The morning commute problem with ridesharing and dynamic parking charges

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The morning commute problem with ridesharing and dynamic parking charges Rui Ma, H.M. Zhang∗ Department of Civil and Environmental Engineering, University of California, Davis, Davis, CA 95616, USA

a r t i c l e

i n f o

Article history: Received 4 October 2016 Revised 7 July 2017 Accepted 10 July 2017 Available online xxx Keywords: Dynamic ridesharing Dynamic parking charges Bottleneck congestion Morning commute

a b s t r a c t This paper studies the traffic flow patterns in a single bottleneck corridor with a dynamic ridesharing mode and dynamic parking charges. Schemes with different ridesharing payments and shared parking prices are investigated. Besides the scheme with constant parking charges and ridesharing payments, dynamic parking charges and ridesharing payments are derived to achieve congestion-free traffic in the corridor. With the dynamic ridesharing ratios, it is found that genuinely nonlinear departure rates and travel time functions can be generated in certain ridesharing cases, which was not observed in the traditional ADL model (Arnott et al., 1990) for the morning commute problems without ridesharing or with constant ridesharing ratios. Moreover, comparing different configurations of ridesharing arrangements and parking charges, the results show that constant parking charges with constant ridesharing payments may not significantly improve system performance over the traditional morning commute with solo-drivers, while dynamic parking charges with properly selected constant ridesharing payments can achieve better system performance in terms of vehicle-miles-traveled, vehicle-hours-traveled and total travel costs, by encouraging ridesharing and spreading vehicular demand over time to eliminate queuing delays. © 2017 Published by Elsevier Ltd.

1. Introduction Traffic congestion often occurs in corridors in modern cities during the morning commute from residential areas to workplaces in the central business district (CBD). The increasing traffic congestion on commuting corridors usually leads to significant travel delays and other externalities, such as extra fuel consumption and air pollution. Among many instruments used to manage auto travel, ridesharing, also referred as carpooling, has been considered as an important demand management tool to reduce travel times, air pollution and other externalities (Kelly, 2007), and has been practiced for decades (Ferguson, 1997). In 1970, 20.4% of American workers commuted to work by carpool, according to the US Census. However, in the past few decades, ridesharing has declined to 10.7% in 2008 (Chan and Shaheen, 2012). Recently, the ridesharing market starts to grow rapidly, thanks to the popularity of GPS-enabled smart devices and the innovative ridesharing services. Paid ridesharing services, extended from the taxi-like ridesourcing services such as Uber and Lyft, are experiencing a rapid growth. Such paid ridesharing services create opportunities to better utilize the empty seats in commuting vehicles, and commuters can easily choose to be a driver or a passenger and be paired to each other in

∗

Corresponding author. E-mail address: [email protected] (H.M. Zhang).

http://dx.doi.org/10.1016/j.trb.2017.07.002 0191-2615/© 2017 Published by Elsevier Ltd.

Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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real-time. Unlike the traditional carpool services, recent paid ridesharing services allow travelers to be paired in real-time instead of in a pre-arranged fashion. Traditional ridesharing/carpooling studies are mainly limited to static analysis and ignore the effects of traffic dynamics and peak spreading in the morning commute. Brownstone and Golob (1992) studied the effectiveness of incentives for ridesharing services using probit discrete choice models to predict commuters’ mode choices in a static fashion. Baldacci et al. (2004) studies a special case of the carpooling as a Dial-a-Ride problem by formulating integer programming formulations. An equilibrium model based on nested logit model was proposed in Viti and Corman (2013), where the ridesharing was modeled based on static link flow and travel times. Recently, Xu et al. (2015) extended the static ridesharing problem for general networks with multiple origin-destination pairs, and formulated the ridesharing problem with pick-up and drop-off choices, in the form of a nonlinear complementarity problem. Enabled by emerging technologies and ridesharing marketplaces, recent ridesharing studies have started to explore dynamic ridesharing problems. A comprehensive review on optimization problems of dynamic ridesharing can be found in Agatz et al. (2012). Qian and Zhang (2011) studies the morning commute problem with transit, driving alone and carpool modes on a homogeneous population. The ridesharing payments were internalized in the homogeneous carpoolers, and thus neglected in the formulation. Viti and Corman (2014) proposed a joint model considering both day-to-day and within-day models for dynamic ridesharing. In their within-day model, roles of travelers and departure-time choices were updated via an iterative solution process. Recently, Liu and Li (2017b) proposed a time-varying compensation scheme for ridesharing user equilibrium, while the ridesharing ratio does not vary and fixed as one. As explained in the pioneer work of Vickrey (1969) and further studies extending Vickrey’s work in the last several decades, traffic congestion in morning commute can be modeled as a deterministic queuing process. Given the early and late arrival penalties, individual drivers compete with each other to reach a user equilibrium with respect to their commuting costs. At the equilibrium, drivers arriving earlier than the preferred time may benefit from less travel times in the queued corridor, but encounter higher early arrival penalties; drivers arriving later than the preferred time may also benefit from reduced travel times, but suffer from higher late arrival penalties. Under such an equilibrium, the overall system cost is increased from its optimum due to the deadweight loss in terms of drivers’ queuing delays. However, without the consideration of such traffic dynamics and departure-time choice (which leads to peak spreading), static models fail to properly capture the congestion phenomena in the morning commute, or develop time-varying pricing policies to relieve traffic congestion. In Vickrey (1969), a dynamic toll was proposed to eliminate the queuing congestion at the bottleneck. A large number of studies (e.g., Arnott et al., 1990; Arnott and Kraus, 1998) have been made to extend Vickrey’s work, but none of these studies, to the best of our knowledge, has included ridesharing in its analysis. Parking is a critical component of daily commuting trips. As pointed out in the literature (e.g. Button, 1995; Johansson and Mattsson, 1995), parking charges have been suggested as an efficient alternative instrument for managing travel demand. Most of the parking pricing studies are limited to static cases (e.g., Spiess, 1996; Vianna et al., 2004). Although in Vickrey’s bottleneck model, it was suggested that congestion toll could be served to eliminate the queuing delays, while in reality, congestion toll has often met with strong public resistance. As a result, only a few cities in the world have successfully adopted congestion charging. On the other hand, people are used to pay low or no parking fees in suburban shopping malls but higher fees in downtown areas. From the perspective of congestion pricing theory, properly designed parking charges have almost identical effectiveness in influencing commuters’ travel choices, while at the same time, they are easier to implement than congestion tolls. The availability of cutting-edge sensing and information technologies enables the dynamic parking pricing and the transmission of price signals to commuters in real-time. For instance, the SFPark program establishes varying parking prices in different time-of-day in the San Francisco downtown area, and the online system is able to broadcast the prices to the travelers via smartphone apps and web pages. Recent studies have integrated parking and morning commute to leverage parking to influence morning commute travel demand. Zhang et al. (2008) integrated both morning and evening commute with road tolls and parking fees in a linear city. Qian et al. (2012) derived the optimal parking fees, capacities and access times in the morning commute scenario to minimize total social costs. Yang et al. (2013) studied the morning commute problem with parking space constraints. Fosgerau and De Palma (2013) developed a dynamic parking charge scheme to a single bottleneck. Qian and Rajagopal (2015) analyzed optimal dynamic pricing for morning commute by considering parking cruise, parking occupancy in the generalized costs. Most studies and practices on parking pricing have not considered the dynamic ridesharing demand in the morning commute, with a handful exceptions in recent studies (Liu and Li, 2017a; Xiao et al., 2016). There are still open questions to explore. For instance, does the combination of dynamic ridesharing and parking charges have an influence on the traffic flow patterns in the morning commute? If so, how to effectively use such combination to manage traffic congestion and reduce the environmental costs? In the traditional morning commute problem, the optimal pricing is uniquely determined once the bottleneck characteristics and the piecewise linear arrival penalty function are given. However, in the ridesharing schemes, parking charges and ridesharing payments may lead to multiple combinations of pricing strategies, if they are simultaneously considered. Due to the lack of studies for pricing strategies with parking charges and ridesharing payments coherently in the ridesharing scheme, it is not clear that whether the traditional pricing schemes for solo drivers can be transferred to the ridesharing schemes. In order to answer the above questions, this paper formulates an analytical continuous-time dynamic ridesharing problem for a single bottleneck in the morning commute, in the context of dynamic user equilibrium of departure-time and ridesharing mode choices. It also designs policies with proper parking charges and ridesharing payments by analyzing and Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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comparing different policy schemes. Unlike traditional morning commute problems without ridesharing or with simplified ridesharing (fixed number of passengers), this paper shows that under certain conditions, even the same piecewise linear arrival penalty function is applied, the resulting dynamic ridesharing payments and parking charges may bring genuine nonlinearity to departure rates and travel time functions, which was not observed in traditional morning commute problems without ridesharing or with fixed ridesharing ratios. By comparing the benchmark scheme with solo drivers and the ridesharing schemes with different model parameters, the results show that dynamic ridesharing payments and parking charges can effectively mitigate the traffic congestion in the ridesharing schemes. It is shown in the numerical results that among all ridesharing schemes, dynamic ridesharing payments with properly selected constant parking charges can achieve better system performance in terms of vehicle-miles-traveled, vehicle-hours-traveled and total travel costs, by encouraging more travelers to join ridesharing as passengers and eliminating the queuing congestion. In this study, we assume all travelers have their own cars, and can freely choose to become a driver or a passenger, which is endogenously determined by the mode/role choice in the equilibrium. This assumption is coherent with the homogeneity assumption for the travelers. It is different from the assumptions in Xiao et al. (2016), in which solo-drivers and ridesharing travelers are both considered in their equilibrium model, and these two types of vehicles travel on separate lanes, namely general purpose (GP) and high-occupancy-vehicle (HOV) lanes, respectively. While in this study, we assume all travelers are homogeneous and participate the ridesharing program, and the only differences are their roles (either a driver or a passenger). We do not distinguish different types of lanes or drivers, as all drivers participate ridesharing program and drive on the same lanes; that is, if there exists a solo-driver, then all travelers are solo-drivers in this framework. In this study, we do not specify any group of solo-drivers, and there are only homogeneous ridesharing travelers. Therefore, explicit ridesharing inconvenience cost to distinguish costs of solo and ridesharing drivers (which were considered explicitly in literature) is not necessary in this study. If the inconvenience is identical to both passengers and drivers, they would cancel out each other; if the inconvenience varies over two roles, it can be combined into the savings of the operational costs for passengers. A passenger would save the operational costs (such as fuel cost, driving and pick-up costs). As a compensation, passengers need to pay ridesharing payments to the driver for the morning commute trips. At the destination, parking charges are imposed to each vehicle, and the costs of such parking charges are shared among the driver and the passengers in a vehicle evenly. To reduce the notational clutter and streamline our subsequent discussions, we assume that all travelers share the same value-of-time, and without loss of generality it is taken as 1 unit of monetary cost per time unit in this paper. We define the ridesharing ratio as the ratio of the number of passengers and drivers entering the bottleneck link at the same time. Note that this ratio and the other flow variables (such as vehicular inflow and exit flow rates) are all treated as aggregated continuous variables of time, which is different from the agent-based models (Nourinejad and Roorda, 2016) or microscopic simulations (Di Febbraro et al., 2013), where individual travelers need to be assigned to a vehicle, so that strict integer ridesharing ratios have to be maintained at every time for each vehicle. On the other hand, since the ridesharing ratios have been recognized as a continuous variable/parameter rather than integer one in the literature of ridesharing modeling with flow variables (Xu et al., 2015), we apply the aggregated continuous ridesharing ratios. The aggregated ridesharing ratio may be non-integer. For instance, ‘the ridesharing ratio is 2.4’ means that on average, among travelers that enter the bottleneck link at time t, the number of passengers is 2.4 times that of drivers. More discussion on the aggregated continuous ridesharing ratio can be found in Section 2. The ridesharing ratio is a key component in the proposed formulation, which is connected to the modeling of pick-up costs for drivers, as well as the calculation of the shared parking charges. In Xiao et al. (2016) and existing literature on morning commute without ridesharing, the travel time is a (piecewise) linear function according to the traditional ADL model (Arnott et al., 1990). One of the main reasons the ADL model still apply in Xiao et al. (2016) is that the ridesharing ratio is assumed to be a fixed value m for all time and is not adjustable. In this paper, we do not assume constant ridesharing ratios. This leads to quite different analysis and solutions. For certain schemes, we show in both analytical and numerical results that ridesharing ratios could vary over time, and can be a piecewise nonlinear function. The nonlinearity introduced to the system is a unique feature that distinguishes it from the traditional morning commute models without ridesharing or with simplified ridesharing. We show in the following sections that the derived ridesharing ratio itself, as well as the demand rates, could be piecewise genuinely nonlinear functions, even when the classic piecewise linear arrival penalty function is applied. We believe such nonlinearity is an important finding with the following reasons. First, to the authors’ best knowledge, the nonlinearity is rarely observed in existing literature, mainly because the ridesharing is either ignored, or assumed to be a fixed constant in the literature. Second, the nonlinear ridesharing ratios, demand rates and travel time functions show the ridesharing travelers adjust their choice behaviors of both departure-time and mode/role choices in a coherent fashion and have their realistic implications. Third, without the ridesharing payment mechanism discussed in this paper, the mode/role choices and the nonlinear results cannot appear. Therefore, the nonlinearity suggests the ridesharing payment mechanism plays as a critical role, apart from the linear, traditional morning commute problems. The system performance of each ridesharing/pricing schemes is represented by three indicators: vehicle-miles-traveled (VMT), vehicle-hours-traveled (VHT) and the total costs (TC) of all travelers. The analytical expressions of these indicators are derived whenever feasible. Due to the nonlinearity of the solutions, explicit expressions may not always be derived for all of the system performance indicators in all schemes. In order to investigate more thoroughly the relations among the parameters of the system, the choices of parking charges and ridesharing payments and the system performance indicators, numerical examples with given characteristics are also analyzed. Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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The rest of this paper is organized as follows. Section 2 introduces a general morning commute problem with dynamic ridesharing on a single bottleneck in continuous-time. Section 3 analyzes schemes with different settings of ridesharing payments and parking charges analytically. Section 4 derives the analytical system performance indicators for all schemes. Section 5 discusses with numerical examples. Section 6 concludes the paper with discussion of future research. 2. Problem statement In this Section, we propose our modeling framework for the morning commute problem with dynamic ridesharing on a single bottleneck in continuous-time, where travelers choose their mode and departure-time to minimize their own commuting cost. 2.1. Notations We first list the notations in this paper as follows. Model parameters (all positive scalars) D C hp n

