${\mathit{K}}_1$	$({\mathit{n}}_1,{\mathit{n}}_2)$	${\mathit{c}}^{\mathit{a}}$	${\mathit{c}}^{\mathit{h}}$
1	(1) $K_1<N^h$	$(K_1,N-K_1)$	$\beta \frac{N^a}{s}+\frac{{\alpha }^a}{{\alpha }^h}(\frac{\beta N^h}{s}+({\eta }^h-\beta )\Delta w-\Delta p)+p_2+{\eta }^a{v}_2$	$\beta \frac{N}{s}+p_1+({\eta }^h-\beta )w_1$
	(2) $N^h\le K_1<N$	$(K_1,N-K_1)$	$\beta \frac{N^a}{s}+\frac{{\alpha }^a}{{\alpha }^h}\frac{\beta N^h}{s}+p_1+{\eta }^a{v}_1$
	(3) $K_1\ge N$	$(N,0)$
2	(4) $K_1$ free	$(0,N)$	$\beta \frac{N^a}{s}+\frac{{\alpha }^a}{{\alpha }^h}\frac{\beta N^h}{s}+p_2+{\eta }^a{v}_2$	$\beta \frac{N}{s}+p_2+({\eta }^h-\beta )w_2$
3-1	(1) $K_1<N^h$	$(K_1,N-K_1)$	$\beta \frac{N^a}{s}+\frac{{\alpha }^a}{{\alpha }^h}(\frac{\beta N^h}{s}+({\eta }^h-\beta )\Delta w-\Delta p)+p_2+{\eta }^a{v}_2$	$\beta \frac{N}{s}+p_1+({\eta }^h-\beta )w_1$
	(1) $N^h\le K_1<N^h+y$
	(5) $N^h+y\le K_1<N$	$(N^h+y,N^a-y)$	$\beta \frac{N^a}{s}+\frac{{\alpha }^a}{{\alpha }^h}\frac{\beta N^h}{s}+p_1+{\eta }^a{v}_1$
	(6) $K_1\ge N$
3-2	(1) $K_1<N^h$	$(K_1,N-K_1)$	$\beta \frac{N^a}{s}+\frac{{\alpha }^a}{{\alpha }^h}(\frac{\beta N^h}{s}+({\eta }^h-\beta )\Delta w-\Delta p)+p_2+{\eta }^a{v}_2$	$\beta \frac{N}{s}+p_1+({\eta }^h-\beta )w_1$
	(7) $K_1\ge N^h$	$(N^h,N^a)$	$\beta \frac{N^a}{s}+\frac{{\alpha }^a}{{\alpha }^h}\frac{\beta N^h}{s}+p_2+{\eta }^a{v}_2$
3-3	(8) $K_1<x$	$(K_1,N-K_1)$	$\beta \frac{N^a}{s}+\frac{{\alpha }^a}{{\alpha }^h}\frac{\beta N^h}{s}+p_2+{\eta }^a{v}_2$	$\beta \frac{N}{s}+p_1+({\eta }^h-\beta )w_1$
	(9) $K_1\ge x$	$(x,N-x)$
4-1	(8) $K_1<x$	$(K_1,N-K_1)$	$\beta \frac{N^a}{s}+\frac{{\alpha }^a}{{\alpha }^h}\frac{\beta N^h}{s}+p_2+{\eta }^a{v}_2$	$\beta \frac{N}{s}+p_1+({\eta }^h-\beta )w_1$
	(2) $x\le K_1<N^a+x$	$(K_1,N-K_1)$	$\beta \frac{N^a}{s}+\frac{{\alpha }^a}{{\alpha }^h}\frac{\beta N^h}{s}+p_1+{\eta }^a{v}_1$
	(10) $K_1\ge N^a+x$	$(N^a+x,N^h-x)$
4-2	(11) $K_1<N^a$	$(K_1,N-K_1)$	$\beta \frac{N^a}{s}+\frac{{\alpha }^a}{{\alpha }^h}\frac{\beta N^h}{s}+p_1+{\eta }^a{v}_1$	$\beta \frac{N}{s}+p_2+({\eta }^h-\beta )w_2$
	(12) $K_1\ge N^a$	$(N^a,N^h)$
4-3	(11) $K_1<y$	$(K_1,N-K_1)$	$\beta \frac{N^a}{s}+\frac{{\alpha }^a}{{\alpha }^h}\frac{\beta N^h}{s}+p_1+{\eta }^a{v}_1$	$\beta \frac{N}{s}+p_2+({\eta }^h-\beta )w_2$
	(13) $K_1\ge y$	$(y,N-y)$
5	(8) $K_1<x$	$(K_1,N-K_1)$	$\beta \frac{N^a}{s}+\frac{{\alpha }^a}{{\alpha }^h}\frac{\beta N^h}{s}+p_2+{\eta }^a{v}_2$	$\beta \frac{N}{s}+p_1+({\eta }^h-\beta )w_1$
	(8) $x\le K_1<x+y$	$(K_1,N-K_1)$
	(14) $K_1\ge x+y$	$(x+y,N-x-y)$

