An equilibrium analysis of commuter parking in the era of autonomous vehicles

Transportation Research Part C 92 (2018) 191–207

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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc

An equilibrium analysis of commuter parking in the era of autonomous vehicles Wei Liu

T

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School of Computer Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia Research Centre for Integrated Transport Innovation, School of Civil and Environmental Engineering, University of New South Wales, Sydney, NSW 2052, Australia

A R T IC LE I N F O

ABS TRA CT

Keywords: Morning commute Bottleneck model Parking location Tolling or pricing Social parking cost

This study is the first in the literature to model the joint equilibrium of departure time and parking location choices when commuters travel with autonomous vehicles (AVs). With AVs, walking from parking spaces to the work location is not needed. Instead, AVs will drop off the commuters at the workplace and then drive themselves to the parking spaces. In this context, the equilibrium departure/arrival profile is different from the literature with non-autonomous vehicles (non-AVs). Besides modeling the commuting equilibrium, this study further develops the first-best time-dependent congestion tolling scheme to achieve the system optimum. Also, a location-dependent parking pricing scheme is developed to replace the tolling scheme. Furthermore, this study discusses the optimal parking supply to minimize the total system cost (including both the travel cost and the social cost of parking supply) under either user equilibrium or system optimum traffic flow pattern. It is found that the optimal planning of parking can be different from the non-AV situation, since the vehicles can drive themselves to parking spaces that are further away from the city center and walking of commuters is avoided. This paper sheds light on future parking supply planning and traffic management.

1. Introduction The limited and costly land supplies in city centers of most large cities, such as Beijing, Hong Kong, London, Shenzhen, and Sydney, have raised big challenges for authorities in parking and transport planning and management. Due to parking limitation, it has been observed that 30% of traffic were cruising for a parking space (Shoup, 2006) based on empirical data of several cities. In this context, many studies have examined the integrated problem of parking and traveling, and proposed parking pricing, parking reservation, and parking permit systems to manage the parking supply/operation and traffic congestion (e.g., Arnott et al., 1991; Zhang et al., 2005; Arnott and Inci, 2006; Qian et al., 2011; Zhang et al., 2011; Qian et al., 2012; Yang et al., 2013; Liu et al., 2014a,b; Inci and Lindsey, 2015; Chen et al., 2015, 2016b; Xiao et al., 2016; Nourinejad and Roorda, 2017). There is also a branch of studies looking into the parking management problem in a network context with either a detailed parking network formulation or aggregate network settings (Boyles et al., 2015; He et al., 2015; Liu and Geroliminis, 2016; Zheng and Geroliminis, 2016). However, all of these studies are focused on the parking problem when vehicles have to be operated by drivers (commuters), and none has considered the situation with fully autonomous vehicles. This means that walking of commuters is a generic component of the travel cost although walking might be ignored in some studies due to different research focuses.

⁎

Address: School of Computer Science and Engineering, University of New South Wales, Sydney, NSW 2052, Australia. E-mail address: [email protected].

https://doi.org/10.1016/j.trc.2018.04.024 Received 20 October 2017; Received in revised form 26 April 2018; Accepted 30 April 2018 0968-090X/ © 2018 Elsevier Ltd. All rights reserved.

Transportation Research Part C 92 (2018) 191–207

W. Liu

Fig. 1. The linear city network.

This study envisions a fully autonomous traffic network in the future and considers the commuter parking problem. Particularly, we consider the linear city network in Fig. 1. All commuters travel from home to work in the morning. To reach their destination in the city center, commuters need to pass through a highway bottleneck, where congestion occurs. Parking spaces are continuously distributed along the traffic corridor in the linear city. This network setting closely follows Arnott et al. (1991). In Arnott et al. (1991), drivers will park their cars at some locations near to the city center (depending on the location and parking fee) and then walk to the center (workplace). Different parking locations will yield different walking times, which further affect the arrival times to work for commuters, as well as the arrival rates to work and the schedule delay costs. However, in this study, given the fully autonomous traffic environment, the commuters do not need to walk any more. Instead, the autonomous vehicles (AVs) will drive the commuters to the city center (for work), and then drive back by themselves (self-driving of AVs) and find a parking space along the corridor. Note that commuters will shift to this behavior pattern when walking is much more expensive than AV self-driving. Since walking time is replaced by additional AV self-driving time, the need to park does not affect the rate at which individuals can arrive at work. This accounts for the main differences in the commuting equilibrium from Arnott et al. (1991). Also, a too far away parking space is not feasible in the non-AV case due to unacceptable walking distance (while park-and-ride might help), which is reflected in Arnott et al. (1991). However, AVs might park far away from the city center to save parking fees (if self-driving cost is cheaper than the parking fee). Similar AV self-driving behaviors can arise later in the afternoon for returning home trips where AVs can pick up commuters, which are not considered in this paper. However, the combined morning and evening commutes with AVs are under the author’s consideration for future research. This paper will serve as a basis for that to be built on. These behavior changes due to the development of autonomous vehicles and connected transport systems can be expected in the future, and can change the commuting and parking pattern in cities substantially. Recent efforts toward planning and operation issues of autonomous vehicles system can be found in, e.g., Zhu and Ukkusuri (2015), Chen et al. (2016a, 2017). Under Vickrey’s single-bottleneck setting, van den Berg and Verhoef (2016) and Lamotte et al. (2017) examined the commuting equilibrium with autonomous vehicles. Specifically, they explored the potentially improved road capacity from driver-less cars and possible reservation services. Shared and autonomous vehicles operations have also attracted more and more attentions in the literature (e.g., Ma et al., 2017; Loeb et al., 2018). Parking issues for AVs and associated behavior patterns have rarely been considered. Only very recently Zakharenko (2016) used a monocentric model and considered that AVs might be parked at home during the day so that parking fees can be avoided. As mentioned in the above, this study adopts the bottleneck-constrained highway model to capture the essentials of the traffic dynamics and commuters’ departure time choices. This approach has been adopted by many studies since Vickrey (1969). Particularly, Smith (1984) and Daganzo (1985) have established the existence and uniqueness of the departure/arrival equilibrium solution in the presence of a single bottleneck, respectively. Later, Lindsey (2004) further discussed the existence, uniqueness, and trip cost properties with heterogeneous commuters in the bottleneck model. Some other studies, e.g., Lindsey (2009) and Xiao et al. (2015), examined the equilibrium flow patterns under stochastic demand or supply settings. Thanks to its analytical tractability, the bottleneck model has been widely adopted to study a number of management and policy issues such as congestion tolling and mobility credits, parking management, and heterogeneous commuters. For a recent review, one can refer to e.g., Small (2015). To capture the spatial dimension of parking, this study adopts a linear, continuous and monocentric approach, where AVs have a parking location to choose. This joint choice framework of parking location and departure time governed by the bottleneck model offers us an analytically tractable way to study the commuter parking problem in a time-and-space-dependent context. Most of the literature using the linear corridor approach often looks at the static traffic case without considering time-of-departure, when examining congestion tolling (Mun et al., 2003; Verhoef, 2005; Li et al., 2014), curbside parking and park-and-ride facilities (Anderson and De Palma, 2004; Wang et al., 2004; Liu et al., 2009), joint equilibrium of land-use and travel (Li et al., 2012b), and bi-modal transport systems (Li et al., 2012a). The rest of the paper is organized as follows. Section 2 presents the problem description and the cost formulations, and explores the dynamic traffic pattern at the departure/arrival equilibrium with AVs, and then compares it with those in Arnott et al. (1991) with non-AVs and walking. In Section 3, the dynamic traffic patterns at the system optimum with autonomous vehicles are discussed. Moreover, time-dependent congestion tolling and location-dependent parking pricing schemes are developed and evaluated to achieve the system optimum, and the optimal parking supply is also discussed to minimize total system cost including the travel cost and the social cost of parking supply. Numerical illustrations are presented in Section 4. Finally, Section 5 concludes the paper.