α γ β R

Total demand of all travelers Bottleneck flow capacity Average pickup travel time for each traveler; h = 0.5h p Maximum ridesharing ratio Early arrival penalty parameter, 0 < α < 1 Late arrival penalty parameter, γ > 1 Parameter for modified bottleneck trip cost of passengers Preferred arrival time

Time-varying variables q(t ) τ (t ) d2 (t ) d3 (t ) w(t ) s(t ) c2 (t ) c3 (t ) p(t ) kˆ (t ) k(t )

Queue on the bottleneck link at time t Travel time at the bottleneck for travelers entering the bottleneck at time t Vehicular demand rate entering the bottleneck at time t Passenger demand rate entering the bottleneck at time t Exit flow rate at time t Ridesharing ratio for travelers entering the bottleneck at time t Cost for drivers entering the bottleneck at time t Cost for passengers entering the bottleneck at time t Ridesharing payment at time t Parking charges for travelers arriving at the destination at time t Parking charges for travelers entering the bottleneck at time t, ki (t ) = kˆ i (t + τ (t ))

Intermediate notations t1 t2 t3

τm

The The The The

earliest departure time critical departure time; t2 + τ (t2 ) = R latest departure time maximum travel time in Scheme 1; τ m = maxt τ (t ) and τ (t2 ) = τ m .

2.2. The single bottleneck model with ridesharing The single bottleneck is represented as a link with a single origin-destination pair. The total demand is D. The link freeflow travel time is τ 0 , the bottleneck capacity is C , which constrains the exit flow rate from the link. The bottleneck is represented by Vickrey’s single bottleneck model in continuous-time, i.e., the continuous-time pointqueue (PQ) model (e.g., Ban et al., 2012) for the link traffic dynamics. The queue dynamic is

q˙ (t ) = u(t − τ 0 ) − w(t ).

(1)

where u(t) is the inflow rate of the vehicular traffic at time t, and w(t) is the exit flow rate of the vehicular traffic at time t. When a positive queue is observed, the exit flow rate should equal to the bottleneck flow capacity, i.e, q(t) > 0, w(t ) = C ; when there is no queue, the exit flow should equal to the inflow rate with a free-flow travel time delay, i.e., q(t ) = 0, w(t ) = u(t − τ 0 ). Since the bottleneck capacity is not restrained by any downstream blockage, the exit flow should keep its maximum utilization without having any holding of flow. The link travel time is

τ (t ) = τ 0 +

q(t + τ 0 ) C

t 

= τ0 +

0



u (ξ ) − w (ξ + τ 0 ) d ξ C

(2)

The actual arrival time is then

τ (t ) = t + τ (t ) = t + τ 0 +

q(t + τ 0 ) C

.

(3)

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Without losing generality, we denote that the free-flow travel speed on the bottleneck link is vf , and thus the length of the bottleneck link is τ 0 vf . Before entering the bottleneck link, we further assume any driver needs to drive on the local street at the speed of vh before entering the bottleneck link. In the ridesharing schemes of this study, all travelers in the network participate the ridesharing program, i.e., a traveler is either a driver or a passenger. The dynamic ridesharing ratio is defined as s(t), which is the ratio of passengers over drivers departing from the entrance of the bottleneck link at time t. Here we note that the time-varying ridesharing ratios are treated as aggregated variables, which should be distinguished from assigning an individual traveler to a specific vehicle as in microscopic simulation. The ridesharing ratio is thus a continuous variable, i.e., non-integer ridesharing ratios may appear. The continuity of such a variable lies in two aspects. First, the value of ridesharing ratio is defined on a set over a continuous range; Second, the value of ridesharing ratio is presented as a continuous function of time. Furthermore, in this research, ridesharing ratio s(t), as a function of time t, is differentiable almost everywhere. The first aspect emphasizes that the value of ridesharing ratio is not necessarily integer values, while the second aspect emphasizes the ridesharing ratio can change smoothly over time under the dynamic equilibrium. The introduction of continuous ridesharing ratios, rather than strict integer ones, follows the literature of ridesharing modeling, where the ridesharing ratio has been recognized as a continuous variable/parameter rather than integer one. For instance, in >Xu et al. (2015), the fraction of ridesharing drivers versus the passengers is treated as a continuous variable, and is bounded by 1/C and 1, where C is the most number of passenger per driver. Instead of enforcing integer ratios, using continuous ratios is much more beneficial to study the equilibrium states and the corresponding solutions under various schemes; more specifically, the shift of ridesharing role choices over time, as well as departure, arrival and queuing dynamics, can be studied with continuous ratios. Counter examples shown later in Section 3 suggest that under integer ratios, there are no equilibrium solutions to be found for some of the ridesharing schemes. In reality, the ridesharing ratio is constrained by the maximum number of passengers in a commute vehicle (vehicle capacity). When homogeneous vehicles are considered, the ridesharing ratio is bounded as 0 ≤ s(t) ≤ N, where N is the maximum number of passengers a vehicle can hold. Note that in this general formulation of the ridesharing problem with a single bottleneck, such constraint is used to decide whether a given set of parameters would lead to infeasible results. The departure rate at time t for all travelers is defined as d(t), which is the sum of the departure rates of drivers d2 (t) and passengers d3 (t), respectively.

d (t ) = d2 (t ) + d3 (t ) = (1 + s(t ))d2 (t ), ∀t .