. 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:

$$\begin{array}{c}\hfill \frac{\mathrm{d}{c}_{t,j}^a}{\mathrm{d}t}=0\Rightarrow \frac{\mathrm{d}Q\left(t\right)}{\mathrm{d}t}=\frac{\beta s}{{\alpha }^a-\beta },\end{array}$$

(5)

$$\begin{array}{c}\hfill \frac{\mathrm{d}{c}_{t,j}^h}{\mathrm{d}t}=0\Rightarrow \frac{\mathrm{d}Q\left(t\right)}{\mathrm{d}t}=\frac{\beta s}{{\alpha }^h-\beta }.\end{array}$$

(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. For each case of parking preference, we can further categorize the equilibrium into subcases based on the relation between the parking capacity and parking demand associated with two clusters. For each possible travel pattern, we examine the equilibrium conditions to provide the explicit number of vehicles parking at each cluster and the commuters’ individual travel costs.

Instead of presenting the derivation procedures for each case, we focus on Case 3-2 with hybrid parking preferences to illustrate the derivation process considering that the procedures for deriving equilibrium solutions are similar for all cases. In Case 3-2, the HV commuters prefer the central parking cluster, and the AV commuters prefer the peripheral parking cluster. All AV commuters can park at the peripheral parking cluster because of its unlimited capacity. Since a fixed and limited parking capacity is assumed for the central cluster, the equilibrium travel patterns rely on the magnitude of the capacity of the central cluster ($K_1$) and the demand for the central cluster ($N_h$). All HV commuters can park at the closer cluster if its capacity is not lower than the demand of HV commuters, i.e., $K_1\ge N^h$. Otherwise, the central cluster can accommodate $K_1$ HV commuters, while the rest of the HV commuters will choose to park at the peripheral one. In the following, we derive the equilibrium patterns under conditions $K_1<N^h$ and $K_1\ge N^h$, separately.

Case 3-2: $K_1<N^h$

The sequence of departures and parking choices is as follows: (i) HV commuters depart from home before AV commuters, and $K_1$ commuters can park at the central cluster; (ii) then, the remaining $N^h-K_1$ HV commuters choose to depart from home, and they will park at the peripheral cluster; (iii) finally, all AV commuters will park at the peripheral cluster. The corresponding parking pattern is $(n_1^h,n_2^h)=(K_1,N^h-K_1)$ for HV commuters and $(n_1^a,n_2^a)=(0,N^a)$ for AV commuters. Thus, there are $K_1$ vehicles parking at the central cluster and $N-K_1$ vehicles parking at the peripheral cluster, which corresponds to a parking pattern $(n_1,n_2)=(K_1,N-K_1)$.

We incorporate the critical times to describe commuters’ departure time choices and parking location choices. Specifically, let $t_0$ and $t_1$ denote the departure times of the first HV commuter and the last HV commuter who park in the central parking cluster, respectively. Let $t_2$ denote the departure time of the first HV commuter who parks at the peripheral parking cluster. Let $t_3$ denote the departure time of the last HV commuter or the first AV commuter who parks at the peripheral parking cluster. Let $t_4$ denote the departure time of the last AV commuter who parks at the peripheral parking cluster. The equilibrium pattern is illustrated in Figure 3a.

Since all commuters have to pass the bottleneck before the desired arrival time and the first $K_1$ HV commuters park at the central cluster, we have the following relations between $t_0$ and $t_1$:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{t}^{*}-t_0=\frac{N}{s},\hfill \\ t_1-t_0=\frac{K_1}{r_h},\hfill \end{array}\right.\Rightarrow \left\{\begin{array}{c}{t}_0=t^{*}-\frac{N}{s}\hfill \\ t_1=t^{*}-\frac{N}{s}+\frac{{\alpha }^h-\beta }{{\alpha }^h}\frac{K_1}{s}\hfill \end{array}\right.\end{array}$$

The first HV commuter parking at the peripheral parking cluster experiences the same generalized travel cost as the last HV commuter parking at the central parking cluster, i.e., $c_{t_2,2}^h=c_{t_1,1}^h$, which gives

$$\begin{array}{c}\hfill t_2=t^{*}-\frac{N-K_1}{s}-\frac{\beta K_1}{{\alpha }^{h}s}+\frac{({\eta }^h-\beta )\Delta w-\Delta p}{{\alpha }^h}.\end{array}$$

(7)

As the remaining $N^h-K_1$ HV commuters choose to depart from home during the time interval $(t_2,t_3)$, we have the condition that

$$\begin{array}{c}\hfill t_3-t_2=\frac{N^h-K_1}{r^h}.\end{array}$$

Thus, the end of the departure times for HV commuters is given by

$$\begin{array}{ccc}\hfill t_3& =& t^{*}-\frac{N^a}{s}-\frac{\beta N^h}{{\alpha }^{h}s}+\frac{({\eta }^h-\beta )\Delta w-\Delta p}{{\alpha }^h}.\hfill \end{array}$$