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2. User equilibrium in a fully autonomous traffic environment This section starts with a thumbnail description of the commuter parking problem in the rush-hour with autonomous vehicles. We consider a linear city depicted in Fig. 1, where there is a highway bottleneck with a capacity of s between home and workplace. Note that the major notations in the paper are summarized in Table 1 (Appendix A). Along the linear corridor, each location is indexed by its distance x to the city center (for the city center x = 0 ). Moreover, following the study of Arnott et al. (1991), employer-provided parking is assumed unavailable, and commuters must use either on-street or off-street parking spaces, which are distributed along the corridor. For location x the parking density is m (x ) (number of parking spaces per unit distance). We assume that m (x ) > 0 and it may vary non-linearly over x. At the commuting equilibrium, the parking location for an AV depends on the departure time t, since earlier arrivals will choose closer or cheaper parking spaces. Every day there is a total number of N commuters traveling from home to the city center. They have a desired arrival time t ∗ at the workplace. Early or late arrivals at the workplace will be penalized. All commuters drive with autonomous vehicles (AVs). It is assumed that AVs are owned privately by individual commuters. Each commuter will need a parking space.1 In the traffic environment with autonomous vehicles (AVs), the commuters will firstly drive through the congested bottleneck, and then reach the destination. After that, the AVs will turn back to find a parking space by themselves (self-driving). The whole traveling process follows the procedure: Leave Home → Pass the Bottleneck → Reach Workplace and Drop Off the Commuter → AVs Turn Back and Find a Parking Space (AV self-driving). Note that as the autonomous vehicles are running in a fully autonomous and connected transport environment, we assume that these vehicles know the nearest/cheapest available spaces, and they do not have to cruise for parking spaces. Based on the above setting, for commuters departing at time t and parking at location x, the travel cost can be written as

c (t ) = α·T (t ) + β ·[t ∗−t −T (t )]+ + γ ·[t + T (t )−t ∗]+ + λ·w·x ,

(1)

where T (t ) is the travel time experienced by the commuters departing from home at time t ,α is the value of time when the commuters are driving the AVs, β and γ are the penalties for a unit time of early and late arrivals at the destination for commuters, λ is the value of a unit AV self-driving time, w is the travel time needed to cover a unit distance by self-driving AVs, and [·]+ = max{0,·} . It is assumed that γ > α > β and α > λ . γ > α > β is a standard assumption in the literature, which is consistent with many empirical evidences. α > λ means that AV self-driving time is less costly than the driving time when commuters operate the vehicles. Note that here parking fee is zero, regardless of the location, and congestion toll is zero, regardless of the departure/arrival time (i.e., we consider the no-fee and no-toll traffic equilibrium). In Eq. (1), without loss of generality, the travel time T (t ) is assumed to have only the queuing delay at the highway bottleneck, q (t ) i.e., T (t ) = s where q (t ) is the queue length experienced by commuters departing from home at time t and s is the bottleneck capacity, and the free-flow time for other road sections is zero. Besides, w is a constant, indicating that there is no flow-dependent congestion for the self-driving of AVs. It is evident that to save the self-driving cost, the AV will park at the nearest available parking space given the same parking fee (parking fee is zero in the current case, and if non-zero parking pricing is considered, AV should park at the cheapest space based on a joint evaluation of distance and fee). This means that there is a determined relationship between t and x, and we let x = x (t ) be differentiable (this will hold since it is assumed that m (x ) > 0 ). Suppose the departure rate from home at time t is f (t ) , when all parking has the same price, it follows that

∫t s

t

f (u) du =

∫0

x

m (y ) dy,

(2)

where ts is the departure time of the first commuter. Eq. (2) simply means flow conservation, as well as that the commuter will park at the closest available parking. Moreover, as m (x ) is positive, an earlier departure time t indicates a smaller x (at equilibrium, the f (t ) dx dx parking location x for an AV is dependent on its departure time t). It is straightforward to see that f (t ) = m (x )· dt or dt = m (x ) . Based on Eqs. (1) and (2), by taking the first-order derivative of c (t ) with respect to departure time t, we have

f (t ) f (t ) f (t )−s dc (t ) = α· −β · + λ·w· , m (x ) s s dt

(3)

for early arrivals at the destination, i.e., t + T (t ) ⩽

t∗

; and

f (t ) f (t ) f (t )−s dc (t ) , = α· + γ· + λ·w· m (x ) s s dt

(4)

t∗

for late arrivals at the destination, i.e., t + T (t ) > . dc (t ) At the user equilibrium, all commuters should have identical travel cost, i.e., dt = 0 for t ∈ [ts,te], where ts is the departure time of the first commuter (as mentioned earlier), and te is the departure time of the last commuter. Therefore, with Eqs. (3) and (4), one can determine the departure rates from home (as well as the arrival rates at the bottleneck), i.e., 1 It is worth mentioning that in the future AVs might be publicly owned and operated to provide Transportation As a Service (TAAS), which is predicted in Godsmark et al. (2015). With a TAAS system, AVs would spend much or most of the day like taxis in motion to serve a series of customers. They would not be parked except perhaps at night when few people want to travel. It is also possible that privately and publicly owned AVs may be present simultaneously. It is of my interest to model these different situations in future research, and plan parking facilities (either for short-term or long-term parking) for AVs accordingly.

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f (t ) =

α λws m (x )

α−β +

s, (5)

for early arrivals at the destination, and

f (t ) =

α α+γ+

λws m (x )

s, (6)

for late arrivals at the destination. Note that we consider f (t ) > s for early arrivals, which requires that following. Assumption 1. It is assumed that

λws m (x )

λws m (x )

< β . This is stated in the

< β for all x.

The physical meaning of Assumption 1 is that for early arrival commuters, by delaying a unit of time (thus more close to t ∗), the marginal saving in early schedule delay cost is larger than the marginal increase in the cost of self-driving AV to find a parking space. Note that there will be a larger self-driving time of the AV if the AV travels later and thus parks further away. As a result, congestion will occur. Note that in the above Eqs. (5) and (6), m (x ) depends on x, and thus also depends on t (refer to Eq. (2) for x = x (t ) ). Together with β ·(t ∗−ts ) = γ ·(te−t ∗) + λ·w·x (te ) (i.e., equal cost for the first and last commuters, where x (te ) can be determined based on the conx (t ) dition N = ∫0 e m (y ) dy ) and (te−ts )·s = N (i.e., flow conservation), we can determine the equilibrium parking flow pattern. Roughly speaking, based on “equal cost” and flow conservation conditions, we can determine the ts and te for the traffic equilibrium, and based on Eq. (2) and Eqs. (5) and (6), we can determine the flow rate at time ts and thereafter. More specifically, we have

ts = t ∗−

β N γ N λw λw x (te ). x (te );te = t ∗ + − − β+γ s β+γ β+γ s β+γ

(7)

