Is It Necessarily Better for More Commuters to Share a Vehicle?

Abstract

Increasing private car ownership has congested urban roads and made parking more difficult, especially during the morning commute. Carpooling offers a new way to address these problems. This paper studies the dynamic departure patterns for both regular and carpooling vehicles with parking space constraints in the morning commute without the carpool lane. The results suggest that as the parking fee of the central cluster increases, the earliest time for the two types of vehicles to enter the central cluster is delayed. The increase in the proportion of regular vehicles delays the earliest time for carpooling vehicles to enter the central cluster. More commuters sharing a vehicle in the morning commute is not necessarily better. Only a reasonable level of carpooling can reduce the peak time and unnecessary time consumption on the road and effectively promote the reduction in parking fees, commuters’ travel costs, and other societal transportation costs. This research gives practical guidance and suggestions on formulating a reasonable parking fee and controlling a reasonable carpooling level.

1. Introduction

Car ownership is rising every year. According to the latest data, China’s car ownership in 2023 is 336 million, and car drivers are 486 million [1]. Increasing private car ownership has made urban roads congested and parking more difficult, especially during the morning commute [2]. During the morning commute, commuters flock to the Central Business District (CBD), making the roads around the CBD more congested. When they arrive in the CBD, commuters are also faced with parking difficulties due to the limited parking spaces available in the CBD [3,4]. Congestion and parking difficulties have put enormous pressure on road traffic around the CBD, also causing a serious impact on the transport efficiency of the city as a whole and further having a huge impact on the efficiency of the urban economy [5,6,7]. In recent years, carpooling has emerged as a new way to address these issues and has gained popularity among some commuters [8,9,10]. When the driver is a solo commuter, we regard this travel behavior as a self-driving behavior and regard this vehicle as a regular vehicle; when commuters choose to carpool to travel, then this travel behavior is regarded as a carpooling behavior, and this vehicle is a carpooling vehicle. In this paper, there is no separate lane for carpooling vehicles. Once in the CBD, both travel modes require further consideration of parking. The parking lots in the CBD are divided into two clusters: the central cluster and the peripheral cluster. The parking lots of the central cluster are close to the CBD. There is no additional cost from the central cluster to the office, but the parking fee is high, and there is a parking space constraint for both types of vehicles. For the peripheral cluster, there is a certain distance between the parking lot of the peripheral cluster and the office. It requires additional costs to return to the office from the peripheral cluster, but the parking fee is low, and the parking space constraint for the two types of vehicles is not considered. How do parking options and space constraints affect commuters’ travel behaviors during the morning commute? Is it necessarily better for more commuters to share a vehicle on morning commutes? The motivation for this paper is to try to address these questions.

The above questions fall within the research field of the travel behaviors of commuters. In this research field, there are two research branches that are closely related to the above questions: one is the impacts of parking on commuters’ travel behaviors, and the other is the impacts of carpooling on commuters’ travel behaviors. Reviewing the literature on these two branches, it can be seen that the following research gaps exist: First, in the research field of the impact of commuters’ parking choices on their travel behaviors, previous studies mostly considered the impact of one parking space choice on commuters’ travel behaviors, while few studies considered the impact of multiple parking choices on commuters’ travel behaviors. According to the different conditions and the parking fee of each parking choice, commuters usually choose the most suitable from multiple parking choices, affecting their travel behavior. Therefore, it is necessary to take into account the impact of multiple parking choices on commuters’ travel behaviors. Second, much of the literature assumes that only regular vehicles consider the parking space constraint. Little attention has been given to the impacts of the parking space constraint for both regular vehicles and carpooling vehicles on commuters’ travel behaviors. In reality, like regular vehicles, carpooling vehicles also need to consider parking constraints. Third, few studies have considered the non-existence of the carpool lane. In the vast majority of cases, carpooling vehicles need to share the same lane as regular vehicles and do not have an exclusive lane.

The objective of our research is to fill in these gaps in the literature. This paper investigates the dynamic departure patterns of regular vehicles and carpooling vehicles when both types of vehicles have parking space constraints without the carpool lane.

Our study makes three important contributions. First, our research narrows the literature gap on the impact of parking choices on commuters’ travel behaviors. Many studies have only discussed the impact of one parking choice on commuters’ travel behaviors. In this paper, we discuss that commuters choose the most suitable parking space from multiple parking choices. Second, the previous studies only discussed parking constraints for regular vehicles, neglecting to consider parking constraints for both types of vehicles and assuming a separate lane for carpooling vehicles. Without the carpool lane, this paper studies the impact of the parking space constraint for both travel modes on commuters’ travel behaviors. Third, from a practical perspective, our study gives practical guidance and suggestions to formulate a reasonable parking fee and control a reasonable carpooling level. This has important implications for achieving sustainable green commuting.

The remainder of the paper is structured as follows: Section 2 reviews the relevant literature. Section 3 formulates the research problem. And Section 4 presents the departure patterns of regular vehicles and carpooling vehicles with and without the parking space constraint. The impacts of the penetration rate of commuters using regular vehicles and the number of commuters in each carpooling vehicle on the departure patterns of the two types of vehicles are further discussed in Section 5. In Section 6, the numerical studies are reported to validate the analytical results. Finally, Section 7 concludes the paper and proposes future research directions.

2. Literature Review

This paper falls within the research scope of the travel behaviors of commuters. In this field, there are two branches of the literature that are closely related to this paper: one is the impact of parking on commuters’ travel behaviors, and the other is the impact of carpooling on commuters’ travel behaviors. Therefore, in the following subsections, the literature on these two branches is reviewed in detail.