(8)

where $t_3$ also represents the start of the departure times for AV commuters. Considering that the AV commuters prefer the farther (peripheral) cluster with sufficient parking slots, we have the relation between the end of the departure times for AV commuters $t_4$ and $t_3$, i.e., $t_4-t_3=\frac{N^a}{r_a}$. Therefore, the critical time $t_4$ is

$$\begin{array}{ccc}\hfill t_4& =& t^{*}-\frac{\beta N^a}{{\alpha }^{a}s}-\frac{\beta N^h}{{\alpha }^{h}s}+\frac{({\eta }^h-\beta )\Delta w-\Delta p}{{\alpha }^h}.\hfill \end{array}$$

(9)

At equilibrium, the individual generalized travel costs of AV and HV commuters are derived below.

$$\begin{array}{ccc}\hfill c^a& =& \beta \frac{N^a}{s}+\frac{{\alpha }^a}{{\alpha }^h}(\frac{\beta N^h}{s}+({\eta }^h-\beta )\Delta w-\Delta p)+p_2+{\eta }^a{v}_2,\hfill \\ \hfill c^h& =& \beta \frac{N}{s}+p_1+({\eta }^h-\beta )w_1.\hfill \end{array}$$

Case 3-2: $K_1\ge N^h$

In this case, the capacity of the central cluster is larger than the demand of HV commuters. The sequence of departures and parking choices is as follows: (i) HV commuters depart from home first, and all HV commuters park at the closer cluster; (ii) then, AV commuters depart from home, and they all park at the farther cluster. The parking pattern is $(n_1,n_2)=(N^h,N^a)$.

The start and end of the departure times from home for HV commuters are denoted by $t_0$ and $t_1$, respectively. $t_2$ is also the departure time of the first AV commuter. The end of the departure time for AV commuters is denoted by $t_3$. We can derive the equilibrium travel pattern by combining three conditions. The first condition states that all commuters will pass the bottleneck during the period $(t_0,t^{*})$, so no commuters will be late. The second condition describes that all HV commuters will depart from home during $(t_0,t_1)$. The last condition enforces that AV commuters depart from home during $(t_1,t_2)$. Thus, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{t}^{*}-t_0=\frac{N}{s},\hfill \\ t_1-t_0=\frac{N^h}{r^h},\hfill \\ t_2-t_1=\frac{N^a}{r^a},\hfill \end{array}\right.\Rightarrow \left\{\begin{array}{c}{t}_0=t^{*}-\frac{N}{s},\hfill \\ t_1=t^{*}-\frac{N^a}{s}-\frac{\beta }{{\alpha }^h}\frac{N^h}{s},\hfill \\ t_2=t^{*}-\frac{\beta }{{\alpha }^h}\frac{N^h}{s}-\frac{\beta }{{\alpha }^a}\frac{N^a}{s}.\hfill \end{array}\right.\end{array}$$

With the derived critical times, the equilibrium travel pattern when $K_1\ge N^h$ in Case 3-2 is depicted in Figure 3b.

Following a similar way, we can obtain the solutions to each case. Based on the critical times, we depict the equilibrium patterns for each case in Appendix A. The equilibrium parking patterns and the individual travel costs at UE are included in Table 3. In this table, x denotes the number of HV commuters who are indifferent between two parking clusters and choose to park at the closer cluster. x does not exceed the travel demand for HVs, i.e., $x\le N^h$. y denotes the number of AV commuters who are indifferent between two parking clusters and choose to park at the closer cluster. And $y\le N^a$.

4. The Impacts of Parking Capacity and Parking Fees

In what follows, we explore the respective effects of parking fees and parking capacity on system performance measured by total travel cost. Our results help us understand how the location of parking lots, the choice of parking lots, and the setting of parking pricing affect the total travel cost. Additionally, we provide insights into urban parking management for the government in the presence of both AV and HV commuters.

4.1. 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,

$$\begin{array}{c}\hfill \pi =N^h{c}^h+N^a{c}^a-n_1{p}_1-n_2{p}_2,\end{array}$$

(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 derivative of total travel cost $\pi$ with respect to the capacity of the closer cluster $K_1$, i.e., $\frac{\partial \pi }{\partial K_1}=-p_1\frac{d{n}_1}{d{K}_1}-p_2\frac{d{n}_2}{d{K}_1}$. The corresponding results of the derivative $\frac{\partial \pi }{\partial K_1}$ are provided in

Table A1. The derivatives of $\pi$ with respect to $K_1$, $p_1$, and $p_2$.