γ N t ∗− β + γ s

It is evident that ts in Eq. (7) is less than if x (te ) > 0 . This means that when parking location plays a role in shaping the traffic pattern and travel cost, congestion will start earlier (when compared to the case with all parking located at the city center, and parking location does not affect cost). This is due to the incentive of commuters to park closer and have less self-driving for AVs. It is noteworthy that the equilibrium departure rate f (t ) for early or late arrival is generally non-linear. This is because the parking density m (x ) could have very different patterns over x, i.e., non-homogeneous and non-regular spatial distribution of parking. For analytical tractability, starting from here, we adopt the following assumption on parking distribution over space. One may also find the same assumption in, e.g., Arnott et al. (1991). Assumption 2. The parking density over x is constant, i.e., m (x ) = m . Under Assumption 2, we can explicitly solve the user equilibrium solution. Particularly, the departure rates for early and late arrivals can be written as

fe =

α α−β +

λws m

s;fl =

α α+γ+

λws m

s. (8) N λ·w· m ,

The “equal cost” condition for the first and last commuters suggests that β ·(t ∗−ts ) = γ ·(te−t ∗) + and again the flow conservation suggests that (te−ts )·s = N . We then can determine the departure times of the first and last commuters as follows:

ts = t ∗−

β N γ N λw N λw N ;te = t ∗ + − − . β+γ s β+γm β+γ s β+γm

(9)

Moreover, the departure time of the on-time arrival commuter is

to = t ∗−

β−

λws m

α

⎛⎜ γ N + λw N ⎞⎟. β + γ m⎠ ⎝β + γ s

(10)

With the above results in mind, the time-dependent equilibrium flow pattern can be determined, which is shown in Fig. 2, where the blue2 solid line represents the cumulative departure from home (also the arrivals at the highway bottleneck), and the black solid line represents the cumulative departure from the highway bottleneck (also the arrival at the city center or the workplace), and the red solid line represents the arrival of the AVs at their parking spaces. We can further determine the total user cost based on the start and ending time points for the peak, which is

βγ N βλw N ⎤ + N, TC = ⎡ ⎢ β + γ m⎥ ⎣β + γ s ⎦

(11)

where the term in the square bracket is the individual travel cost at equilibrium. Moreover, the total schedule delay cost is 2

For interpretation of color in Figs. 2, 8, and 12, the reader is referred to the web version of this article.

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Fig. 2. The departure/arrival pattern at User Equilibrium. 2

2

β N γ N λw N ⎤ λw N ⎤ + , + 0.5γs ⎡ − TS = 0.5βs ⎡ ⎢β + γ s ⎢ β + γ m⎥ ⎣ ⎦ ⎣β + γ s β + γ m⎥ ⎦

(12)

where the first term in the right-hand side is the total schedule delay cost of early arrivals, and the second term is for late arrivals. The total queuing delay cost is

λws ⎞ ⎛ γ N λw N ⎞ ⎜ ⎟N; + TQ = 0.5 ⎛β− m ⎠⎝ β + γ s β + γ m⎠ ⎝

(13)

and the total AV self-driving time cost is

TD = 0.5λw

N2 . m

(14)

Besides, the total numbers of early arrival and late arrival commuters are

Ne =

β γ λws N λws N ;Nl = N− . N+ β+γ β+γm β+γ β+γm

(15)

Note that based on Assumption 1, one can readily verify that Nl > 0 and Ne < N . We now analyze those system efficiency measures in the above against the parking density m, and compare them with the user equilibrium case when there is no parking consideration or parking is all located at x = 0 . Generally speaking, a larger m suggests a more concentrated parking supply around the city center, and on average, cars will park closer (i.e., a smaller N ). m

Proposition 2.1. At the commuting equilibrium, we have

dTC dTS dTQ dTD dN dN < 0; < 0; > 0; < 0; e < 0; l > 0 dm dm dm dm dm dm for m >

λws , β

(16)

where TC ,TS,TQ,TD,Ne , and Nl are given in Eqs. (11)–(15).

The above results can be readily verified by checking the first-order derivatives with respect to m of Eqs. (11)–(15). It indicates that overall providing a higher density of parking m would be beneficial for commuters, as the total travel cost TC decreases with m (and so does the individual travel cost). This is expected as the average trip distance of AV self-driving is reduced (i.e., a larger m indicates a smaller N ). However, it is noteworthy that total queuing delay cost is indeed increasing over m. This is explained as m follows. The parking location is governed by the departure time, one can refer to Eq. (2). Being earlier will yield a smaller AV selfdriving time due to parking closer to the city center. Therefore, commuters are less willing to queue for a smaller schedule delay cost, which leads to smaller departure rates (see Eq. (8)). More intuitively speaking, a smaller m indicates a less concentrated parking distribution over space, which leads to less concentrated departures and less congestion. In addition, a larger m indicates a smaller AV self-driving distance on average, and the commuters are less motivated to depart from home earlier (refer to departure interval defined by Eq. (9) and the numbers of early and late arrivals in Eq. (15)). Following the above discussion, we further compare the considered case against the case without parking consideration or when parking is all located at x = 0 and m → +∞. We could notice an increased total travel cost and total schedule delay cost, but a γ N reduced congestion delay cost, i.e., TQ < 0.5·β · β + γ s ·N (the right-hand side of this inequality is exactly equal to the total queuing delay cost at the equilibrium without parking consideration or m → +∞, one may refer to Arnott et al. (1990) or Liu et al. (2015)). Similar to the above, we can derive the relationships of different efficiency measures with respect to the value of self-driving time λ and the AV self-driving time w to cover a unit of distance, which are summarized in the following. Proposition 2.2. At the commuting equilibrium, we have 195

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dTC dTS dTQ dTD dN dN > 0; > 0; < 0; > 0; e > 0; l < 0; dλ dλ dλ dλ dλ dλ

(17)

dTC dTS dTQ dTD dN dN > 0; > 0; < 0; > 0; e > 0; l < 0; dw dw dw dw dw dw

(18)

and

where TC ,TS,TQ,TD,Ne , and Nl are given in Eqs. (11)–(15). The above results in Proposition 2.2 indicate that when the AV self-driving is more costly, and slower, commuters will have a larger travel cost, and there will be more early arrivals and less late arrivals. However, there will be less congestion delays. To this point, we have obtained the major features of the commuting equilibrium with AVs given the parking supply. We now briefly discuss its differences from those in Arnott et al. (1991) where non-autonomous vehicles (non-AVs) and walking are considered. Firstly, in Arnott et al. (1991), as drivers will park at some locations near to the city center and then walk to the final destination in the city center, the walking time will affect commuters’ arrival time at the destination. One may notice that the commuters’ arrival rate at workplace will be s at the user equilibrium in this paper, while in Arnott et al. (1991) it is less than s due to the walking and parking further away for a later commuter. Secondly, as walking affects the arrival times of commuters, walking time is then included in the calculation of schedule delay cost in Arnott et al. (1991). Therefore, the parking location indeed will affect the schedule delay of commuters. With AVs, however, walking is avoided and the self-driving time of AVs as well as the parking location will not directly affect commuters’ arrival time and the schedule delay cost. Thirdly, the user equilibrium considered in this paper will be more efficient than that in Arnott et al. (1991) when the walking time is much more costly than the AV self-driving time (in terms of total travel cost). In the next section, we will further discuss the difference between the system optimum with AVs and that with non-AVs in Arnott et al. (1991). 3. System optimum, tolling/pricing, and optimal parking supply under AV behavior patterns This section firstly discusses the system optimum traffic flow pattern (in terms of minimizing total travel cost excluding congestion toll and parking fee), and then compares it against the optimal case in Arnott et al. (1991) where the city has a nonautonomous traffic environment. Secondly, this section develops and evaluates the first-best congestion tolling and parking pricing schemes to achieve the system optimum. Thirdly, this section will discuss the optimal parking supply to minimize total system cost including the travel cost and the social cost of parking. Note that we will focus on the first-best tolling or pricing scheme to support the system optimum as an equilibrium, but will not go to detailed modeling for equilibrium flow patterns under general or diversified tolling or pricing schemes. 3.1. System optimum This subsection discusses the System Optimum (SO) flow pattern given the parking supply density m, where the total travel cost excluding congestion tolls and parking fees should be minimized. At the system optimum, the following conditions should hold: (1) the queuing delay should be completely eliminated (i.e., minimum queuing delay is equal to zero); (2) the schedule delay cost should be minimized (travels are concentrated around t ∗); (3) the total AV self-driving time cost is minimized (parking locations should be as close to the city center as possible given the parking supply). To satisfy the above three conditions, firstly, the departure rate of commuters from home should be equal to the bottleneck capacity s (and then there is zero queuing delay); and secondly, the departure times of commuters should be within [tsso,teso] where γ N β N tsso = t ∗− β + γ s and teso = t ∗ + β + γ s (this is standard in the bottleneck model literature to minimize schedule delays); and thirdly, the N