2.1. Impacts of Parking on Commuters’ Travel Behaviors

Around the 1970s, parking-related studies emerged, which focused on the impacts of different parking attributes on parking decisions [11,12]. Linking the morning and evening commutes, Zhang et al. [13] propose a time-varying road toll and location-dependent parking fee regime to reduce the cost of morning schedule delays. Qian et al. [14] investigated how to design the parking fee and the parking supply to alleviate traffic congestion and reduce the total social costs. They suggest that the parking fee and capacity should be set so that commuters would prefer to park further away when they arrive early. Considering the situation where some commuters have reserved parking spots while others have to compete for public spots on a first-come, first-served basis, Yang et al. [7] found that an appropriate combination of reserved and unreserved parking spaces can effectively alleviate traffic congestion and reduce social costs. Liu et al. [15] further investigated that commuters with a parking reservation have to arrive at parking spaces for the reservation before a predetermined expiration time. They found that the total travel cost can be further reduced compared to the reservation scheme without an expiration time. Liu et al. [16] investigated the impact of parking restrictions and parking reservations for two modes of travel (driving travel and public travel) and found that when the supply of parking spaces is less than the potential demand but relatively large, it is favored to reserve some parking spaces for competition. Zhang et al. [17] investigated the daily commuting and parking problems with autonomous vehicles and suggested an optimal parking space supply under the minimum total system cost, including parking costs.

It can be found that previous studies mostly considered the impact of one parking space choice on commuters’ travel behaviors, while few studies considered the impact of multiple parking choices on commuters’ travel behaviors.

2.2. Impacts of Carpooling on Commuters’ Travel Behaviors

Due to the advantages of carpooling, it has gradually become a travel choice for many commuters. Some researchers have conducted a series of studies on the impacts of carpooling attributes on commuters’ travel behaviors. Discussing the impact of providing a special lane for carpoolers, Qian and Zhang [18] found that enlarging the special lane may reduce transit ridership and increase auto travel. Similarly, assuming that carpooling vehicles have a dedicated lane, the research of Xiao et al. [19] was under the condition of the carpool lane being available only within a reserved time window. They propose a joint temporal–spatial capacity allocation on the morning commute and further found that an optimal temporal allocation scheme requires prior knowledge of the extra carpool costs.

Some researchers are concerned about how to incentivize travelers to choose to carpool. Xu et al. [20] investigated how to incentivize travelers to participate in carpooling and the impact of congestion on carpooling. They found that fewer travelers participate in carpooling when the congestion cost decreases or the carpooling inconvenience cost increases. Considering a carpooling platform, a variable-ratio charging-compensation scheme, to provide the incentive for carpooling behavior, Wang et al. [21] investigated the impacts of this scheme and found the optimal VCS uniquely exists when the platform maximizes its profit. Similarly, considering the subsidy for carpooling, Yu et al. [22] investigated the impacts of the heterogeneity of commuters on the efficiency of the carpooling subsidy. And they found that users benefit from the second-best subsidization policy regardless of whether they arrive in the peak or in the shoulder. Combining the tolling-rewarding scheme with the temporal capacity allocation scheme for carpooling, Wei et al. [23] suggest that carpooling should not always be rewarded. Liu and Li [24] derived a time-varying toll combined with a flat carpooling price and found that under the derived optimum toll, carpooling can attract more commuters.

Other researchers have focused on the impact of parking constraints on carpooling. Assuming that carpooling vehicles have a dedicated lane, Xiao et al. [25] investigated commuters’ behavior under three modes, which are solo driving, carpooling, and using public transit, under the parking space constraint. And they found that with all three modes of travel coexisting and the decrease in parking spots, the number of carpoolers increases first and then decreases. Researching dynamic parking fees and carpooling payments in a situation where only carpooling travel exists, Ma and Zhang [26] found that dynamic parking fees and carpooling payments mitigate traffic congestion and reduce the vehicle miles traveled and vehicle hours traveled.

By reviewing the literature in this research branch, it can be found that little attention has been given to the impacts of parking space constraints for both regular vehicles and carpooling vehicles on commuters’ travel behaviors. And few studies discuss the absence of a dedicated lane for carpooling vehicles; that is, carpooling vehicles use the same lane as regular vehicles.

2.3. The Gaps

By reviewing the relevant literature on two research branches of commuters’ travel behaviors, it can be found that the current research has the following gaps: First, few studies discuss the impact of multiple parking choices on commuters’ travel behaviors. Previous studies mostly assumed that commuters only have one parking choice and ignored the commuters’ parking choice between multiple parking spots. Second, previous studies only focus on parking constraints for regular vehicles. In reality, carpooling vehicles also require the consideration of parking constraints. There is a lack of research on the impact of parking space constraints for both types of vehicles, regular vehicles and carpooling vehicles, on commuters’ travel behaviors. Third, few studies consider the absence of a carpool lane. In reality, carpooling vehicles need to use the same lane as regular vehicles.

This paper fills these above gaps. The differences between this paper and existing research is detailed in

Table 1. The differences between this paper and existing research.

References	Multiple Parking Choices	Parking Constraints for Regular Vehicles and Carpooling Vehicles	Carpooling Vehicles Use the Same Lane as Regular Vehicles
[13]	✕	/	/
[14]	✓	/	/
[7]	✕	/	/
[15]	✕	/	/
[16]	✕	/	/
[17]	✕	/	/
[18]	✕	/	✕
[19]	✕	/	✕
[20]	✕	/	✕
[21]	✕	/	✓
[22]	/	/	✓
[23]	/	/	✕
[24]	/	/	✓
[25]	✕	✓	✕
[26]	✕	✕	✕
This paper	✓	✓	✓

.

Therefore, this paper investigates the dynamic departure patterns of regular vehicles and carpooling vehicles when both types of vehicles have parking space constraints without the carpool lane.

3. Problem Formulation

In this section, we extend the Vickrey’s bottleneck model [27] to investigate the dynamic departure patterns of regular vehicles and carpooling vehicles when both types of vehicles have parking space constraints without the carpool lane. The Vickrey’s bottleneck model is the most commonly used model to study the dynamics of rush-hour traffic congestion [28,29,30,31]. Therefore, we extend the Vickrey’s bottleneck model.