Cases	Condition on ${\mathit{K}}_1$	$\frac{\mathit{\partial }\mathit{\pi }}{\mathit{\partial }{\mathit{K}}_1}$	$\frac{\mathit{\partial }\mathit{\pi }}{\mathit{\partial }{\mathit{p}}_1}$	$\frac{\mathit{\partial }\mathit{\pi }}{\mathit{\partial }{\mathit{p}}_2}$
1	(1) $K_1<N^h$	$p_2-p_1$	$N^h-N^a\frac{{\alpha }^a}{{\alpha }^h}-K_1$	$-N^h+N^a\frac{{\alpha }^a}{{\alpha }^h}+K_1$
	(2) $N^h\le K_1<N$	$p_2-p_1$	$N-K_1$	$K_1-N$
	(3) $K_1\ge N$	0	0	0
2	$K_1$ free	0	0	0
3-1	$K_1<N^h+y$	$p_2-p_1$	$N^h-N^a\frac{{\alpha }^a}{{\alpha }^h}-K_1$	$-N^h+N^a\frac{{\alpha }^a}{{\alpha }^h}+K_1$
	$N^h+y\le K_1<N$	$p_2-p_1$	$N^a-y$	$y-N^a$
	$K_1\ge N$	0	$N^a-y$	$y-N^a$
3-2	$K_1<N^h$	$p_2-p_1$	$N^h-N^a\frac{{\alpha }^a}{{\alpha }^h}-K_1$	$-N^h+N^a\frac{{\alpha }^a}{{\alpha }^h}+K_1$
	$K_1\ge N^h$	0	0	0
3-3	$K_1<x$	$p_2-p_1$	$N^h-K_1$	$K_1-N^h$
	$K_1\ge x$	0	$N^h-x$	$x-N^h$
4-1	$K_1<x$	$p_2-p_1$	$N^h-K_1$	$K_1-N^h$
	$x\le K_1<N^a+x$	$p_2-p_1$	$N-K_1$	$K_1-N$
	$K_1\ge N^a+x$	0	$N^h-x$	$x-N^h$
4-2	$K_1<N^a$	$p_2-p_1$	$N^a-K_1$	$K_1-N^a$
	$K_1\ge N^a$	0	$N^h$	$-N^h$
4-3	$K_1<y$	$p_2-p_1$	$N^a-K_1$	$K_1-N^a$
	$K_1\ge y$	0	$N^a-y$	$y-N^a$
5	$K_1<x$	$p_2-p_1$	$N^h-K_1$	$K_1-N^h$
	$x\le K_1<x+y$	$p_2-p_1$	$N^h-K_1$	$K_1-N^h$
	$K_1\ge x+y$	0	$N^h-x-y$	$x+y-N^h$

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 π 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 $N^h$. When the parking capacity of the closer cluster is greater than or equal to its potential parking demand, i.e., $K_1\ge N^h$, the first-order derivative of the total travel cost with respect to $K_1$ is 0. The total travel cost does not change with the capacity of the closer cluster. When the parking capacity of the closer cluster is lower than its potential parking demand, i.e., $K_1<N^h$, the first-order derivative of the total travel cost with respect to $K_1$ is $p_2-p_1<0$. The total travel cost decreases with $K_1$.

We can further elaborate on the tendency of the total travel cost against the parking capacity of the central cluster. All commuters who prefer the central cluster can park at this cluster when its parking capacity is sufficient, i.e., $K_1\ge N^h$. Moreover, in this case, it is impossible to increase the number of commuters who choose to park at the central cluster. The effective parking capacity of the central cluster, i.e., the actual number of vehicles parking at the central cluster, is the same as the potential parking demand for this cluster. In this situation, the total travel cost remains constant when we increase $K_1$. Otherwise, when the parking capacity of the central cluster is insufficient to meet the potential parking demand ($K_1<N^h$), we can increase the effective parking capacity by enlarging $K_1$. According to Equation (10), with more vehicles parking at the central cluster, the total travel cost becomes lower as the total schedule delay cost and self-driving time cost decrease. At the same time, more revenue is collected from commuters, which also contributes to the decrease in the total travel cost. It implies that the public parking manager can attract more vehicles to choose the central cluster by setting more parking slots so as to improve system performance.

4.2. Parking Pricing Scheme

Furthermore, we investigate the influence of parking prices on the total travel cost to understand how the parking manager can improve social welfare by adjusting parking prices. To show the impacts of parking fees on the total travel cost, we take the first-order partial derivatives of total travel cost $\pi$ in Equation (10) with respect to $p_1$ and $p_2$, respectively. The results of $\frac{\partial \pi }{\partial p_1}$ and $\frac{\partial \pi }{\partial p_2}$ are included in Table A1 in Appendix B. Obviously, the parking prices charged at the central cluster and the peripheral cluster impose opposite impacts on the total travel cost. Thus, we focus on the analysis regarding the parking price $p_1$ for simplicity.

Case 5 represents a special case since the conditions ${\eta }^a\Delta v=({\eta }^h-\beta )\Delta w$ and ${\eta }^a\Delta v=\Delta p$ are very strict and may hardly be observed in practice. In this case, both AV and HV commuters are indifferent with the two parking clusters, which leads the equilibrium problem to the classic morning commute problem with a single cluster. Thus, we only focus on Cases 1–4 in the following analysis. By excluding Case 5, we obtain the tendency of the total travel cost against the parking price at the central cluster in Proposition 4.