(4)

where d2 (t ) = u(t ) is the demand of ridesharing driver (equal to the inflow rate u(t)), and d3 (t) is the demand of passengers. The total demand follows the conservation law as



D=

T 0

d (t )dt =



0

T

(1 + s(t ) )d2 (t )dt.

(5)

We define the beginning and terminal times of the demand are t1 and t3 , respectively, i.e., the first traveler entering the bottleneck link at time t1 , and the last one entering at time t3 . All demand enters the bottleneck during the time period of (t1 , t3 ). We define two critical time instants, the beginning and terminal times of the demand. At these two time instants, the bottleneck link has no queue, which means

q(t1 ) = q(t3 ) = 0.

(6)

We assume a common value of time for all travelers as 1. For each driver entering the link at time t, the cost of traveling on the bottleneck link is τ (t). In order to represent the operational costs saved by the passengers, the cost of traveling on the bottleneck link of a passenger entering the link at time t is modeled as βτ (t), (0 < β < 1), which is less than that of the corresponding driver, since passengers save from operation costs and can gain productivity during their travel time. We note that such assumptions simply the analysis, while other realistic operating cost terms can still be integrated into the proposed formulation in the future study, while current methodology and framework for deriving the equilibrium solutions in this paper should still apply. 2.3. Pick-up trips We consider the case where a driver needs to pick up every passenger before entering the bottleneck link. For simplicity, we assume that any pickup trip between two sites in the local area takes a unique time hp ≥ 0. Each pickup trip lasting time hp is assumed to be after the corresponding pickup time instant, and the drivers would ‘pickup’ themselves first. Under this rule, when the ridesharing ratio is s, the driver experiences a in-vehicle pickup time of (1 + s ) · h p , the first passenger experiences a in-vehicle pickup time of s · hp , and the last passenger experiences a in-vehicle pickup time of hp ; In general, the nth passenger experiences a pickup time of (1 + s − n ) · h p . On average, the pickup time experienced by a passenger is (1+s )·h

p half of that of the corresponding driver, which is = (1 + s ) · h, h = 0.5h p . Notice that under this pickup assumption, 2 even a driver travels alone (i.e., s = 0), there is still a pickup time hp , representing the local trip from the drivers’ home and the entrance of the bottleneck link.

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On the passengers’ side, we only consider the in-vehicle pickup time experienced by the passengers (in an average sense), and omit the waiting times for pick-ups, since recent dynamic ridesharing services such as Uber can provide passengers fairly accurate estimated pick-up times in real-time, which means planned schedules may enable the waiting times to provide positive utility, instead of negative ones from traditional ridesharing services without notice of estimated pick-up times in advance. 2.4. Arrival penalties Travelers share a common preferred arrival time R. Early or late arrival results penalties to the travelers. Here we use the piecewise linear penalty function F (τ (t )), where τ (t ) is the actual arrival time for vehicles that entering the bottleneck link at time t.

 α ( R − τ (t ) ) if τ (t ) ≤ R F (τ (t ))  γ ( τ (t ) − R ) if τ (t ) ≥ R

(7)

where α , γ are non-negative parameters. Generally, 0 < α < 1 < γ since late arrival usually brings more penalty. 2.5. Parking charges and ridesharing payments A dynamic parking charge k(τ (t )) is imposed to each vehicle at the destination at arrival time τ (t ). Since the arrival time τ (t ) is a one-to-one mapping from departure time t under the first-in-first-out (FIFO) condition (which always holds for single-destination Vickrey’s bottleneck model, see Ban et al., 2012), for simpler notations, we note the parking charge as k(t) in the rest of this paper. The ridesharing payment paid by the each passenger to the driver entering the link at time t is defined as p(t). While in Liu and Li (2017a) and Xiao et al. (2016) and most literature on morning commute problem, ridesharing payment and the equilibrium of the role choices (driver or passenger) are largely neglected, such payment is critical for the equilibrium of ridesharing drivers and passengers in this study. Note that the ridesharing payment and the parking charges are two distinct terms in this study. The parking charge is charged by the parking facilities for all ridesharing travelers, while the ridesharing payment is paid to the driver by the passengers in the same vehicle. The ridesharing payment can compensate the operating cost by drivers, including pick-up trips and the travel cost difference (modeled by the parameter β in Section 2.2) on the bottleneck link. The parking charges are imposed to each vehicle. In this study, we assume the parking charges are shared evenly among the driver and passengers. We note that in real-world applications, the parking charges do not necessarily follow such assumptions. In this case, other options of splitting the parking costs also work within our ridesharing equilibrium framework, if the generalized costs of a driver and a passenger in the same vehicle are equal. In fact, no matter who pays the parking charges initially, one can find that the parking charges are eventually transferred and paid by all the travelers at the equilibrium, as a portion of the ridesharing payment. 2.6. A general formulation of dynamic user equilibrium With the costs of pickup trips, bottleneck link traveling, arrival penalty, the parking charge and the ridesharing payment (costs for passengers and gains for drivers), the travel cost for a ridesharing driver entering the bottleneck link at time t is

c2 (t ) = (1 + s(t ))h p + τ (t ) + F (τ (t )) +

k(t ) − s(t ) p(t ). 1 + s(t )

(8)

The travel cost for a ridesharing passenger at the same time is

c3 (t ) = 0.5(1 + s(t ))h p + βτ (t ) + F (τ (t )) +

k(t ) + p(t ). 1 + s(t )

(9)

Under an user equilibrium, when the demand is positive, the traveling cost for travelers entering the bottleneck link at different times should equal to each other, regardless whether the travelers are drivers or passengers, which is interpreted as

d2 (t )