Carpooling is gradually becoming a new option for commuters and is changing commuters’ habits and travel patterns. When the driver is a solo commuter, we regard this travel behavior as a self-driving behavior and regard this vehicle as a regular vehicle; when commuters choose to carpool to travel, then this travel behavior is regarded as a carpooling behavior, and this vehicle is a carpooling vehicle. In the morning commute period, we can assume that there are two types of vehicles on the road: one type is regular vehicles, where the driver is a solo commuter; another type is carpooling vehicles. The number of commuters carried in each carpooling vehicle can be represented by $\rho$, where $\rho$ is bounded within the range of [2, 5]. We can assume that the population of all commuters is $N$ and the penetration rate of commuters using regular vehicles can be represented by $\theta$; then, $N_p=\theta N$ is the number of commuters using regular vehicles and also denotes the total number of regular vehicles; $N_g=(1-\theta )N$ is the number of commuters using carpooling vehicles, and ${\displaystyle \frac{(1-\theta )N}{\rho }}$ denotes the number of carpooling vehicles. Thus, the total number of vehicles on the road $N_{A0}$ can be expressed as $N_{A0}=\theta N+{\displaystyle \frac{(1-\theta )N}{\rho }}$. Notably, $\theta =0$ and $\theta =1$ denote extreme cases in which there are only carpooling vehicles or only regular vehicles on the road. What is more, the desired arrival time at the CBD workplace for commuters by both types of vehicles is assumed to be identical and equal to $t^{*}$.

After arriving in the CBD, both types of vehicles need to consider parking issues. The parking lots in the CBD are divided into two clusters: the central cluster and the peripheral cluster. The central cluster refers to the parking lots that are closer to the CBD but have more expensive parking spaces. And there is a parking space constraint for both types of vehicles, $P$. The peripheral cluster refers to the parking lots that are far away from the CBD but have cheaper parking spaces. The parking space constraint for the two types of vehicles in the peripheral cluster is not considered. If there are no parking spaces available in the central cluster, commuters only park vehicles in the peripheral cluster. The choice of parking will affect the dynamic travel behaviors of both types of commuters. In order to distinguish the parameters with or without the parking space constraint, superscripts are added to the relevant parameters. The superscript $o$ indicates the parameter situation when there is no parking space constraint; the superscript $c$ indicates the parameter situation when there is a parking constraint.

Without the loss of generality, the conventional bottleneck model is used to capture the traffic dynamics in the morning peak hours, in which the travel cost is linear to the travel time and the schedule delay [5,27,32,33]. By incorporating the pickup fee, the parking fee, and other costs, Ma and Zhang [26] provided a generalized travel cost function. Thus, the travel cost for commuters whose departure time is using regular vehicles $C_p\left(t\right)$ and carpooling vehicles $C_g\left(t\right)$ at time $t$ is defined as

$$C_p\left(t\right)=\left\{\begin{array}{c}{\alpha }_1\times T_p\left(t\right)+{\beta }_1\times \max \{t^{*}-t-T_p\left(t\right)-l,0\}+{\gamma }_1\times \{t+T_p\left(t\right)+l-t^{*},0\}+R_p^c, if parking in central cluster\\ {\alpha }_1\times T_p\left(t\right)+{\beta }_1\times \max \left\{t^{*}-t-T_p\left(t\right)-l,0\right\}+{\gamma }_1\times \left\{t+T_p\left(t\right)+l-t^{*},0\right\}+R_p^p+\lambda \times l, if parking in peripheral cluster\end{array}\right.$$

(1)

$$C_g\left(t\right)=\left\{\begin{array}{c}{\alpha }_2\times T_g\left(t\right)+{\beta }_2\times \max \left\{t^{*}-t-T_g\left(t\right)-l,0\right\}+{\gamma }_2\times \left\{t+T_g\left(t\right)+l-t^{*},0\right\}+J+R_g^c, if parking in central cluster\\ {\alpha }_2\times T_g\left(t\right)+{\beta }_2\times \max \left\{t^{*}-t-T_g\left(t\right)-l,0\right\}+{\gamma }_2\times \left\{t+T_g\left(t\right)+l-t^{*},0\right\}+J+R_g^p+\lambda \times l, if parking in peripheral cluster\end{array}\right.$$

(2)

where ${\alpha }_1$ and ${\alpha }_2$ are the values of the unit travel time for regular vehicles and carpooling vehicles, where ${\alpha }_1=\alpha +\xi$, ${\alpha }_2=\alpha +{\displaystyle \frac{\xi }{\rho }}$, and ${\alpha }_1\ge {\alpha }_2$. $\xi$ is a parameter related to the fuel cost. And the larger the $\xi$, the more expensive the fuel cost. And the travel time of regular vehicles $T_p\left(t\right)$ and the travel time of carpooling vehicles $T_g\left(t\right)$ contain only the queuing time at the bottleneck, where the service capacity is fixed at $s$. Let ${\beta }_1$ and ${\beta }_2$ denote the values of unit time of regular vehicles and carpooling vehicles for early arrival, where ${\beta }_2$ < ${\beta }_1$. And ${\gamma }_1$ and ${\gamma }_2$ denote the values of unit time of regular vehicles and carpooling vehicles for late arrival, where ${\gamma }_2<{\gamma }_1$. Without the loss of generality, $0<{\beta }_i{{<\alpha }_i<\gamma }_i$ ($i=1,2$) always holds. In addition, the fee for carpooling vehicles to pick up commuters is $J=\rho \times {\alpha }_2\times t_h$, where $t_h$ is the unique constant pickup time between two commuters who commute to work using carpooling vehicles. If the vehicle is parked in the peripheral cluster, the time from the peripheral cluster return to the office is $l$, and the unit cost in the return time is $\lambda$. Thus, the return cost from the peripheral cluster to the office is $\lambda \ast l$.

4. Departure Patterns with Regular and Carpooling Vehicles

In this section, we discuss the general situation where there are two types of vehicles on the road simultaneously. The departure patterns of two types of vehicles on the road without and with the parking space constraint are studied in detail in the following subsection:

4.1. Departure Patterns for Two Types of Vehicles without the Parking Space Constraint

In this subsection, the case is that $0<\theta <1$ and is without the parking space constraint. In this case, $t_{qp}^o$ is the earliest departure time for commuters using regular vehicles, $t_{qg}^o$ is the earliest departure time for commuters using carpooling vehicles, $t_{q{p}^{\prime }}^o$ indicates the latest departure time for commuters using regular vehicles, $t_{q{g}^{\prime }}^o$ denotes the latest departure time for commuters using carpooling vehicles, ${\tilde{t}}_p$ represents the departure time for commuters using regular vehicles who arrive at the destination on time, and ${\tilde{t}}_g$ is the departure time for commuters using carpooling vehicles who arrive at the destination on time.