Proposition 4.

The impacts of parking prices at two clusters on the total travel cost can be categorized into the following scenarios.

(a) 

For Case 1 with $K_1\ge N$ and Case 2 and Case 3-2 with $K_1\ge N^h$, the total travel cost does not vary with $p_1$.

(b) 

For Case 1 with $K_1<N^h$, Case 3-1 with $K_1<N^h+y$, and Case 3-2 with $K_1<N^h$, the total travel cost may increase or decrease with $p_1$, depending on the specific level of $K_1$.

(c) 

Otherwise, the total travel cost does not decrease with $p_1$.

Proposition 4 shows that the system performance can be improved by adjusting the parking prices at the central. In particular, for cases in Scenario (c) as stated in Proposition 4, the total travel cost becomes higher when the parking price at the closer cluster $p_1$ is higher. Meantime, the total travel cost will decrease with the parking price at the farther cluster $p_2$. In other words, increasing $p_2$ helps to improve social welfare while increasing $p_1$ will harm system performance in most cases. To achieve a better system performance, it is required to set the parking price at the closer cluster at a lower level or set the parking price at the farther cluster at a higher level. For Scenario (a), the total travel cost does not vary with $p_1$ or $p_2$. So, we could not reduce the total travel cost by adjusting the prices. The reason is that the potential parking demand for the central cluster can be met. In other words, the capacity of the central cluster is sufficient to accommodate all potential users. For Scenario b, the total travel cost may increase with $p_1$ if $K_1$ is low, but decreases with $p_1$ if $K_1$ is high. The joint effects of parking prices $\pi$ and parking capacity $K_1$ on the total travel cost are demonstrated in Section 6.2. Proposition 4 provides insights for public parking managers in order to better regulate parking pricing schemes and improve system performance.

The findings provided in Proposition 4 describe the tendency of the total travel cost for each equilibrium case. It is still unclear that how the total travel cost differs among different equilibrium cases. Therefore, we further investigate how the minimized total travel cost can be achieved through the optimal design of parking prices in the next section.

5. Optimal Parking Pricing Scheme

Based on the equilibrium results derived in the previous sections, we further examine the optimal design of parking prices charged at the central cluster to minimize the total travel cost. Recalling the analysis in Section 2.2, the parking preferences and the occurrence of equilibrium cases depend on $\Delta p$, $\Delta v$ and $\Delta w$. Additionally, we can obtain the transition patterns of equilibrium cases based on the relative magnitudes of $\Delta p$, ${\eta }^a\Delta v$ and $({\eta }^h-\beta )\Delta w$. Here, we investigate the optimal design of parking management policies following the same way. When ${\eta }^a\Delta v=({\eta }^h-\beta )\Delta w$, i.e., in Case 5, both AV commuters and HV commuters are indifferent between the closer and the farther parking clusters. The equilibrium problem is reduced to a simplified version with only one parking cluster. In the following, we mainly focus on the scenario under condition ${\eta }^a\Delta v<({\eta }^h-\beta )\Delta w$. For the scenario under condition ${\eta }^a\Delta v>({\eta }^h-\beta )\Delta w$, we can perform the analysis following similar procedures.

The optimal parking pricing scheme for the central cluster can be achieved by exploring the tendency of the total travel cost $\pi$ when the parking capacity $K_1$ and the difference in parking fees $\Delta p$ jointly vary. Thus, given fixed $p_2$, $v_j$ and $w_j$, we derive the first-order derivative of the total travel cost $\pi$ (Equation (10)) with respect to the parking price $p_1$, i.e., $\frac{\partial \pi }{\partial p_1}$. We find that the sign of $\frac{\partial \pi }{\partial p_1}$ is affected by the relation between x and $N^h-\frac{{\alpha }^a}{{\alpha }^h}{N}^a$, in addition to the ranges of $K_1$ and $\Delta p$. To facilitate the analysis, we summarize the sign of the first-order derivative $\frac{\partial \pi }{\partial p_1}$ in

Table 4. The sign of $\frac{\partial \pi }{\partial p_1}$ when ${\eta }^a\Delta v<({\eta }^h-\beta )\Delta w$ and $x<N^h-\frac{{\alpha }^a}{{\alpha }^h}{N}^a$.