>0

t ∈ (t1 , t3 )

d2 (t )

=0

t ∈ (−∞, t1 ) ∪ [t3 , +∞ )

(10)

For t ∈ (t1 , t3 ),

c2 (t ) = c3 (t )

⇒ (1 − β )τ (t ) = (1 + s(t ))( p(t ) − h ), and

c˙ 3 (t ) = hs˙ (t ) + β τ˙ (t ) +

 

∂ F (t + τ (t )) k(t ) · (1 + τ˙ (t )) + + p˙ (t ) = 0. ∂τ 1 + s(t )

(11)

(12)

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7

Table 1 Schemes defined by exogenous terms. Scheme

p(t)

k(t)

Benchmark, Case 1 Benchmark, Case 2 Scheme 1 Scheme 2 Scheme 3

N/A N/A p(t ) = p0 p(t ) = p0

k(t ) = 0 k(t ) = k0 k(t ) = k0

τ (t)

s(t)

τ (t ) = τ 0

s(t ) = 0 s(t ) = 0

τ (t ) = τ 0 τ (t ) = τ 0

Here we define t2 so that t2 + τ (t2 ) = R. It is obvious that t1 + τ (t1 ) < t2 + τ (t2 ) < t3 + τ (t3 ). From the FIFO condition, we τ (t )) have t1 < t2 < t3 . According to the piecewise linear arrival penalty function (7), for t1 < t < t2 , ∂ F (t+ = −α ; for t2 < t < t3 , ∂τ ∂ F (t+τ (t )) = ∂τ

γ.

The system defined by (1)–(12) formulates the general morning commute problem with ridesharing under the dynamic user equilibrium of mode and departure-time choices. It is noted that when the dynamic parking charges and the ridesharing payments are both dynamic without further constraints, the system is under-determined and it leads to multiple solutions. To narrow down the solution sets with practical consideration, added constraints are proposed in the next section for schemes with different ridesharing payments and parking charges. 3. Ridersharing and parking charge schemes with realistic constraints The general formulation in Section 2.6 suggests that there are at least four independent terms of time-varying variables (or functional), namely, ridesharing payments p(t), parking charges k(t), travel time τ (t) and ridesharing ratios s(t). Vehicular demand rates d2 (t) are solely dependent to travel times τ (t), and passengers’ demand rates d3 (t) can be derived from ridesharing ratios and vehicular demand rates. Therefore, both demand rates are dependent on parts of the above-mentioned four terms. There are only two sets of equilibrium constraints (8) and (9). Each of the other constraints, such as demand conservation, FIFO and boundary conditions, is served for a single term. Apparently, the general formulation is an underdetermined problem, if less than two terms are exogenously given. To have a unique equilibrium solution (if it exists), one needs to have at least two exogenous terms out of these four terms. Without given at least two terms exogenously, it would be very difficult, if not impossible, to solve all feasible sets of these four terms for equilibrium states, as the number of feasible solutions could be infinite for such under-determined problems. In this Section, we aim to derive two unknown variables from the general ridesharing formulation endogenously, with the other two exogenous terms. Different choices of exogenous terms would lead to different schemes. Table 1 shows five schemes in total, with their exogenous terms given as constants. Note that for the benchmark cases, since there are no ridesharing passengers, the constraints are reduced to only one set and p(t) is eliminated, so that one still needs two exogenous terms to solve one unknown. The above schemes are defined with physical merits. The benchmark scheme (including two cases) is the traditional morning commute problem without ridesharing, i.e., with only solo drivers. The other three represent the following schemes, respectively. Scheme 1 Ridesharing with constant parking charges and ridesharing payment; Scheme 2 Ridesharing with dynamic parking charges and constant ridesharing payment; Scheme 3 Ridesharing with constant parking charge and dynamic ridesharing payments. In each scheme, we will derive the endogenous terms accordingly in this Section. We recall the following expressions as the critical parts of the solutions. Variable

Definition

t1 t2 t3

The earliest departure time The critical departure time; t2 + τ (t2 ) = R The latest departure time Travel time on the corridor with a single bottleneck; a function of departure time t Time-varying parking charge that eliminates the queue at the bottleneck (if applicable) Time-varying ridesharing payment from a passenger to the corresponding driver Time-varying ridesharing ratio

τ (t ) k(t ) p(t ) s(t )

We will show that Scheme 1 will lead to a queued bottleneck, and in Schemes 2 & 3, proper dynamic parking charges or ridesharing payments can be applied, so that the traffic flow is optimized to yield the free-flow that enhances the system performance. 3.1. Benchmark scheme: no ridesharing The benchmark scheme with no ridesharing, i.e., all travelers are solo drivers, have been studied in the classical morning commute prolems; detailed discussion can be found in Vickrey (1969), Arnott et al. (1990) and Qian et al. (2012). We briefly review this scheme under two different parking charges. Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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Fig. 1. The dynamic parking charges for solo drivers for free-flow traffic.

Case 1: no parking charges This is the no-toll scheme in the traditional single bottleneck problem. The earliest, critical and latest departure times are derived as

⎧ γ ⎪ t1 = R − τ 0 − ⎪ ⎪ α + γ ⎪ ⎪ ⎨ αγ t2 = R − τ 0 − α+γ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩t3 = R − τ 0 + α α+γ

D C D C D C

, , .

The travel time is piecewise linear.

⎧ α αγ ⎪ (t − t2 ) + τ 0 + ⎨ 1−α α+γ τ (t ) = ⎪ ⎩ −γ (t − t2 ) + τ 0 + αγ 1+γ α+γ

D C D C

,

t ∈ (t1 , t2 ); (13)

, t ∈ (t2 , t3 ).

The demand rate (departure rate) is piecewise constant.

⎧ 1 ⎪ ⎨ C , t ∈ (t1 , t2 ); 1−α d (t ) = 1 ⎪ ⎩ C , t ∈ (t2 , t3 ). 1+γ

(14)

Case 2: dynamic parking charges for free-flow traffic This is the tolling scheme in the traditional single bottleneck problem. The starting and ending times of the positive demand is the same as the case where no parking charges are imposed. The travel time in this case is constantly equal to the free-flow travel time τ 0 .

⎧ γ ⎪ ⎪ t1 = R − τ 0 − ⎪ ⎪ α+γ ⎨ t2 = R − τ 0 ⎪ ⎪ ⎪ α ⎪ ⎩t3 = R − τ 0 + α+γ

D C (15) D C

According to the derivation, the first and last drivers do not pay parking charges, which is k(t1 ) = k(t3 ) = 0, the dynamic parking charges are derived as a piecewise linear function with respect to the departure time t, and is illustrated in Fig. 1.