For commuters using regular vehicles, the earliest departure time, the latest departure time, and the departure time for on-time arrival are given by

$$t_{qp}^o=t^{*}-{\displaystyle \frac{{\gamma }_1}{{\beta }_1+{\gamma }_1}}{\displaystyle \frac{\theta N}{s}}$$

(3)

$$t_{q{p}^{\prime }}^o=t^{*}+{\displaystyle \frac{{\beta }_1}{{\beta }_1+{\gamma }_1}}{\displaystyle \frac{\theta N}{s}}$$

(4)

$${\tilde{t}}_p=t^{*}-{\displaystyle \frac{{\beta }_1{\gamma }_1}{{\alpha }_1\left({\beta }_1+{\gamma }_1\right)}}{\displaystyle \frac{\theta N}{s}}$$

(5)

At equilibrium, the total travel cost for all regular vehicles from the household to the CBD workplace ${TC}_p$ is

$${TC}_p={\displaystyle \frac{{\beta }_1{\gamma }_1}{{\beta }_1+{\gamma }_1}}{\displaystyle \frac{{\left(\theta N\right)}^2}{s}}$$

(6)

The total travel time cost for all regular vehicles ${TTC}_p$ is

$${TTC}_p={\displaystyle \frac{{\beta }_1{\gamma }_1}{2({\beta }_1+{\gamma }_1)}}{\displaystyle \frac{{\left(\theta N\right)}^2}{s}}$$

(7)

The total schedule delay cost for all regular vehicles ${SDC}_p$ is

$${SDC}_p={\displaystyle \frac{{\beta }_1{\gamma }_1}{2({\beta }_1+{\gamma }_1)}}{\displaystyle \frac{{\left(\theta N\right)}^2}{s}}$$

(8)

For commuters using carpooling vehicles, the earliest departure time, the latest departure time, and the departure time for on-time arrival are given by

$$t_{qg}^o=t^{*}-{\displaystyle \frac{{\gamma }_2}{{\beta }_2+{\gamma }_2}}{\displaystyle \frac{(1-\theta )N}{\rho s}}$$

(9)

$$t_{q{g}^{\prime }}^o=t^{*}+{\displaystyle \frac{{\beta }_2}{{\beta }_2+{\gamma }_2}}{\displaystyle \frac{(1-\theta )N}{\rho s}}$$

(10)

$${\tilde{t}}_g=t^{*}-{\displaystyle \frac{{\beta }_2{\gamma }_2}{{\alpha }_2\left({\beta }_2+{\gamma }_2\right)}}{\displaystyle \frac{(1-\theta )N}{\rho s}}$$

(11)

At equilibrium, the total travel cost for all carpooling vehicles from their household to the CBD workplace ${TC}_g$ is

$${TC}_g={\displaystyle \frac{{\beta }_2{\gamma }_2}{{\beta }_2+{\gamma }_2}}{\displaystyle \frac{{\left(\right(1-\theta \left)N\right)}^2}{{\rho }^{2}s}}$$

(12)

The total travel time cost for all carpooling vehicles ${TTC}_g$ is

$${TTC}_g={\displaystyle \frac{{\beta }_2{\gamma }_2}{2({\beta }_2+{\gamma }_2)}}{\displaystyle \frac{{\left(\right(1-\theta \left)N\right)}^2}{{\rho }^{2}s}}$$

(13)

The total schedule delay cost for all carpooling vehicles ${SDC}_g$ is

$${SDC}_g={\displaystyle \frac{{\beta }_2{\gamma }_2}{2({\beta }_2+{\gamma }_2)}}{\displaystyle \frac{{\left(\right(1-\theta \left)N\right)}^2}{{\rho }^{2}s}}$$

(14)

According to the range situation of $\theta$, the departure patterns for two types of commuters without the parking space constraint are shown in Figure 1. The detailed derivation process is shown in Appendix A.

4.2. Departure Patterns for Two Types of Vehicles with the Parking Space Constraint

In this subsection, the case with $0<\theta <1$ and with the parking space constraint in the central cluster is discussed. Due to the parking space constraint in the central cluster, only a certain number of regular vehicles and carpooling vehicles can be parked here. In order to park the vehicle in the central cluster, it is necessary to arrive early to occupy the parking space in the central cluster. After all parking spaces in the central cluster are occupied, vehicles need to be parked in the peripheral cluster. The parking fee for the central cluster is ${\rho }_{qg}^c$, and the parking fee for the peripheral cluster is $\Phi {\rho }_{qg}^c$, $0<\Phi <1$; that is, the parking fee for the peripheral cluster is less than the parking fee for the central cluster.

The earliest time for commuters using regular vehicles to enter the central cluster is

$$t_p^{+}=t_{qp}^c+{\displaystyle \frac{{\rho }_{qg}^c}{{\beta }_1}}$$

(15)

The latest time for commuters using regular vehicles to enter the central cluster is

$$t_p^{-}=t_{qp}^c+{\displaystyle \frac{\theta N}{s}}-{\displaystyle \frac{2{\rho }_{qg}^c}{{\alpha }_1+{\gamma }_1}}$$

(16)

The earliest time for commuters using carpooling vehicles to enter the central cluster is

$$t_g^{+}=t_{qg}^c+{\displaystyle \frac{{\rho }_{qg}^c}{{\beta }_2}}$$

(17)

The latest time for commuters using carpooling vehicles to enter the central cluster is

$$t_g^{-}=t_{qg}^c+{\displaystyle \frac{(1-\theta )N}{\rho s}}-{\displaystyle \frac{2{\rho }_{qg}^c}{{\alpha }_2+{\gamma }_2}}$$

(18)

The earliest departure time for regular vehicles is

$$t_{qp}^c=t^{*}-{\displaystyle \frac{{\gamma }_1}{{\beta }_1+{\gamma }_1}}{\displaystyle \frac{\theta N}{s}}+{\displaystyle \frac{\left({\gamma }_1-{\alpha }_1\right){\rho }_{qg}^c}{\left({\beta }_1+{\gamma }_1\right)\left({\alpha }_1+{\gamma }_1\right)}}$$