commuters should park their cars at x ∈ [0, m ] (minimum total self-driving distance and time are then achieved). It is noteworthy that N within [0, m ],

the parking order of commuters departing at different times does not affect the system as long as all the AVs are parked efficiency. This paper simply focuses on the case where the commuters departing earlier will park closer to the city center. To achieve this, the toll and parking fee schemes discussed in the next subsection have to be designed in an appropriate way (commuters or AVs will choose the cheapest parking based on both the location and fee). Under these considerations and with Eq. (2), one can readily t − t so obtain that under the system optimum, a commuter departing at time t ∈ [tsso,teso] should park at x = x so (t ) = ms s . The departure/ arrival pattern is then shown in Fig. 3, where the solid lines correspond to the departure/arrival at the system optimum, and the dotted lines correspond to the departure/arrival at the user equilibrium. As one may realize, the departure time of the first commuter under the system optimum is later than that under the user equilibrium, i.e., tsso > ts . This is due to the necessity for a system optimum to reduce schedule delay cost and to avoid too early arrivals due to the incentive to park closer to the city center (save AV self-driving distance). In Fig. 3, as can be seen, the first and last commuters will both have zero congestion delay, and have equal schedule delay cost (please also refer to tsso and teso ). However, the N last commuter will experience an AV self-driving cost of λw m . To support this flow pattern as an equilibrium, we need to appropriately design the tolling or pricing mechanism, which will be discussed in the next subsection. Before we move to the tolling or pricing schemes, we compare the system optimum flow pattern in the current situation with those in Arnott et al. (1991). As mentioned earlier, since in the fully autonomous traffic environment, walking is eliminated for the commuting and is absent in the travel time of commuters, walking then does not affect the arrival times of commuters, and thus does 196

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Fig. 3. The departure/arrival pattern at System Optimum.

not affect the schedule delay costs of commuters. Therefore, firstly, the arrival rate at the final destination for commuters under the system optimum will still be s (similar to the user equilibrium case), while it is different for commuters in Arnott et al. (1991) due to βγ N2

the non-identical walking times of different parking locations. Secondly, the minimum schedule delay cost of 0.5 β + γ s in the original bottleneck model without parking consideration or with all parking located at x = 0 will still be achieved, although parking might be far away from the city center, i.e., a small m and a large N . However, a too far away parking space is not feasible in the non-AV case m due to unacceptable walking distance (while park-and-ride might help). Thirdly, as in Arnott et al. (1991) parking location would affect arrival time, the parking locations of commuters departing at different times should be optimized (e.g., in a reverse order, and those who arrive early should park further away). However, in this paper, the order of parking locations for AVs departing at different times will not affect the overall system optimality (note that it is adopted in this study that earlier commuters will park closer when deriving the tolls and parking fees). These features mentioned in the above are totally different from the case in Arnott et al. (1991) where walking is inevitable. 3.2. Congestion tolling and parking pricing Section 3.1 has discussed the system optimum. This section will discuss the time-dependent congestion toll and/or locationdependent parking pricing to achieve the system optimum, and then evaluate the efficiency of these tolling and pricing schemes. In particular, it will be shown that the first-best time-dependent congestion toll can be replaced by a location-dependent parking fee. 3.2.1. Congestion tolling To achieve the system optimum, the time-dependent toll τ (t ) can be derived. Without loss of generality, we let τ (teso) = τ0 , and we then have

⎧ τ0 ⎪ λws ⎪ τ0 + β− m τ (t ) =

t ∈ (−∞,tsso)

( ) (t−t + ) + (β − ) −(γ + ) (t−t )

⎨ τ0 ⎪ ⎪ τ − λws N ⎩ 0 m s

λws m

∗

γ N β+γ s

γ N β+γ s

λws m

∗

t ∈ [tsso,t ∗) t ∈ [t ∗,teso]

,

t ∈ [teso,+∞)

(19)

which is also shown in Fig. 4. Note that how to derive the toll is omitted here to save space while interested readers may refer to e.g., Arnott et al. (1991). t − t so Under this first-best toll, the commuter departing at time t will park at x = x so (t ) = ms s (departing earlier indicates parking closer), and nobody can reduce his or her cost by unilaterally changing his or her departure time and parking location of his or her AV. As can be seen in Eq. (19) and in Fig. 4, τ (teso) < τ (tsso) . This is explained as follows. At the system optimum, the first and last commuters should both have zero congestion delay and have equal schedule delay cost, as mentioned earlier. However, the first N commuter will park at location x = 0 while the last commuter will park at location x = m . The last commuter will experience a higher N

AV self-driving cost due to parking further away, which is equal to λw m . This is exactly the difference of τ (tsso)−τ (teso) , which is the amount that the tolling authority should “compensate” the last commuter for his or her less preferable parking location when compared with the first commuter. In this setting, the commuters have no incentive to depart from home earlier than tsso , and additional schedule delay cost for the system is avoided. This is consistent with the observations in Zhang et al. (2011), Yang et al. (2013), Liu and Geroliminis (2016), where the parking supply in the city center is limited.

(

0.5βγ 0.5λws Furthermore, the total toll revenue is RT = ⎡τ0 + β + γ − m ⎣ λws RT are based on m > β (which comes from Assumption 1).

)

N ⎤N . s

⎦

197

We have the following results for RT. Note that the bounds of

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W. Liu

Fig. 4. The time-dependent congestion toll.

Proposition 3.1. Under the time-dependent toll in Eq. (19), (i) the total toll revenue increases with parking density m, i.e., total toll revenue is bounded above and below,

0.5β2 N i.e., ⎡τ0− β + γ s ⎤ N ⎣ ⎦

< RT < ⎡τ0 + ⎣

dRT dm

> 0 ; (ii) the

0.5βγ N ⎤N . β+γ s ⎦

It is noteworthy that this paper considers an isolated single-mode system and a fixed total travel demand. Therefore, the same N system optimum will be achieved even if we vary τ0 . If we let τ (teso) = 0 , we can then determine τ0 = λw m . Furthermore, if we let N

τ0 = λw m , we have RT =

(

0.5βγ β+γ

+

0.5λws m

)

N2 s

> 0.5·TC , where TC is the total travel cost at the user equilibrium given in Eq. (11). If we

let τ0 = 0 , the upper bound for RT is exactly equal to the total toll revenue (under the first-best tolling scheme) when parking is not considered in Arnott et al. (1990). If we consider an elastic travel demand or a multi-modal system, the optimal value of τ0 should be further examined. 3.2.2. Parking pricing Alternatively, we can set a location-dependent fee p (x ) to replace the time-dependent congestion toll in order to achieve the t − t so

system optimum. Note that for commuters departing at time t ∈ [tsso,teso], again we let him or her to park at x = x (t ) = ms s (i.e., departing earlier indicates parking closer). We then, without loss of generality, by letting p (0) = p0 , can determine p (x ) as follows:

( (

⎧ p0 + β− λws m ⎪ ⎪ λws p (x ) = p0 + β− m ⎨ ⎪ λws N ⎪ p0 − m s ⎩

) )

γ

mx s

N

x ∈ [0, β + γ m ]

(γ + ) (

γ N − β+γ s

λws m

γ N mx −β + γ s s

)

γ

N N

x ∈ [ β + γ m , m ], N

x ∈ [ m ,+∞)

(

(20)

)

0.5βγ 0.5λws N ⎤ N . We then have Proposition 3.2 for the parking which is also shown in Fig. 5. The total fee revenue is RP = ⎡p0 + β + γ − m s ⎣ ⎦ pricing scheme, which is consistent with Proposition 3.1 for the congestion tolling scheme.