${\mathit{K}}_1$	$\mathbf{\Delta }\mathit{p}$
	$(\mathbf{0},{\mathsf{\eta }}^{\mathit{a}}\mathbf{\Delta }\mathit{v})$	${\mathsf{\eta }}^{\mathit{a}}\mathbf{\Delta }\mathit{v}$	$({\mathsf{\eta }}^{\mathit{a}}\mathbf{\Delta }\mathit{v},({\mathsf{\eta }}^{\mathit{h}}-\mathsf{\beta })\mathbf{\Delta }\mathit{w})$	$({\mathsf{\eta }}^{\mathit{h}}-\mathsf{\beta })\mathbf{\Delta }\mathit{w}$		$(({\mathsf{\eta }}^{\mathit{h}}-\mathsf{\beta })\mathbf{\Delta }\mathit{w},+\infty )$
	Case 1	Case 3-1	Case 3-2	Case 3-3		Case 2
				${\mathit{K}}_{\mathit{1}}<\mathit{x}$	${\mathit{K}}_{\mathit{1}}\ge \mathit{x}$
$(0,x)$	$>0$	$>0$	$>0$	$>0$	-	0
$[x,N^h-\frac{{\alpha }^a}{{\alpha }^h}{N}^a)$	$>0$	$>0$	$>0$	-	$\ge 0$	0
$[N^h-\frac{{\alpha }^a}{{\alpha }^h}{N}^a,N^h)$	$<0$	$<0$	$<0$	-	$\ge 0$	0
$[N^h,N^h+y)$	$>0$	$<0$	0	-	$\ge 0$	0
$[N^h+y,N)$	$>0$	$\ge 0$	0	-	$\ge 0$	0
$[N,+\infty )$	0	$\ge 0$	0	-	$\ge 0$	0

and

Table 5. The sign of $\frac{\partial \pi }{\partial p_1}$ when ${\eta }^a\Delta v<({\eta }^h-\beta )\Delta w$ and $x\ge N^h-\frac{{\alpha }^a}{{\alpha }^h}{N}^a$.

${\mathit{K}}_1$	$\mathbf{\Delta }\mathit{p}$
	$(\mathbf{0},{\mathsf{\eta }}^{\mathit{a}}\mathbf{\Delta }\mathit{v})$	${\mathsf{\eta }}^{\mathit{a}}\mathbf{\Delta }\mathit{v}$	$({\mathsf{\eta }}^{\mathit{a}}\mathbf{\Delta }\mathit{v},({\mathsf{\eta }}^{\mathit{h}}-\mathsf{\beta })\mathbf{\Delta }\mathit{w})$	$({\mathsf{\eta }}^{\mathit{h}}-\mathsf{\beta })\mathbf{\Delta }\mathit{w}$		$(({\mathsf{\eta }}^{\mathit{h}}-\mathsf{\beta })\mathbf{\Delta }\mathit{w},+\infty )$
	Case 1	Case 3-1	Case 3-2	Case 3-3		Case 2
				${\mathit{K}}_{\mathit{1}}<\mathit{x}$	${\mathit{K}}_{\mathit{1}}\ge \mathit{x}$
$(0,N^h-\frac{{\alpha }^a}{{\alpha }^h}{N}^a)$	$>0$	$>0$	$>0$	$>0$	-	0
$[N^h-\frac{{\alpha }^a}{{\alpha }^h}{N}^a,x)$	$<0$	$<0$	$<0$	$>0$	-	0
$[x,N^h)$	$<0$	$<0$	$<0$	-	$\ge 0$	0
$[N^h,N^h+y)$	$>0$	$<0$	0	-	$\ge 0$	0
$[N^h+y,N)$	$>0$	$\ge 0$	0	-	$\ge 0$	0
$[N,+\infty )$	0	$\ge 0$	0	-	$\ge 0$	0

. The two tables also illustrate the transition of equilibrium cases when $\Delta p$ changes.

From Table 4 and Table 5, we can find that the tendency of $\pi$ relies on the range of $k_1$ and $\Delta p$. Given the fixed parking price at the peripheral cluster $p_2$, each row in Table 4 and Table 5 reflects how the total travel cost $\pi$ changes with the parking price at the central cluster $p_1$. In this case, we are able to determine the minimum point of the total travel cost within an interval of $\Delta p$ and an interval of $K_1$. However, given $K_1$, the total travel cost $\pi$ may be discontinuous at the boundaries of each interval of $\Delta p$. Thus, we need to combine both the sign of the first-order partial derivative and the level of $\pi$ at each boundary to obtain the minimized total travel cost. We provide the optimal price at the central cluster $p_1$ when its parking capacity varies in Proposition 5.

Proposition 5.

(The optimal parking price $p_1$ when ${\eta }^a\Delta v<({\eta }^h-\beta )\Delta w$). At equilibrium, the optimal parking price charged at the central cluster depends on the level of its parking capacity $K_1$, specifically as follows:

• When $K_1\in [N^h-\frac{{\alpha }^a}{{\alpha }^h}{N}^a,N^h)$, the minimized total travel cost is observed in Case 3-3 and the optimal parking price at the central cluster is given by $p_1^{*}=p_2+({\eta }^h-\beta )\Delta w$.
• Otherwise, the total travel cost is minimized when $p_1$ approaches $p_2$.