(19)

The earliest departure time for carpooling vehicles is

$$t_{qg}^c=t^{*}-{\displaystyle \frac{{\gamma }_2}{{\beta }_2+{\gamma }_2}}{\displaystyle \frac{(1-\theta )N}{\rho s}}+{\displaystyle \frac{\left({\gamma }_2-{\alpha }_2\right){\rho }_{qg}^c}{\left({\beta }_2+{\gamma }_2\right)\left({\alpha }_2+{\gamma }_2\right)}}$$

(20)

The parking fee of the central cluster for regular and carpooling vehicles is

$${\rho }_{qg}^c={\displaystyle \frac{1}{2s}}\times {\displaystyle \frac{N{\beta }_1{\beta }_2{\Psi }_1{\Psi }_2}{\left[2{\beta }_1{\beta }_2\left({\alpha }_1+{\alpha }_2+{\gamma }_1+{\gamma }_2\right)+{\left({\beta }_1{+\beta }_2\right)\Psi }_1{\Psi }_2\right]}}\times [{\displaystyle \frac{{\beta }_1\left({\gamma }_1-{\alpha }_1\right)}{{\Phi }_1{\Psi }_1}}+{\displaystyle \frac{{\beta }_2\left({\gamma }_2-{\alpha }_2\right)}{{\Phi }_2{\Psi }_2}}+{\displaystyle \frac{1+(\rho -1)\theta }{\rho }}]$$

(21)

where ${\alpha }_1+{\gamma }_1={\Psi }_1$, ${\alpha }_2+{\gamma }_2={\Psi }_2$, ${\beta }_1+{\gamma }_1={\Phi }_1$, and ${\beta }_2+{\gamma }_2={\Phi }_2$.

For commuters using regular vehicles, the parking cost for regular vehicles in the central cluster $R_p^c$ is

$$R_p^c={\rho }_{qg}^c\times h\times P_p$$

(22)

where $P_p$ represents the number of regular vehicles in the central cluster.

The total schedule delay cost for all regular vehicles in the central cluster ${SDC}_p^c$ is

$${SDC}_p^c={\displaystyle \frac{{\beta }_{1}s}{2}}{(t^{*}-t_{qp}^c)}^2+{\displaystyle \frac{{\gamma }_{1}s}{2}}{(t_{qp}^{\prime }-t^{*})}^2$$

(23)

The total travel cost for all regular vehicles from the household to the CBD workplace in the central cluster ${TC}_p^c$ is

$${TC}_p^c=\theta N{\beta }_1\left(t^{*}-t_{qp}^c\right)$$

(24)

At equilibrium, the total travel time cost for all regular vehicles in the central cluster ${TTC}_p^c$ is calculated as

$${TTC}_p^c={TC}_p^c-{SDC}_p-R_p^c$$

(25)

For commuters using carpooling vehicles, the parking cost in the central cluster $R_g^c$ is

$$R_g^c={\rho }_{qg}^c\times h\times P_g$$

(26)

where $P_g$ represents the number of carpooling vehicles in the central cluster.

The total schedule delay cost for all carpooling vehicles in the central cluster ${SDC}_g^c$ is

$${SDC}_g^c={\displaystyle \frac{{\beta }_{2}s}{2}}{(t^{*}-t_{qg}^c)}^2+{\displaystyle \frac{{\gamma }_{2}s}{2}}{(t_{q{g}^{\prime }}-t^{*})}^2$$

(27)

The total travel cost for all carpooling vehicles from their household to the CBD workplace in the central cluster ${TC}_g^c$ is

$${TC}_g^c={\displaystyle \frac{(1-\theta )N}{\rho }}{\beta }_2\left(t^{*}-t_{qg}^c\right)$$

(28)

At equilibrium, the total travel time cost for all carpooling vehicles in the central cluster ${TTC}_g^c$ is readily calculated as

$${TTC}_g^c={TC}_g^c-{SDC}_g^c-R_g^c-J$$

(29)

In the case of the reasonable parking fee in the central cluster, no one will choose to park in the peripheral cluster at first. After all parking spaces are occupied in the central cluster, two types of vehicles need to park in the peripheral cluster.

For commuters using regular vehicles, the parking cost in the peripheral cluster $R_p^p$ is

$$R_p^p=\Phi \times {\rho }_{qg}^c\times h\times (\theta N-P_p)$$

(30)

where $\Phi \times {\rho }_{pg}^c$ represents the parking fee in the peripheral cluster, and $0<\Phi <1$.

The total schedule delay cost for all regular vehicles in the peripheral cluster ${SDC}_p^z$ is

$${SDC}_p^z={\displaystyle \frac{{\beta }_1{\gamma }_1}{2({\beta }_1+{\gamma }_1)}}{\displaystyle \frac{{(\theta N-M_p)}^2}{s}}$$

(31)

The total travel cost for all regular vehicles in the peripheral cluster from their household to the CBD workplace ${TC}_p^z$ is

$${TC}_p^z={TTC}_p^z+{SDC}_p^z+R_p^p+\lambda \times l$$

(32)

At equilibrium, the total travel time cost for all regular vehicles in the peripheral cluster ${TTC}_p^z$ is calculated as

$${TTC}_p^z={\displaystyle \frac{{\beta }_1{\gamma }_1}{2({\beta }_1+{\gamma }_1)}}{\displaystyle \frac{{(\theta N-P_p)}^2}{s}}$$

(33)

For commuters using carpooling vehicles, the parking fee in the peripheral cluster $R_g^p$ is

$$R_g^p=\Phi \times {\rho }_{qg}^c\times h\times ({\displaystyle \frac{(1-\theta )N}{\rho }}-P_g)$$

(34)

The total schedule delay cost for all carpooling vehicles in the peripheral cluster ${SDC}_g^z$ is

$${SDC}_g^z={\displaystyle \frac{{\beta }_2{\gamma }_2}{2({\beta }_2+{\gamma }_2)}}{\displaystyle \frac{1}{s}}[{\displaystyle \frac{(1-\theta )N}{\rho }}-P_g{]}^2$$

(35)

The total travel cost for all carpooling vehicles in the peripheral cluster from their household to the CBD workplace ${TC}_g^z$ is

$${TC}_g^z={TTC}_g^c+{SDC}_g^z+R_g^p+J+\lambda \times l$$

(36)

At equilibrium, the total travel time cost for all carpooling vehicles in the peripheral cluster ${TTC}_g^z$ is

$${TTC}_g^z={\displaystyle \frac{{\beta }_2{\gamma }_2}{2({\beta }_2+{\gamma }_2)}}{\displaystyle \frac{1}{s}}[{\displaystyle \frac{(1-\theta )N}{\rho }}-P_g{]}^2$$

(37)

According to the range situation of θ, the departure patterns for the two types of commuters with the parking space constraint are shown in Figure 2. The detailed derivation process is shown in Appendix B.