Proposition 3.2. Under the space-dependent fee in Eq. (20), (i) the total parking fee revenue increases with parking density m, i.e., (ii) the total parking fee revenue is bounded above and below,

0.5β2 N i.e., ⎡p0 − β + γ s ⎤ N ⎣ ⎦

< RP < ⎡p0 + ⎣

Fig. 5. The location-dependent parking fee. 198

0.5βγ N ⎤N . β+γ s ⎦

dRP dm

> 0;

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Proposition 3.3. If τ0 = p0 , then (i) p (x ) = τ (t ) where x = x so (t ) = ms for t ∈ [tsso,teso]; (ii) the total congestion toll revenue RT under the first-best time-dependent congestion tolling in Eq. (19) is identical to the total parking fee revenue RP under the first-best locationdependent parking pricing. Proposition 3.3 indicates that we can construct the first-best location-dependent parking pricing scheme to replace the first-best time-dependent congestion tolling scheme in order to achieve the system optimum. Moreover, we can further design the joint scheme of congestion tolling and parking pricing to support the System Optimum as an equilibrium flow pattern, where the joint scheme is not unique. We denote the joint scheme by (τ ′ (t ),p′ (x )) . We let τ ′ (t ) = τ (t ) for t < tsso , and let τ ′ (t ) + p′ (x so (t )) = τ (t ) for t ⩾ tsso . Note that to this point, either under User Equilibrium or System Optimum, we are considering that the commuters departing earlier will park closer. In case we want to force the earlier commuters to park further away, we can design a joint scheme of time-dependent congestion tolling and location-dependent parking pricing as well. These considerations are feasible while tedious. As our objective is to achieve the system optimum, we thus omit discussions on these less relevant technical issues. However, in Arnott et al. (1991), such discussion is necessary as parking order would affect the system cost. It is also worth mentioning that integrated parking pricing and congestion tolling might be necessary when one wants to integrate morning and evening commutes (e.g., Zhang et al., 2008). 3.2.3. Efficiency At the system optimum (under the first-best tolling and/or pricing), the total system cost or social cost (where toll or fee is excluded) is

TSC = 0.5

βγ N 2 N2 + 0.5λw . β+γ s m

It is noteworthy that if we let τ0 =

N λw m ,

(21) we have RT =

(

0.5βγ β+γ

+

0.5λws m

)

N2 s

= TSC > 0.5·TC , where TC is the total travel cost at user

equilibrium. We can establish the relative efficiency of the first-best time-dependent congestion tolling or location-dependent parking pricing scheme, which is defined as

θ=

TSC , TC

(22)

where TSC is given in Eq. (21) and TC is given in Eq. (11). Note that a smaller θ means that the system optimum is more efficient against the user equilibrium. The properties of θ are further discussed in the following. Proposition 3.4. Under Assumption 1, the relative efficiency defined in Eq. (22) should satisfy the following inequalities:

0.5 < θ <

0.5β + γ . β+γ

(23)

Proof. Firstly, with Eqs. (11), (21) and (22), we can write the relative efficiency as follows:

θ=

0.5βγm + 0.5(β + γ ) λws . βγm + βλws dθ dm

(24) λws β

< 0 . As m > It can be readily verified that holds given Assumption 1, by letting m → Eq. (24) with these two values, we then have Eq. (23). This completes the proof. □

λws β

and m → +∞ and replacing m in

Proposition 3.4 states that the relative efficiency of the system optimum when compared to the user equilibrium is bounded above dθ and below. Its proof further indicates that dm < 0 , which means that when m is larger, the relative efficiency gain from system optimum (when compared to the user equilibrium) is larger (a smaller θ ). Mathematically, this is because TSC decreases faster than TC with respect to m. It is further explained as follows. The user equilibrium is inefficient in two respects: (i) queuing occurs, and (ii) departures begin too early because commuters want to secure a parking space close to the center. However, when evaluating the efficiency of the system optimum against the user equilibrium, the queuing cost reduction is dominant. Specifically, the queuing delay cost can be βγ N2

reduced from TQ to 0, where TQ is given in Eq. (13), while the minimum total schedule delay cost is 0.5 β + γ s (rather than zero). Therefore, how m might affect the queuing reduction (comparing the system optimum with the user equilibrium) dominates the sign of dθ . When m is larger, the parking distribution at equilibrium will be more concentrated around the city center. This concentration over dm space will lead to traffic concentration over time. As a result, the equilibrium departure rates fe and fl are larger. This indicates a larger total queuing delay cost (please refer to Eq. (13) and Proposition 2.1), as well as a larger queuing delay cost reduction in the system optimum. It follows that θ becomes smaller (a smaller θ indicates a relatively more efficient system optimum against user equilibrium). 3.3. Optimal parking supply In previous sections, it has already been shown that the parking supply will affect the traffic equilibrium and efficiency (e.g., Proposition 2.1). Furthermore, the parking supply has a social cost as it consumes land, especially in city centers. This subsection will optimize the parking supply in order to minimize the total system cost including the travel cost and the social cost of parking supply. It is noteworthy that the congestion management strategies (e.g., the tolling or pricing scheme) adopted in the system can affect the optimal planning decisions. In particular, two cases will be examined: (i) congestion tolling or parking pricing is not present (user 199

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equilibrium traffic flow pattern) and (ii) it is present (system optimum traffic flow pattern). We consider that the social cost for a unit of parking at location x is r (x ) per day, which is differentiable over x. We let dr (x ) r ′ (x ) = dx . The parking spaces closer to the city center are generally more socially costly, which means that r ′ (x ) ⩽ 0 . Note that a larger r ′ (x ) (closer to zero) means that the social cost of parking varies less with respect to its distance to the city center. The social parking cost (per day) to serve the total travel demand N can be written as

TPC =

∫0

x (te )

r (x )·m (x ) dx ,

(25)

where as mentioned earlier in Section 2, x (te ) can be determined by the equation N = N m

∫0

x (te )

m (y ) dy . Under Assumption 2, we then

have TPC = ∫0 r (x )·mdx . When congestion tolling or parking pricing is not present, i.e., Case (i), at the user equilibrium, the total system cost including parking supply cost is (26)

TSC ue = TC + TPC. By taking the first-order derivative of TSC ue with respect to m, we have

βλw N 2 dTSC ue =− + dm β + γ m2

N

∫0 m r (x ) dx−r ⎛⎝ mN ⎞⎠ mN .