Proposition 5 indicates that the optimal parking price at the central cluster should be determined based on its capacity. First, when the parking capacity of the central cluster is at the intermediate level, i.e., $K_1\in [N^h-\frac{{\alpha }^a}{{\alpha }^h}{N}^a,N^h)$, the corresponding parking price should be set as $p_1^{*}=p_2+({\eta }^h-\beta )\Delta w$. That is, a higher parking price should be charged from commuters using the central cluster compared with the peripheral one. By changing the parking price charged at the central cluster, we may observe different equilibrium patterns with varying total commuting cost $N^h{c}^h+N^a{c}^a$ and total revenue collected from commuters $n_1{p}_1+n_2{p}_2$. In Case 3-3, the difference between the total commuting cost and the total collected revenue arrives at a minimum point. Second, the optimal parking price charged at the central cluster should be close to that imposed at the peripheral parking in all other equilibrium cases, which is consistent with the conclusion drawn in Proposition 4. In other words, there is no need to impose the discriminative parking prices at the two different parking clusters. Specifically, when the parking capacity $K_1$ is very high, it is sufficient to meet the potential parking demand at the central cluster. Thus, the increase in the parking price at the central cluster may not affect the parking pattern or help to improve social welfare. When the parking capacity $K_1$ is very low, some potential demand for parking at the central cluster will have to choose the peripheral one. Then, the adjustment of the parking price charged at the central cluster may not help improve the system performance. Proposition 5 provides the policy guidance for designing an optimal parking price scheme considering the capacity of the central cluster near the workspace in order to improve social welfare in the morning commute.

6. Numerical Analysis

In this part, we conduct numerical experiments to demonstrate the findings achieved in previous sections. Section 6.2 presents the joint impacts of parking capacity and parking pricing on the total travel cost. We show the impacts of the AV penetration on system performance in Section 6.3. Finally, we perform sensitivity analysis to show the influences of the network configuration on system performance in Section 6.4.

6.1. Parameter Setting

We first introduce the inputs for parameters used in this study. Following the work by [19], we adopt the value of travel time as ${\alpha }^a=15$ USD/h for AV commuters and ${\alpha }^h=20$ USD/h for HV commuters. And the unit cost of late schedule delay time is $\beta =7$ USD/h. We assume the bottleneck capacity is $s=1000$ vehicles/h. The demand of AV commuters is $N^a=1000$ vehicles, and the demand of HV commuters is $N^h=1500$ vehicles.

In the base case, we consider the fixed network configuration where the distance between each cluster and the workplace is given. Under a stable travel environment, the walking time from cluster j to the workplace for HV users and the self-driving time from the workplace to cluster j for AV users are assumed fixed and constant. Here, the walking time for HV commuters is $w_1=0.3$ h from the central cluster to the workplace and is $w_2=1.2$ h from the peripheral cluster to the workplace. As for AV commuters, the self-driving times from the workplace to the central and the peripheral clusters are $v_1=0.2$ h and $v_2=0.8$ h, respectively. We adopt $p_1=2$ $/h and $p_2=0$ in the base case. As discussed in Section 6.2, we vary the parking price at the central cluster $p_1$ and keep the price charged at the peripheral cluster $p_2$ fixed. The input parameters are given in

Table 6. Input.

Parameter	Value	Unit
${\alpha }^a$	15	USD/h
${\alpha }^h$	20	USD/h
${\eta }^a$	14	USD/h
${\eta }^h$	25	USD/h
$\beta$	10	USD/h
s	1000	vehicles/h
$N^a$	1000	vehicles
$N^h$	1500	vehicles
$p_2$	0	USD

.

6.2. 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 $p_1$ 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 4a and Figure 4b, respectively.

Figure 5a 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 5b. 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 4b, is positive. Later, when $K_1$ increases, the total travel cost in Figure 5b remains constant. The corresponding unfulfilled demand is zero, as shown in Figure 4b. 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 5b 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 5b. 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 5a, is positive. Later, when $K_1$ increases, the total travel cost in Figure 5b remains constant. The corresponding unfulfilled demand is zero, as shown in Figure 5a. 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.

Furthermore, we discuss the influences of the parking price at the central cluster $p_1$ on the total travel cost $\pi$. According to Proposition 4, we can divide the equilibrium cases into two scenarios: $\pi$ does not vary with $p_1$, and $\pi$ increases with $p_1$. From Figure 5b, the total travel cost does not change with $p_1$ in Case 2, which is the same as Case 1 and Case 3 when $K_1$ is large. Specifically, $\pi$ is constant and independent of the parking price $p_1$ in Case 1 when $K_1$ is above 2500, which is the total demand N in Pattern (3). When $K_1$ is larger than or equal to $N^h=1500$ in Case 3-2, we also find that $\pi$ becomes constant. In addition, the total travel cost keeps constant with respect to $p_1$ in Case 2. Those findings correspond to scenario (a) in Proposition 4. However, we observe that the total travel cost $\pi$ increases with parking price $p_1$ in all other cases, which is consistent with Scenario (b) in Proposition 4.