5. Discussion

In this section, we further analyze the impacts of the penetration rate of commuters using regular vehicles and the number of commuters in each carpooling vehicle on the departure patterns of the two types of vehicles in detail. All proofs in this section can be found in Appendix C.

According to the results of the equilibrium, taking the first-order derivatives of the earliest departure time of regular vehicles and the latest departure time of regular vehicles to the penetration rate of commuters using regular vehicles θ yields the following results, as shown in Proposition 1:

Proposition 1.

(1) 

The impact of the penetration rate of commuters using regular vehicles on the earliest departure time of regular vehicles with and without the parking space constraint is as follows: ${\displaystyle \frac{{\partial t}_{qp}^0}{\partial \theta }}\le {\displaystyle \frac{{\partial t}_{qp}^c}{\partial \theta }}<0$.

(2) 

The impact of the penetration rate of commuters using regular vehicles on the latest departure time of regular vehicles with and without the parking space constraint is as follows: $0<{\displaystyle \frac{{\partial t}_{q{p}^{\prime }}^0}{\partial \theta }}\le {\displaystyle \frac{{\partial t}_{q{p}^{\prime }}^c}{\partial \theta }}$.

Proposition 1(1) shows that when there is no parking space constraint, the greater the penetration rate of commuters using regular vehicles, the earlier the earliest departure time of regular vehicles. And when there is a parking space constraint, the increase in the penetration rate of commuters using regular vehicles will still advance the earliest departure time of regular vehicles. However, compared with the case of no parking space constraint, when there is a parking space constraint, the advance range of the earliest departure time of regular vehicles is reduced. This is because when there is a parking space constraint, the parking fee of the central cluster also affects the travel decisions of regular vehicles. Therefore, the influence of the penetration rate of commuters using regular vehicles on the earliest departure time of regular vehicles is reduced.

Proposition 1(2) shows that regardless of whether there is a parking space constraint, as the penetration rate of commuters using regular vehicles increases, the latest departure time of regular vehicles will be delayed. This is because as the penetration rate of commuters using regular vehicles increases, the number of vehicles on the road will increase, which will make the queue longer, thereby inevitably making the latest departure time of regular vehicles later.

Taking the first-order derivatives of the earliest departure time of carpooling vehicles and the latest departure time of carpooling vehicles with and without the parking space constraint to $\theta$ yields the following results, as shown in Proposition 2:

Proposition 2.

(1) 

The impact of the penetration rate of commuters using regular vehicles on the earliest departure time of carpooling vehicles with and without the parking space constraint is as follows: $0<{\displaystyle \frac{{\partial t}_{qg}^0}{\partial \theta }}\le {\displaystyle \frac{{\partial t}_{qg}^c}{\partial \theta }}$.

(2) 

The impact of the penetration rate of commuters using regular vehicles on the latest departure time of carpooling vehicles with and without the parking space constraint is as follows: ${\displaystyle \frac{{\partial t}_{q{g}^{\prime }}^0}{\partial \theta }}\le {\displaystyle \frac{{\partial t}_{q{g}^{\prime }}^c}{\partial \theta }}<0$.

Proposition 2(1) indicates that when there is no parking space constraint, the greater the penetration rate of commuters using regular vehicles, the later the earliest departure time of carpooling vehicles. And when there is a parking space constraint, as the penetration rate of commuters using regular vehicles increases, the earliest departure time of carpooling vehicles will be even later. Combined with Proposition 1(1), this is because the earliest departure time of regular vehicles advances as the penetration rate of commuters using regular vehicles increases, which makes the earliest departure time of carpooling vehicles relatively late to avoid waiting too long in the queue.

Proposition 2(2) shows that regardless of whether there is a parking space constraint, as the penetration rate of commuters using regular vehicles increases, the latest departure time of carpooling vehicles will be advanced. Combined with Proposition 1(2), this is because the latest departure time of regular vehicles delays as the penetration rate of commuters using regular vehicles increases, which makes the latest departure time of carpooling vehicles advance relatively to avoid the peak period.

According to Proposition 1 and Proposition 2, it can be seen that whether there is a parking space constraint or not, the penetration rate of commuters using regular vehicles $\theta$ has a great impact on the departure patterns of regular vehicles and carpooling vehicles. Therefore, it is necessary to discuss the equilibrium value of θ, as shown in Proposition 3.

Proposition 3.

(1) 

The equilibrium penetration rate of commuters using regular vehicles ${\theta }^{*}$ is ${\theta }^{*}=1-{\displaystyle \frac{\rho N_{A0}}{N}}+{\displaystyle \frac{c{\rho }^2{\alpha }_1({\alpha }_2-{\beta }_2)}{N{\alpha }_2{\beta }_1{\beta }_2+N{\alpha }_1({\alpha }_2{\beta }_1-({\alpha }_2+{\beta }_1\left){\beta }_2\right)}}+{\displaystyle \frac{sc{\rho }^2{\alpha }_2{\beta }_1({\alpha }_2-{\beta }_2)({\alpha }_1+{\gamma }_1)}{N({\alpha }_2{\beta }_1{\beta }_2+{\alpha }_1({\alpha }_2{\beta }_1-({\alpha }_2+{\beta }_1){\beta }_2\left)\right){\gamma }_1({\alpha }_2+{\gamma }_1)}}$.