(27) N m

will When there is a marginal increase in parking density (as well as a parking distribution closer to the city center, i.e., decrease), in Eq. (27), the first term in the right-hand side is the marginal decrease of total travel cost (negative), and the summation N N of the second and third terms is the marginal increase in parking supply cost (positive as r (x ) ⩾ r m for x ⩽ m ), which is the shadow area in Fig. 6. Furthermore, the second-order derivative of TSC ue with respect to m is

( )

2βλw d 2TSC ue N N2 =⎡ + r′ ⎛ ⎞ ⎤ 3 . ⎢ ⎥ dm2 β + γ m m ⎝ ⎠ ⎣ ⎦

(28)

Based on the above, we then have Proposition 3.5. Proposition 3.5. For the total system cost TSC ue , we have: (i) when r ′ (x ) >

−2βλw , β+γ

−2βλw −2βλw λws , it is socially preferable to let m → β ; (iii) when r ′ (x ) = β + γ , β+γ ue ue dTSC dTSC λws +∞ if dm < 0 , and it is indifferent to set any m > β if dm = 0 .

r ′ (x ) < m→

it is socially preferable to let m → +∞; (ii) when

it is socially preferable to let m →

λws β

if

dTSCue dm

> 0 , and

Proof. Proposition 3.5 can be verified readily with Eqs. (27) and (28). −2βλw d2TSCue > 0 , i.e., TSC ue is convex over m. , we have When r ′ (x ) > 2 β+γ

dm

Also, it can be easily verified that dTSCue dm

dTSCue dm

d2TSCue dm2 TSC ue

→ 0 when m → +∞. As

< 0 . Therefore, it is socially preferable to let m → +∞ where

Similar derivations can be done for the other two cases with r ′ (x ) <

> 0 , it implies that when m < + ∞, we should have

will be minimized.

−2βλw β+γ

and r ′ (x ) =

−2βλw . β+γ

We omit the details here. □

Proposition 3.5 indicates that when the social cost of parking supply does not decrease with the distance to the city center x too −2βλw N sharply (i.e., r ′ (x ) > β + γ ), we should provide all the parking in the city center, i.e., m → +∞ and m → 0 , since doing so would reduce commuters’ travel cost (less self-driving for AVs). In contrast, when land value and social cost of providing parking will −2βλw λws decrease with the distance to the city center relatively more sharply (i.e., r ′ (x ) < β + γ ), we should let m → β (note that due to Assumption 1, m is bounded below), since doing so would save the land for other purposes (social and economic activities). We would

Fig. 6. The location-dependent social cost for parking. 200

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like to highlight this possibility under AV traffic environment since parking is now not constrained by walking distance of commuters. Instead, the AVs can drive to places further away from the city center, where the social cost of parking can be negligible. We now turn to Case (ii) where the first-best congestion tolling or parking pricing is present. In this case, the total system cost including the social cost of parking supply is (29)

TSC so = TSC + TPC. Similarly, we can take the first-order derivative of TSC so with respect to m, i.e.,

dTSC so N2 = −0.5λw 2 + dm m

N

∫0 m r (x ) dx−r ⎛⎝ mN ⎞⎠ mN ,

(30)

where given a marginal increase in the parking density, in Eq. (30), the first term in the right-hand side is the marginal decrease of total travel cost under the system optimum (negative), and again the summation of the second and third terms is the marginal increase in parking supply cost. Furthermore, the second-order derivative of TSC so with respect to m is

d 2TSC so N N2 = ⎡λw + r ′ ⎛ ⎞ ⎤ 3 , dm2 m m ⎝ ⎠⎦ ⎣ where again r ′ (x ) =

dr (x ) . dx

(31)

We summarize an observation from Eqs. (27), (28), (30), (31) in the following.

Proposition 3.6. For TSC ue and TSC so , (i)

dTSCue dm

>

dTSC so ; dm

(ii)

d2TSCue dm2

<

d2TSC so . dm2 2βλw

βλw

dTSCue

dTSC so

Proposition 3.6 can be readily verified given that β < γ (− β + γ > −0.5λw and β + γ < λw ). In particular, dm > dm follows similar reasonings for Proposition 3.4, i.e., TSC decreases faster than TC with respect to m, and it follows that TSC so = TSC + TPC decreases faster than TSC ue = TC + TPC with respect to m. Furthermore, it can be readily verified that d2TSCue

d2TSC so

d2TC dm

<

d2TSC , dm

and thus

< . dm2 If we have an interior optimal solution for m, Proposition 3.6 suggests that it will be higher in the system optimum than in the (notoll) user equilibrium. This is because parking concentration over space (a smaller m) can result in traffic concentration over time, and thus more congestion at the user equilibrium. Therefore, we need to decrease m to reduce queuing delay in the user equilibrium case (please refer to Proposition 2.1 for dTQ ). However, in the system optimum, as queuing is eliminated by the tolling scheme, dm decreasing m cannot bring any benefit in terms of queuing reduction. Therefore, we have a larger optimal m under the system optimum traffic pattern. This observation is also numerically tested for more complicated cases in Appendix B. It is noteworthy that without time-dependent congestion, Anderson and De Palma (2004) showed that the user equilibrium involves tighter parking than is optimal (with pricing) because of the uninternalized externality associated with parking (one may also found discussions in Small and Verhoef (2007)), i.e., at equilibrium, drivers selecting a parking space close to the CBD will increase the search cost of a large number of other drivers trying to park there. However, the case in this paper is different. As a more dispersed parking distribution over space can lead to a less concentrated departure over time and less congestion, the optimal parking dTSCue dTSC so supply might be less tighter in the user equilibrium case ( dm > dm ). dm2

Proposition 3.7. For the total system cost TSC so , we have: (i) when r ′ (x ) > −λw , it is socially preferable to let m → +∞; (ii) when dTSC so λws λws r ′ (x ) < −λw , it is socially preferable to let m → β ; (iii) when r ′ (x ) = −λw , it is socially preferable to let m → β if dm > 0 , and

m → +∞ if

dTSC so dm

< 0 , and it is indifferent to set any m >

λws β

if

dTSC so dm

= 0.

Similar to Proposition 3.5, we have the results for TSC so in Proposition 3.7. The interpretation is also similar, i.e., if the social cost of parking does not vary too much over distance to city center, we should plan parking in the city center to accommodate travel demand and reduce self-driving of AVs; if the social cost of parking decreases over space significantly, we should plan parking in locations further away from the city center (AVs make this more feasible, as walking will not be necessary now). It is also noticed −2βλw −2βλw that, for r ′ (x ) > −λw but r ′ (x ) < β + γ (note that β < γ , and thus −λw < β + γ ), we have different optimal solutions for TSC ue and λws

TSC so , i.e., to minimize TSC ue we should let m → β , and to minimize TSC so we should let m → +∞. This indicates that under no-fee and no-toll equilibrium traffic pattern with AVs, it is more likely that we should let m to be small and plan parking further away from the city center. This is because decreasing m indicates a less concentrated parking distribution over space, which leads to less concentrated traffic over time (smaller departure rates fe and fl ) and less congestion (Proposition 2.1) at the user equilibrium. However, under the system optimum traffic pattern, as congestion is already eliminated by tolling/pricing, decreasing m does not help reduce congestion. This reasoning is similar to that for Proposition 3.6. 4. Numerical studies This section presents some numerical experiments to illustrate the proposed model and analysis. Particularly, a uniform parking distribution over space is assumed, i.e., a constant m (however, different values of m might be tested for sensitivity analysis). Under this setting, firstly, this section shows how different system efficiency measures vary with the parking density m. Secondly, the shapes of the first-best congestion tolling and parking pricing schemes are displayed and discussed. In the end, the optimal parking supply is examined to minimize the total system cost including travel cost and social cost of parking supply. 201

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104

2

Costs (EUR$)

1.5 TC

TS

TQ

TD

TSC

1

0.5

0

0

1

2

3

4

5

Parking density m (space per meter) Fig. 7. Different efficiency measures against the parking density.