⎧ αγ D α (t − R + τ 0 ) + if t ∈ (t1 , t2 ), ⎪ ⎪ α+γ C ⎨ k(t ) = −γ (t − R + τ 0 ) + αγ D if t ∈ (t , t ), 2 3 ⎪ α+γ C ⎪ ⎩ 0

(16)

otherwise.

3.2. Scheme 1: ridesharing with constant parking charges and ridesharing payment In this scheme, we fix the parking charges as a constant k0 . The ridesharing payment is fixed as a constant of p0 . Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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9

Scheme 1 is similar to the equilibrium case in the benchmark scheme, where the queuing delays (and the travel time) can be derived with a given parking charge (which is zero in Case 1 of the benchmark scheme); On the other hand, with dynamic ridesharing ratios, the queuing delays (i.e., travel time on the bottleneck link) and the resulting demand rates in Scheme 1 have significant differences from that in the benchmark scheme without ridesharing. We show below that the demand rates and queuing delays in Scheme 1 are genuinely (piecewise) nonlinear, which is quite different from the (piecewise) constant demand rates and linear queuing delays in the benchmark scheme. As defined in the general case in Section 2, the first vehicle departs at time t1 , the last vehicle departs at time t3 , and the peak time (vehicles departs at this time would have zero arrival penalty) is t2 . k0 Drivers’ cost: c2 (t ) = (1 + s(t ))h p + τ (t ) + F (t + τ (t )) + − s(t ) p0 . 1 + s(t ) k0 Passengers’ cost: c3 (t ) = 0.5(1 + s(t ))h p + βτ (t ) + F (t + τ (t )) + + p0 . 1 + s(t ) Under equilibrium, drivers and passengers share the same cost, for t ∈ (t1 , t3 ), c2 (t ) = c3 (t )

⇒ (1 + s(t ))( p0 − h ) = (1 − β )τ (t ).

(17)

Passengers that depart at different time share the same cost in the equilibrium, c˙ 3 (t ) = 0.

⇒

h (1 − β ) τ˙ (t ) + β τ˙ (t ) + p0 − h

 1 ∂F p −h (1 + τ˙ (t )) + k0 0 − 2 τ˙ (t ) = 0. ∂τ 1−β τ (t )

(18)

For t ∈ (t1 , t2 ), ∂∂ τF = −α , so that (18) gives



h (1 − β ) p0 − h 1 + β − α − k0 p0 − h 1 − β τ 2 (t )

 τ˙ (t ) − α = 0

(19)

p −h β) We here define A  h(p1−−h + β − α , B  k0 0 . Then the differential equation (19) for t ∈ (t1 , t2 ) can be solved by the 0 1−β following integration.

  A − τB2 dτ = α dt + C1 , ⇒ Aτ + τB = αt + C1 , 



(20)

where C1 is a constant. Since τ (t) ≥ τ 0 > 0, we then have an explicit solution of τ (t) by solving the quadratic term in (20). For t ∈ (t1 , t2 ),

τ (t ) =

(αt + C1 ) ±



(αt + C1 )2 − 4AB 2A

, t ∈ (t1 , t2 ),

(21)

with the boundary conditions τ (t1 ) = τ 0 > 0 and τ (t2 ) = τ m > τ 0 . By definition of t2 , τ m is the maximum travel time on the bottleneck. Since for t ∈ (t1 , t2 ), the queue on the bottleneck is building up, which means that τ (t) is an increasing function for t ∈ (t1 , t2 ), so that in (21) the ± sign should be ‘ + ’. In fact, τ (t) is an increasing concave function during time t ∈ (t1 , t2 ). Similarly, for t ∈ (t2 , t3 ), ∂∂ τF = γ , so that (18) gives



h (1 − β ) p0 − h 1 + β + γ − k0 p0 − h 1 − β τ 2 (t )

 τ˙ (t ) + γ = 0

(22)

β) We further define  = h(p1−−h + β + γ . Then the differential equation (22) for t ∈ (t2 , t3 ) can be solved by integration.







0



 − τ 2 dτ = −γ dt + C2 , ⇒  τ + τB = −γ t + C2 , B

(23)

where C2 is a constant. Given that the queue on the bottleneck is discharging during time t ∈ (t2 , t3 ), and the boundary conditions τ (t3 ) = τ 0 > 0 and τ (t2 ) = τ m > τ 0 , we have the explicit expression of τ (t) for t ∈ (t2 , t3 ) as

τ (t ) =

(−γ t + C2 ) +



(−γ t + C2 )2 − 4 B , t ∈ (t2 , t3 ), 2

(24)

which is a decreasing convex function during time t ∈ (t2 , t3 ). We notice that if and only if B = 0, the travel time function τ (t) can be a piecewise linear function. This is equivalent to k0 = 0 or p0 = h. However, according to (17), p0 − h = 0, otherwise τ (t ) = 0 for all the time, which contradicts the queuing dynamics. So it is clear that the if and only if the parking charge is fixed as zero, i.e., k0 = 0, the travel time function τ (t) is a piecewise linear function. The nonlinearity is introduced by the parking cost sharing among the driver and the passengers. This distinguishes our model from the traditional single bottleneck models without ridesharing, where a constant toll, no Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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matter it is zero or a positive value, would only lead to a piecewise linear travel time functions, given a piecewise linear arrival penalty function. We assume the inflow rate d2 (t) is well-defined so that the travel time function τ (t) is continuous everywhere. So at time t2 , the above implicit expressions of travel time (20) and (23) should both hold for t = t2 , i.e.,

γ t2 + C2 = αt2 + C1 = Aτ m +

B

τm

,

(25)

where t2 + τ m = R. The boundary condition τ (t1 ) = τ 0 leads to

αt1 + C1 = Aτ 0 +

B

τ0

.

(26)

The boundary condition τ (t3 ) = τ 0 leads to

−γ t3 + C2 =  τ 0 +

B

τ0

.

(27)

We can then express t1 , t2 and t3 , respectively, by the known parameters and C1 from (25), (26) and (27) as

⎧  0 B  1 ⎪ ⎨t1 = α Aτ + τ 0 − C1 , t2 = R − τ m , ⎪ ⎩t = 1  τ 0 + B − C ,  −γ

3

τ0

where

τm =

(C1 +α R )+

√

(28)

2

(C1 +α R ) −4B(A+α ) , 2 ( A+α ) 2

C2 =  τ m + τBm + γ t2 .