6.3. Impacts of AV Penetration

The penetration of AVs in the morning commute may vary with the development of AV technology. It is essential to understand how AV penetration affects the system performance measured by the total travel cost. Thus, we proceed to study the change in the total travel cost against the parking price at the central cluster ($p_1$) and the demand for AVs ($N_a$). Specifically, we fix the total travel demand at $N=3000$ and vary the demand for AVs in the range of $[0,3000]$ in the following numerical example. And parking price $p_1$ takes three different values: 1, 4, and 7 USD/h. The parking capacity of the central cluster is fixed at $K_1=1000$. The total travel cost is depicted in Figure 6.

As shown in Figure 6, the total travel cost under a high level of demand for AVs ($N_a\ge 2000$) is lower than that under a low or medium demand for AVs ($N_a<2000$). The results reveal that the system performance can be improved when the demand for AVs is high, or the demand for HVs does not exceed $K_1$. Additionally, we find that the effects of parking prices on the system performance depend on the AV penetration. When the demand for AVs varies at the intermediate level, we can improve the system performance by increasing the parking price at the central cluster. In contrast, when the demand for AVs is at a low or high level, the system performance can be improved by setting a low parking price $p_1$.

6.4. Sensitivity Analysis

The spatial distribution of parking slots affects commuters’ decisions and thereby influences system performance. In this part, we perform sensitivity analysis to evaluate the changes in the total travel cost to explore the effects of network configuration on system performance. We vary the walking time from the peripheral cluster to the workplace and the capacity of the central cluster to show how the total travel cost $\pi$ changes with $w_2$ and $K_1$. When the peripheral cluster is farther from the workplace, the HV commuters experience a longer walking time to arrive at the workplace. At the same time, AV commuters experience a relatively longer self-driving time after dropping off at the workplace. By assuming that the ratio between the walking and the self-driving speeds is fixed, we also vary the level of $v_2$ in this example. Specifically, we vary $w_2$ in the range of $[1.5,3.0]$, and the ratio between $w_2$ and $v_2$ is 1.5. The capacity of the central cluster $K_1$ takes values in the range of $[0,3000]$. The parking price at the central cluster is set as $p_1=\$5$. Regarding other parameters, we follow the parameter setting described in Section 6.1.

Figure 7 depicts the total travel cost against $w_2$ and $K_1$. As shown in this figure, the total travel cost does not change when the capacity of the central cluster $K_1$ is high. When $K_1$ is around or below 1500, the total travel cost increases with $w_2$ and decreases with $K_1$. When the distance between the peripheral cluster and the workplace is large, the total travel cost of commuters also becomes more significant. It indicates that when the capacity of the central cluster is insufficient, it is better to deploy more parking slots near the workplace to improve system performance.

7. Conclusions

This paper investigates the decision making of commuters and the impacts of parking management policies on the morning commute problem under a partially automated environment. We consider a single-corridor network with two discrete parking clusters near the workplace. The commuters make joint decisions on departure time choices and parking location choices for completing the trip from home to work. To formulate the joint decisions of commuters, we extend Vickrey’s bottleneck model by incorporating the influences of parking locations and the travel modes used by travelers. We derive the equilibrium conditions to solve the dynamic user equilibrium problem. The equilibrium state is obtained when no commuters can further reduce their travel costs by unilaterally altering departure times or parking location choices. Moreover, we examine the impacts of parking fees and parking capacity provision on system performance to provide parking management implications.

Considering that parking slots located in the central area and the peripheral area are offered, we examine the parking preferences of AV commuters and HV commuters for commuting trips separately. Considering that commuters make parking location choices to minimize their travel costs, we analyze the parking preferences by comparing the individual travel costs of a commuter between choosing two different parking clusters. We find that the parking preferences of commuters depend on the range of the difference in parking prices and the travel time between the workplace and parking location. Specifically, our analytical results reveal that all commuters prefer the central cluster when the difference in parking fees between the two clusters is low. When the difference in parking fees is very high, all commuters prefer the peripheral cluster in order to reduce their travel costs. Otherwise, the parking preferences of commuters are characterized by both the range of differences in parking prices and the relation between the self driving time of AV commuters and the walking time of HV commuters for commuting between the workplace and the parking cluster.

Furthermore, we investigate the impacts of the parking fee and the parking capacity of the central cluster on the total travel cost. With insufficient parking capacity at the central cluster, the total travel cost will decrease if we offer more parking slots at this cluster. This finding implies that we can attract more commuters to parking at the central cluster if we can offer more parking slots to the commuters who prefer this cluster. As for the effects of parking fees, we find that the total travel cost increases with the parking price at the central cluster, whereas it decreases with the parking price at the peripheral cluster. Thus, it is possible to improve system performance by lowering the parking price at the central cluster when needed. Based on the aforementioned equilibrium analysis, we investigate the optimal design of the parking price at the central cluster in order to minimize the total travel cost. The results regarding the parking capacity and the parking fee of the central cluster provide managerial insights to the public parking manager in order to better regulate the parking market for the morning commute.

In future research, we may extend this work in the following directions. First, we may consider more realistic parking location scenarios and investigate the optimal parking provision schemes. Second, the long-term impacts of the introduction of autonomous vehicles should be incorporated by formulating the mode choice of commuters.