We start with listing the major common numerical settings. Following Liu et al. (2015), the value of time α is 9.91 (EUR$/h), the early arrival penalty β is 4.66 (EUR$/h), and the late arrival penalty γ is 14.48 (EUR$/h). Moreover, we let λ = 8 (EUR$/h) (note that this is an assumed value, it might be smaller, but should not be larger than α ), w = 0.025 (h/km), the total demand be N = 3000 , and λws the capacity of the highway bottleneck be s = 2000 (veh/h). Based on Assumption 1, we then have m > β = 0.0858 (space per meter). We then vary m from the lower bound to large values and examine how different system efficiency measures will vary. While we may vary m for sensitivity analysis, the benchmark value for m is 0.5 space per meter. It is noteworthy that m = 0.5 (space per meter) does not mean that a parking space has a length of 2 m. Instead, it means that on average for each 2 m along the corridor there will be one parking space. There can be multiple rows of parking spaces along the road. For instance, suppose the size of one parking space is 3 m wide and 6 m long, which is parallel to the road. m = 0.5 (space per meter) means that there are 3 rows of parking spaces along the road, and thus 3/6 = 0.5 (space per meter). For a very large m, it may reflect the case with multi-storey car park, which is not further explained here to save space. Fig. 7 displays the total user cost TC, total schedule delay cost TS, total queuing delay cost TQ, total AV self-driving cost TD, and total system optimum cost TSC (which is achieved under the first-best congestion tolling or parking pricing scheme) against the parking density m. It is evident that all other cost measures decrease with m except the total queuing delay cost TQ. This verifies Proposition 2.1. Moreover, as can be seen in Fig. 7, those efficiency measures decreasing with m are also convex over m, and the total self-driving cost decreases with the sharpest slope. This is partly due to that m would affect the parking distance to the city center non-linearly. The total system optimum cost TSC for given m can be achieved by setting the time-dependent toll in Fig. 8a or the locationdependent parking fee in Fig. 8b (where m = 0.5 space per meter). For numerical illustration, we simply let τ0 = 0 and p0 = 0 . In Fig. 8a, the time-dependent toll exhibits some similarities to those where parking is not considered or parking is all located at x = 0 or m → ∞. For example, the toll is still piece-wise linear. This is due to that we have assumed uniform parking distribution over space. However, differently, as commuters departing later would have a larger AV self-driving cost, the toll at the start of the peak would be larger than the toll at the end of peak (refer to the yellow solid line). This is consistent with our analysis in Section 3.2. Similar observations have been made in Fig. 8b for the parking pricing scheme, where the parking fee and other costs of users are shown against their AV parking locations. 9 8 7

(a) Congestion Tolling

7 6

Costs (EUR$)

Costs (EUR$)

Fee Schedule delay AV self-driving User cost

8

6 5 4 3

5 4 3

2

2

1

1

0

0

7:00

(b) Parking Pricing

9

Toll Schedule delay AV self-driving User cost

8:00

9:00

Time (clock time)

0

0.2

0.4

0.6

Location x (km)

Fig. 8. The time-dependent congestion tolling and location-dependent parking pricing. 202

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104 RT

RT(L)

RT(U)

RP

RP(L)

RP(U)

Total revenue (EUR$)

2.5

2

1.5

1

0.5 0

1

2

3

4

5

Parking density m (space per meter) Fig. 9. Congestion toll and parking fee revenues against the parking density.

Fig. 9 shows how the total toll revenue RT and the total parking fee revenue RP would vary with the parking density m, and also compares them with the lower and upper bounds (in Fig. 9, L and U correspond to the lower and upper bounds, respectively). Note that to distinguish between these two cases (for illustration purpose), we let τ0 = 5 and p0 = 3. As one can see, when m → +∞, both revenues approach their own upper bounds. This is consistent with our analysis for Propositions 3.1 and 3.2. Furthermore, we display the relative efficiency θ in Eq. (22) against the parking density m in Fig. 10. Note that for both the first-best congestion tolling and parking pricing schemes, the same system optimum flow pattern will be achieved, and the efficiencies of both schemes would be identical. As can be seen in Fig. 10, the relative efficiency approaches 0.5 when m increases. This is a numerical illustration of our analysis for Proposition 3.4. Also, 0.5β + γ one can verify that the highest value of θ in Fig. 10 is equal to 0.72 , which is less than β + γ = 0.88 (the upper bound for θ ). We now move to the case when the social cost of parking supply is also taken into account, i.e., we are examining TPC ,TSC ue and TSC so . Planning of parking might become different when AVs are considered, as the vehicles can drive themselves further away from the city center and walking of commuters is avoided. We let r (x ) = 10−r0·x where r0 = 0.5,1.5,3( ×10−4) for Cases (1), (2), and (3). Particularly, Fig. 11 shows how the three measures TPC ,TSC ue and TSC so vary with the parking density for the three cases with different r ′ (x ) . In Case (1), when r ′ (x ) is relatively small (more negative), all three costs TPC ,TSC ue and TSC so increase with m. This is because that locating parking closer to the city center would be relatively more costly as the social cost of parking r (x ) decreases sharply over x. In Case (3), when r ′ (x ) is relatively large (less negative), TPC increases with m. However, TSC ue and TSC so will both decrease with m. This is because that locating parking closer to the city center would be not too costly (when compared to locating parking further away) as the social cost of parking r (x ) decreases less sharply over x. In Case (2) with a medium level of r ′ (x ),TPC and TSC ue increase with m but TSC so decreases. This suggests that depending on the social cost of parking supply, we might have very different optimal parking supply patterns for the system when traffic pattern is at no-toll user equilibrium and tolled system optimum. This highlights the importance of taking into account existing traffic management strategies when planning for infrastructures and facilities. These observations are also in line with the analysis in Section 3.3. Note that in the above cases, the optimal parking density either approaches the lower bound or infinity (or an upper bound if we set one) for TSC ue and TSC so . A more complicated case with a bounded optimal m is further illustrated in Appendix B.

0.75 Relative efficiency 0.7

0.65

0.6

0.55

0.5

0

1

2

3

4

Parking density m (space per meter) Fig. 10. The relative efficiency θ against the parking density. 203

5

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104

5

Case 1: small r'(x)

Case 2: medium r'(x)

4.5

Costs (EUR$)

Costs (EUR$)

4.5 4 3.5 3 TPC

4

2.5 0

1

2

3

4

5

104

0

1

2

3

4

5

Parking density m (space per meter)

Parking density m (space per meter)

5

TPC TSCue TSCso

3.5

3

TSCue TSCso

2.5 2

104

5

Case 3: large r'(x)

Costs (EUR$)

4.5

4 TPC TSCue TSCso

3.5

3

2.5

0

1

2

3

4

5

Parking density m (space per meter)

Fig. 11. The parking cost and total system costs against the parking density.