(29)

During the arrival time window t ∈ (t1 + τ 0 , t3 + τ 0 ), the exit flow rate of the vehicular flow is at flow capacity, i.e., for t ∈ (t1 , t3 ), v(t + τ 0 ) = C due to the positive queue lengths. According to (2), we can derive the vehicular inflow rate for t ∈ (t1 , t3 ) as



d2 (t ) = C + C (τ (t ) − τ 0 )



= C (1 + τ˙ (t ) )

(30)

According to the travel time functions (21) and (24), we know that the travel time functions on both sides of early and late arrivals are nonlinear functions. Further, if B > 0, the derivative of the travel time function is also a nonlinear function on either side. In this scheme with constant ridesharing payments and constant parking charges, (30) shows that the vehicular demand rate, i.e., the demand rate of the drivers, is a piecewise nonlinear function, if the parameter B is positive. This is quite different from the traditional single bottleneck morning commute model without ridesharing, as discussed in the benchmark scheme. Whether there is a parking charge, the vehicular demand rates are always constant, or piecewise constants in the latter model. The nonlinearity observed in this scheme is in fact introduced by the cost sharing for the parking charges. The shared parking charge is a term with ridesharing ratio as its denominator, so that (18) leads to a nonlinear differential equation for travel time τ (t), rather than a linear one as in (13) in the benchmark scheme. We substitute the above demand rate (30) and the ridesharing ratio (17) into the demand conservation (5), then we have

D=

1−β C p0 − h



t3

t1

τ (t )(1 + τ˙ (t ) )dt .

(31)

By substituting (28) into (31), we get an equation with only one unknown variable C1 as F (C1 ) = 0. Ideally, if such an equation can be analytically solved to have an explicit expression of C1 , other critical variables can be shown explicitly by substituting C1 into the above formulations. However, since we have a nonlinear function τ (t), which makes it difficult to obtain the explicit analytical expression of C1 in general cases. When the parameters are specifically given, one can solve for C1 using Newton’s method or any iterative/numerical methods. We will show a numerical case in Section 5, where C1 , along with other variables, is obtained numerically. Feasibility The feasibility condition of Scheme 1 is determined by the parameters of p0 − h, β , τ 0 , as well as C1 for (31). In order to have the ridesharing ratio bounded by zero and n, i.e.,

1 ≤ 1 + s(t ) =

1−β τ (t ) ≤ n + 1 p0 − h

Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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11

The following two conditions must hold simultaneously.

 1 −β

p0 −h 1 −β p0 −h

τ m ≤ n + 1, τ0

(32)

≥ 1,

or

( 1 − β )τ m 1+n

≤ p0 − h ≤ ( 1 − β )τ 0 ,

which implies that the difference between the constant ridesharing payment and the pick-up cost for one passenger should be in a properly bounded region. If the ridesharing payment was too high, then the ridesharing ratio would fall below zero; if it was too low, then the ridesharing ratio would be too high to exceed the maximum passenger capacity. Since

τm =

(C1 +α R )+ (C1 +α R )2 −4B(A+α ) , we need to have one more condition to ensure 2 ( A+α )

τ m ∈ IR, which is (C1 + α R ) − 4B(A + α ) ≥ 0. For the special case where the parking charge is a constant of zero, explicit analytical solution in fact can be derived. Since the travel time function is piecewise linear in this specific case, the vehicular demand rate function is made up by piecewise constants. Detailed explicit solution of this case can be found in Appendix A. 2

Discussion on strict integer ridesharing ratio In Scheme 1, according to (17), if the ridesharing ratio is enforced to be integer values, then τ (t) is also limited as integer values, so that τ˙ (t ) (the derivative of τ (t)) is zero at almost everywhere if it exists. This leads to α = 0 from (19), which contradicts α > 0. This proves that if s(t) is limited as integer values, no feasible solution can be found in Scheme 1. Extension to schemes with step-wise parking charges The parking charge in Scheme 1 is an exogenously given constant. In practice, step-wise parking charges are usually applied, i.e., parking charges are varied over time periods, but keep a constant value within each time period. An example of the step-wise charges is the coarse toll discussed in Xiao et al. (2011). Schemes with such types of parking charges, if also exogenously given, can be treated as a variation of the above-mentioned Scheme 1. More specifically, in the step-wise parking charges schemes, one needs to replace p0 by a time period specific value pi for period i in (17). For each time period, a solution process similar to the above-mentioned one is expected. There will be some necessary variations for step-wise parking charges, though. In Scheme 1, there are only three boundary conditions for the travel time continuity at time instants t1 , t2 and t3 . The more the steps in the parking charges, the more boundary conditions are expected for step-wise parking charges. In fact, if there are n steps in the parking charges for each of early and late arrival times, then there will be 2n + 1 boundary conditions for travel time continuity. Within each time period with a constant parking charge, the travel time function follows a quadratic term that is similar to (21) and (24). Therefore, overall the travel time function for the entire time would be a piecewise genuinely nonlinear function as well. 3.3. Scheme 2: ridesharing with dynamic parking charges and constant ridesharing payment In this scheme, the fixed terms are the travel time τ (t) and ridesharing payment p(t), and the unknown terms are dynamic parking charges k(t) and ridesharing ratios s(t). As we show in the following, this scheme is in fact similar to Case 2 in the benchmark scheme, and the ridesharing ratio is a constant for a given ridesharing payment. At the free-flow state, τ (t ) = τ 0 , and the vehicular flow demand should be a constant during the time period t ∈ (t1 , t3 ), i.e., d2 (t ) = C . The costs of drivers and passengers, respectively, are simplified due to the free-flow condition as follows. k(t ) −s t 1+s(t ) k(t ) + 1+s(t ) + p t

c2 (t ) = 2h(1 + s(t )) + τ 0 + F (t + τ 0 ) + c3 (t ) = h(1 + s(t )) + βτ + F (t + τ ) 0

0

( ) p(t ) ()

(33)

For t ∈ (t1 , t3 ), (11) is simplified as

c2 (t ) = c3 (t ) ⇒ (1 − β )τ 0 = (1 + s(t ))( p(t ) − h ).