(2) 

When $\rho <{\displaystyle \frac{N_{A0}({\alpha }_2{\beta }_1{\beta }_2+{\alpha }_1({\alpha }_2{\beta }_1-({\alpha }_2+{\beta }_1){\beta }_2\left)\right){\gamma }_1({\alpha }_2+{\gamma }_1)}{c({\alpha }_2-{\beta }_2)(2s{\alpha }_2{\beta }_1{\gamma }_1+{\alpha }_1({\gamma }_1^2+{\alpha }_2(2s{\beta }_1+{\gamma }_1)\left)\right)}}$, the equilibrium penetration rate of commuters using regular vehicles ${\theta }^{*}$ is an increasing concave function of the number of commuters carried in each carpooling vehicle $\rho$; otherwise, it is a decreasing concave function of the number of commuters carried in each carpooling vehicle $\rho$.

(3) 

When ${\displaystyle \frac{1}{1-\rho }}>L$, the total number of vehicles on the road $N_{A0}$ is an increasing function of the number of commuters carried in each carpooling vehicle $\rho$; when ${\displaystyle \frac{1}{1-\rho }}\le L$, the total number of vehicles on the road $N_{A0}$ is a decreasing function of the number of commuters carried in each carpooling vehicle $\rho$.

Through derivation and solving, we obtain the equilibrium penetration rate of commuters using regular vehicles ${\theta }^{*}$. According to Proposition 3(2) and Proposition 3(3), the equilibrium penetration rate of commuters using regular vehicles ${\theta }^{*}$ is not a monotonic function of the number of commuters carried in each carpooling vehicle ρ. Moreover, the number of vehicles on the road $N_{A0}$ is also not a monotonic function of the number of commuters carried in each carpooling vehicle $\rho$. In other words, only when $\rho$ is within a certain range, the increase in $\rho$ will reduce the value of $\theta$ and reduce the number of vehicles on the road. Therefore, the government should guide carpooling platforms to control a reasonable level of carpooling to relieve the traffic pressure and reduce social costs effectively.

6. Numerical Analysis

In this section, we validate the main findings mentioned above. Based on the parameter settings of the existing literature [30], the parameters are set as follows: ${\alpha }_1$ varies in the range of [0.5, 1.5], ${\alpha }_2$ varies in the range of [0.5, 1.5], ${\gamma }_1$ varies in the range of [2, 2.5], ${\gamma }_2$ varies in the range of [1.7, 2.2], ${\beta }_1$ varies in the range of [0.5, 1], ${\beta }_2$ varies in the range of [0.2, 0.7], $s$ varies in the range of [100, 300], $N$ varies in the range of [2000, 6000], and $\rho$ varies in the range of [1, 4].

6.1. The Impact of the Time for Two Types of Vehicles to Enter the Central Cluster

In this subsection, we first discuss the impact of the penetration rate of commuters using regular vehicles on the earliest time for regular vehicles to enter the central cluster when ${\gamma }_1=2$, ${\gamma }_1=2.25$, and ${\gamma }_1=2.5$. As shown in Figure 3, the increase in θ advances the earliest time for regular vehicles to enter the central cluster. This is because the larger the penetration rate of commuters using regular vehicles, the more vehicles are competing for the limited parking spaces of the central cluster, thereby promoting regular vehicles to reach the central cluster earlier. What is more, it can be found that the greater the cost of regular vehicles being late, the greater the advancement. This is because if vehicles cannot park in the central cluster, they have to go to the peripheral cluster, which will increase the risk of being late. Therefore, the greater the cost of the regular vehicles being late, the earlier the regular vehicles arrive at the central cluster to avoid the risk of being late.

We further discuss the impact of the penetration rate of commuters using regular vehicles on the latest time for regular vehicles to enter the central cluster. As shown in Figure 4, the increase in $\theta$ delays the latest time for regular vehicles to enter the central cluster. Considering an extreme situation where there is only one carpooling vehicle, the last vehicle to enter the central cluster is likely to be a regular vehicle. Therefore, the larger the penetration rate of commuters using regular vehicles, the later the time for regular vehicles to enter the central cluster will be. Moreover, as the cost of late arrivals for regular vehicles increases, the extent of the delay in the latest time for regular vehicles to enter the central cluster will decrease. This is because when the penetration rate of commuters using regular vehicles increases, the number of vehicles increases, which leads to more intense competition for parking spaces in the central cluster. Therefore, in the case of a further increase in the cost of regular vehicles being late, the latest time for regular vehicles to enter the central cluster will not be delayed too much.

Next, consider the impact of the penetration rate of commuters using regular vehicles on the earliest time for carpooling vehicles to enter the central cluster when $\rho =1$, $\rho =2$, $\rho =3$, and $\rho =4$. According to Figure 5, it can be found that the larger the penetration rate of commuters using regular vehicles, the later the earliest time for carpooling vehicles. According to the impact of the penetration rate of commuters using regular vehicles on the earliest time for regular vehicles to enter the central cluster ${\displaystyle \frac{{dt}_p^{+}}{d\theta }}$, if the proportion of regular vehicles is larger, the first vehicle to enter the central cluster is likely to be a regular vehicle. Therefore, the earliest time for carpooling vehicles to enter the central cluster will be delayed. What is more, as the number of carpoolers increases, the delay in the earliest time for regular vehicles to enter the central cluster decreases. This is because the more people need to be carried and the increase in $\theta$, so in order to be able to enter the central cluster, the carpooling vehicles not be delayed much.

We further discuss the impact of the penetration rate of commuters using regular vehicles on the latest time for carpooling vehicles to enter the central cluster when $\rho =1$, $\rho =2$, $\rho =3$, and $\rho =4$, as shown in Figure 6.

According to Figure 6, it can be found that the increase in θ advances the latest time for carpooling vehicles to enter the central cluster. Combined with the impact of the penetration rate of commuters using regular vehicles on the latest time for regular vehicles to enter the central cluster ${\displaystyle \frac{{dt}_p^{-}}{d\theta }}$, it can be known that because the latest time for regular vehicles to enter the central cluster is delayed as $\theta$ increases, the latest time for carpooling vehicles to enter the central cluster has to be earlier. What is more, as the number of carpoolers increases, the advance in the latest time for carpooling vehicles to enter the central cluster decreases. This is because the more people there are in a carpool vehicle, the more time it takes to carry people, so compared to a carpool vehicle with a small number of people, the latest time to enter the central cluster will not be earlier.