5. Conclusion This study formulates the morning commute problem when commuters travel with AVs in a fully autonomous environment. The major behavior patterns under AV environment, i.e., AVs can drive themselves to parking and walking of commuters is avoided, have been modeled and analyzed. In particular, theses behavior patterns change the joint equilibrium of departure time and parking location choices. Such temporal and spatial analysis of the commuter parking problem in a fully autonomous environment has shed light on future traffic management and parking planning with AVs. It is found that the user equilibrium flow patterns will be different from those without AVs due to the different behavior patterns. Moreover, the system optimum is also different from those in the literature when parking and walking is considered (e.g., Arnott et al., 1991). It is because, when AVs are present, walking can be avoided and will not affect travel times and arrival times of commuters any more. Congestion tolling and parking pricing schemes have been examined to achieve the system optimum flow pattern, and efficiencies of these schemes have been evaluated. Furthermore, this paper examines different system performance measures against the parking density. It is noteworthy that with a lower parking density and a more dispersed parking distribution over space, the total queuing delay cost will be smaller at the no-fee and no-toll user equilibrium. This is due to that the parking location would affect the self-driving cost of AVs and thus affect the equilibrium departure rates from home, which increases with the parking density. When taking into account the social cost of parking supply to serve the travel demand, this paper further examines the total system cost including the travel cost and the social cost of parking (for both user equilibrium and system optimum time-dependent flow patterns). It is found that if the social cost of unit parking supply does not vary too much over space, we should plan parking in the city center as doing so saves self-driving of AVs; and we should plan the parking further away from the city center if the social cost of parking in city center is much more expensive than that for locations further away, which saves the social cost of parking. However, in the medium case, we have different optimal parking supply solutions for the system when the traffic pattern is in no-fee and no-toll user equilibrium and tolled system optimum. This highlights the importance of taking into account existing traffic 204

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management strategies when planning for infrastructures and facilities. This paper can be extended in several ways. Firstly, as a first step to model commuter parking problem with AVs, we have assumed a fully autonomous traffic environment. However, it can be expected that at least for some periods, there might be a mixed traffic environment with both AVs and non-AVs. Therefore, it is necessary and interesting to consider a mixed case with both types of vehicles and behavior patterns. This would be challenging as the interaction among different vehicle types through the shared road network and parking facilities are complex. Secondly, further research might also consider that a parking AV can impose (flowdependent) congestion on other parking vehicles and on inbound vehicles, which might be modeled through a macroscopic flowdependent traffic model (Chu, 1995; Liu and Geroliminis, 2016). Thirdly, the adoption of AVs will not only affect the congestion dynamics in the morning commute, but also in the evening commute. The author is currently considering how the adoption of AVs would reshape the combined morning and evening commutes, where similar considerations have been presented in Zhang et al. (2008) for transportation systems with non-AVs. In line with this, it will also be interesting to consider the pre-peak duration when AVs go from the parking spaces at the edge of the living area to the homes to pick up passengers. Since the AVs can drive themselves, it may be socially preferable to plan the parking spaces not near to home locations. This might allow for larger homes, gardens, and communal areas. Fourthly, this study focuses on an isolated driving system, while it can be integrated into a multi-modal transportation system framework, where AVs might affect mode choices significantly as the parking limitation in city center might not hold for AVs (especially when transit service is responsive to road traffic conditions, such as in the studies of Zhang et al. (2014) and Zhang et al. (2016)). Last but not least, this paper focuses on the parking issue related to AVs. However, it does not incorporate the potentially larger capacity by AV platooning. Future study may take this into account as well. Acknowledgments I would like to thank Professor Yafeng Yin from the University of Michigan, Ann Arbor and Dr. Fangni Zhang from the University of Leeds, UK for their helpful comments on an earlier version of this paper. I am also very grateful for the anonymous reviewers’ comments for improving both the technical quality and exposition of the paper. All remaining errors are my own. Appendix A. Notations See Table 1.

Table 1 List of notations. Notation

Specification

t x m (x ) N t∗ α β γ λ w s T (t ) q (t ) f (t ) fe ,fl

The The The The The The The The The The The The The The The

c (t ) ts,te,to tsso,teso TC ,TS,TQ,TD Ne,Nl τ (t ) τ0

The cost for commuters departing at time t The departure times for the first, the last, and the on-time commuters at the user equilibrium The departure times for the first and the last commuters at the system optimum

RT p (x ) p0 RP TSC θ r (x ) TPC TSC ue,TSC so

clock time location along the traffic corridor (distance to the city center) parking density at location x total number of commuters desired arrival time for work value of time when commuters are driving with the AVs penalty for a unit time of early arrival penalty for a unit time of late arrival value of a unit AV self-driving time travel time needed to cover a unit distance by self-driving AVs capacity of the highway bottleneck travel time experienced by the commuters departing at time t queue length for the commuters departing at time t departure rate from home at time t equilibrium departure rates for early and late arrival commuters

The total travel cost, the total schedule delay cost, the total queuing delay cost, and the total AV self-driving time cost The total number of early and late arrival commuters The toll charge for the commuters arriving at the bottleneck at time t The The The The The The The The The The

toll charge at time tsso , i.e., τ (tsso) total toll revenue parking fee for the parking space at location x parking fee for the parking spaces at location x = 0 total parking fee revenue total system cost at the system optimum ratio of TSC to TC social cost for a unit of parking at location x social parking cost (per day) to serve the total demand N total system costs including parking supply cost at the user equilibrium and at the system optimum

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Appendix B. Extended numerical studies

TSCue: optimal m

(a) r0 =0.002: TSC ue or TSCso 2.5 2 1.5

Decrease

1 0.5 0

0.2

0.4

0.6

0.8

1

1.2

The parking location nearest to the CBD xs (km) (c) r0 =0.01: TSC ue or TSCso 2.5 2

Decrease

1.5 1 0.5 0

0.2

0.4

0.6

0.8

1

TSCso

TSCso: optimal m (b) r0 =0.004 : TSCue or TSCso

The parking density m (space per meter)

The parking density m (space per meter)

The parking density m (space per meter)

TSCue

The parking density m (space per meter)

For analytical tractability, most of the analysis in this paper is based on the consideration that parking is distributed along the traffic corridor (starting from x = 0 ) with a parking density of m. This part further numerically examines a more general case. In N particular, it is considered that parking is provided between xs and xs + m with a parking density of m. In this context, we can explore how the total system costs (including the social cost of parking) TSC ue (for the user equilibrium flow pattern) and TSC so (for the system optimum flow pattern) would vary with m and xs . The numerical setting is identical with that in Section 4 except that r (x ) = 50·exp−r0·x (EUR per unit parking space) is adopted here. Furthermore, four values for r0 have been tested (0.002,0.004,0.01, and 0.03) to reflect different situations for parking cost. For the four different cases, the total system costs contours are displayed in the domain of (xs ,m) in Fig. 12. In particular, the blue and red lines correspond to TSC ue and TSC so , respectively. Moreover, the blue and red dotted lines represent the optimal parking density m under given xs for the user equilibrium and system optimum cases, respectively. Firstly, it is evident that for any given xs , the red dotted line is always above the blue dotted line, indicating that with the system optimum flow pattern, the optimal parking density should be larger. This result is consistent with Proposition 3.6. Secondly, when r0 increases, indicating that the social parking cost decreases more sharply with respect to the distance to the city center, the optimal parking density m increases (the blue and red dotted lines move leftward and upward when we shift from Fig. 12a to b, and then to Fig. 12c, and finally to Fig. 12d). This is because when parking becomes less costly near the city center (r0 increases), one should plan more and tighter parking close to the city center. Thirdly, it is evident in Fig. 12 that the optimal m under given xs is bounded, which is different from those in Propositions 3.5 and 3.7, where we can have unbounded corner solutions.

1.2

The parking location nearest to the CBD xs (km)

2.5 2 1.5 1

Decrease 0.5 0

0.2

0.4

0.6

0.8

1

(d) r0 =0.03: TSC ue or TSCso

Decrease 2.5 2 1.5 1 0.5 0

0.2

0.4

0.6

0.8

1

1.2

The parking location nearest to the CBD xs (km)

Fig. 12. Total system cost contours in the domain of (xs ,m) .

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