6.2. The Impact of the Departure Time of Two Types of Vehicles

In this subsection, we first focus on the impact of the penetration rate of commuters using regular vehicles on the earliest departure time of regular vehicles when ${\gamma }_1=2$, ${\gamma }_1=2.25$, and ${\gamma }_1=2.5$. As shown in Figure 7, the increase in θ advances the earliest departure time of regular vehicles. This is because the higher the penetration rate of commuters using regular vehicles, the more vehicles there are on the road. However, parking spaces in the central cluster are limited. To secure a parking space in the central cluster, ordinary vehicles will leave earlier. What is more, the greater the cost of the regular vehicles being late, the more regular vehicles will depart earlier. This is because if the vehicle cannot be parked in the central cluster, it needs to be parked in the peripheral cluster. However, it takes extra time to get and return to the peripheral cluster, so there is a risk of being late. Therefore, the greater the cost of being late for regular vehicles, the earlier the regular vehicles leave.

We further discuss the impact of the penetration rate of commuters using regular vehicles on the latest departure time of regular vehicles when ${\gamma }_1=2$, ${\gamma }_1=2.25$, and ${\gamma }_1=2.5$. According to Figure 8, it can be found that the increase in θ delays the latest departure time of regular vehicles. This is because as the penetration rate of commuters using regular vehicles increases, the road congestion will be serious, and the waiting time in the queue will become longer. Therefore, the latest departure time of regular vehicles is delayed. But when the regular vehicles’ cost of being late increases, the extent of the delay in the latest departure time of regular vehicles will decrease. This is because when the regular vehicles’ cost of being late is increased, the vehicle leaves early to avoid the risk of being late. Thus, the latest time for regular vehicles to leave will not be delayed too much.

Next, we focus on the impact of the penetration rate of commuters using regular vehicles on the earliest departure time of carpooling vehicles when $\rho =1$, $\rho =2$, $\rho =3$, and $\rho =4$. According to Figure 9, it can be found that the increase in θ advances the earliest departure time of carpooling vehicles. According to the impact of the penetration rate of commuters using regular vehicles on the earliest departure time of carpooling vehicles ${\displaystyle \frac{{dt}_{qp}^c}{d\theta }}$, if the proportion of regular vehicles is larger, the first vehicle to leave is likely to be a regular vehicle. Therefore, the earliest time for carpooling vehicles to leave will be delayed. What is more, as the number of carpoolers increases, the delay of the earliest time for regular vehicles to leave decreases. This is because the more people there are being carried, the more time it takes to carry people. Therefore, to avoid the risk of being late, if more people need to be carried, the departure will not be delayed too much.

We further focus on the impact of the penetration rate of commuters using regular vehicles on the latest departure time of carpooling vehicles when $\rho =1$, $\rho =2$, $\rho =3$, and $\rho =4$.

As shown in Figure 10, the increase in $\theta$ advances the latest departure time of carpooling vehicles. Combined with the impact of the penetration rate of commuters using regular vehicles on the latest departure time of regular vehicles ${\displaystyle \frac{{dt}_{q{p}^{\prime }}^c}{d\theta }}$, the larger the proportion of regular vehicles, the latest the departure vehicle may be a regular vehicle. Therefore, the latest departure time for carpooling vehicles is advanced. What is more, the greater the number of carpoolers, the later the latest departure time of carpooling vehicles. This is because the more people there are in a carpool vehicle, the more time it takes to carry people, so compared to a carpool vehicle with fewer people, a carpool vehicle with more people will not leave sooner.

7. Conclusions

The ever-increasing number of private car owners has put urban traffic operations to the test. Not only do the roads become congested, but parking also becomes more difficult. This becomes particularly evident during the morning commute. In recent years, the emergence of carpooling as a form of shared travel has provided a new way to address these issues. This paper investigates the dynamic departure patterns of regular vehicles and carpooling vehicles when both types of vehicles have parking space constraints without the carpool lane. The following important conclusions are obtained:

An increase in the parking fee of the central cluster delays the earliest time for both types of vehicles to enter the central cluster. And an increase in the proportion of regular vehicles advances the earliest time for regular vehicles to enter the central cluster, while the earliest time for carpooling vehicles to enter the central cluster is delayed. In addition, only when the number of commuters carried in each carpooling vehicle is within a certain range will the number of vehicles on the road reduce with the increase in the number of commuters carried in each carpooling vehicle. This implies that more commuters sharing a vehicle in the morning commute is not necessarily better. Only under a reasonable level of carpoolers can carpooling reduce the peak time and unnecessary time consumption on the road and effectively promote the reduction of parking fees, commuters’ travel costs, and other societal costs of transportation. Therefore, the government should guide carpooling platforms to control a reasonable level of carpooling to relieve the traffic pressure and reduce social costs effectively. For example, a credit charging scheme considering carpooling and carbon emissions can be implemented [34].

Our study makes three important contributions. First, our research narrows the literature gap of the impact of parking choices on commuters’ travel behaviors. Many studies have only discussed the impact of one parking choice on commuters’ travel behaviors. In this paper, we discuss that commuters choose the most suitable parking space from multiple parking choices. Second, the previous studies only discussed parking constraints for regular vehicles, neglecting to consider parking constraints for both types of vehicles and assuming a separate lane for carpooling vehicles. Without the carpool lane, this paper studies the impact of the parking space constraint for both travel modes on commuters’ travel behaviors. Third, from a practical perspective, our study gives practical guidance and suggestions to formulate a reasonable parking fee and control a reasonable carpooling level.

In this paper, we extend the Vickrey’s bottleneck model [27] to investigate the dynamic departure patterns of regular vehicles and carpooling vehicles when both types of vehicles have parking space constraints without the carpool lane. In the future, it could be an interesting direction to extend Vickrey’s bottleneck model further to enable it to solve more traffic congestion problems. In addition, further consideration of travel behaviors in the case of reserved private parking spaces is a worthwhile direction for research [35]. What is more, this paper only considers the travel behaviors of commuters using regular vehicles and carpooling vehicles. Simultaneously considering more patterns of transportation, such as e-bikes, is one of the directions worth studying.