(34)

(12) is reduced to the following equation,

c˙ 3 (t ) = hs˙ (t ) +

 

∂F k(t ) + + p˙ (t ) = 0. ∂ τ (t ) 1 + s(t )

(35)

In Scheme 2, the ridesharing payment is fixed as a constant p0 . Substitute p(t ) = p0 into (34), it is derived that the ridesharing ratio is a constant as well,

s(t ) =

( 1 − β )τ 0 p0 − h

− 1  s0 .

(36)

Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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Given the constant ridesharing payment and ratio, we can derive the dynamic parking charges from (35) as

∂ F (t + τ 0 ) (1 − β )τ 0 k˙ (t ) = − , ∂τ p0 − h

(37)

where ∂∂ τF (t + τ 0 ) = −α for t ∈ (t1 , t2 ), and ∂∂ τF (t + τ 0 ) = γ for t ∈ (t2 , t3 ). t2 = R − τ 0 . Note that similar to the free-flow case with solo drivers in the benchmark scheme, the dynamic parking charge is a piecewise linear function. However, the slopes in two schemes are different. With ridesharing, the slopes of the parking charges are steeper than that of the benchmark scheme. Further, the starting and ending times of the demand are also different from the benchmark scheme, since the ridesharing decreases the overall vehicular demand. The vehicular demand rate equals to the flow capacity of the bottleneck during the departure time window, i.e., d2 (t ) = C , t ∈ (t1 , t3 ). Since the ridesharing ratio is constantly 1 + s0 , the demand rate of all travelers is d (t ) = d2 (t )(1 + s0 ), t ∈ (t1 , t3 ). The demand conservation for this scheme is

D = C (1 + s0 )(t3 − t1 ).

(38)

Given the boundary conditions k(t1 ) = k(t3 ) = 0, the slopes of the piecewise linear parking charges in (37) and the demand conservation (38), we can calculate the starting and ending times of the demand, respectively, as

⎧ p0 − h γ ⎪ ⎪ t1 = R − τ 0 − ⎪ ⎪ ( 1 − β )τ 0 α + γ ⎨ t2 = R − τ 0 , ⎪ ⎪ ⎪ p0 − h α ⎪ ⎩t3 = R − τ 0 + ( 1 − β )τ 0 α + γ

D C

, (39)

D C

.

Similar as in (16), we have the dynamic parking charges for the free-flow traffic, given a constant ridesharing payment p0 , as

⎧ ( 1 − β )τ 0 αγ ⎪ ⎪ + α (t − R + τ 0 ) + ⎪ ⎪ p0 − h α+γ ⎨ k(t ) = ( 1 − β )τ 0 αγ − γ (t − R + τ 0 ) + ⎪ ⎪ p − h α +γ ⎪ 0 ⎪ ⎩ 0,

D C D C

,

if t ∈ (t1 , t2 ),

, if t ∈ (t2 , t3 ),

(40)

otherwise.

Feasibility of Scheme 2 is determined by parameters τ 0 , p0 − h and β . The ridesharing ratio is constrained as follows,

1≤

( 1 − β )τ 0 p0 − h

≤ n + 1,

(41)

which is equivalent to

( 1 − β )τ 0 n+1

≤ p0 − h ≤ ( 1 − β )τ 0 .

(42)

Such feasibility condition suggests that given the other parameters, the constant ridesharing payment should be properly selected in order to keep the ridesharing ratio within reasonable bounds. Discussion on strict integer ridesharing ratio In Scheme 2, according to (34) and (36), p(t ) = p0 is limited as certain discrete values, if s(t ) = s0 is limited as integer values. In fact, the solution set of feasible p0 is a discrete subset of that without integer limitation of s(t ) = s0 , which are trivial cases. 3.4. Scheme 3: Ridesharing with constant parking charge and dynamic ridesharing payments Scheme 3 is rarely discussed in the traditional bottleneck model studies, as it involves using dynamic ridesharing payments from passengers to drivers to achieve SO, which is fundamentally different from Case 2 of the benchmark scheme with only solo-drivers. In Scheme 3, we show that in order to have SO (free-flowing) traffic for a fixed parking charge satisfying certain conditions, the dynamic ridesharing payments are piecewise linear functions. At first glance, such functions seem to share the similar pattern with the dynamic parking charges in Scheme 2 and Case 2 of the benchmark scheme, but in fact they have quite different implications. We will show in the following analysis and in the numerical examples that the ridesharing ratios are genuinely (piecewise) nonlinear functions in Scheme 3, which suggests that the dynamic ridesharing payments encourage more ridesharing in the early and late times, and discourage ridesharing in the center time. Such patterns of ridesharing ratios lead to nonlinear passengers’ demand, and fixed drivers demand rate as the bottleneck flow capacity. In Scheme 3, the fixed terms are the parking charge k(t) and the travel time τ (t). The constant parking charge is k(t ) = k0 , so that k˙ (t ) = 0. The unknown terms are the ridesharing ratios s(t) and the dynamic ridesharing payments p(t). Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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From (11), we have

( 1 − β )τ 0 p(t ) − h 1 p(t ) − h ⇒ = 1 + s(t ) ( 1 − β )τ 0  1 

p˙ (t ) ⇒ = 1 + s(t ) ( 1 − β )τ 0 1 + s(t ) =

(43)

Substituting k(t ) = k0 and (43) in (35), we have an ODE of p(t) as

h ( 1 − β )τ

0





1 p(t ) − h

  ∂ F (t + τ 0 ) k0 + + 1+ p˙ (t ) = 0. ∂τ ( 1 − β )τ 0

(44)

Under certain conditions, the above nonlinear ODE (44) can be approximated by a linear one. Details of the conditions and the approximation can be found in Appendix B. We then have a linear ODE for p(t), and the slope of the dynamic ridesharing payment is

p˙ (t ) = −

∂ F (t + τ 0 ) 1 ∂τ 1 + (1−kβ0 )τ 0

(45)

for t ∈ (t1 , t3 ). As known, the derivative of piecewise linear arrival penalty function is

∂ F (t + τ 0 ) = −αδt+τ 0 R ∂τ Assume the ridesharing payment is equal to P1 for the first and last passengers, i.e., p(t1 ) = p(t3 ) = P1 , and the peak payment is given as p(t2 ) = PR , where t2 = R − τ 0 . Then we have



p(t ) =

P1 −PR t 1 − ( R −τ 0 ) P2 −PR t 2 − ( R −τ 0 )

· (t − (R − τ 0 )) + PR , t ≤ R − τ 0

(46)

· (t − (R − τ 0 )) + PR , t > R − τ 0

where P1 −PR t 1 − ( R −τ 0 )

=

P2 −PR t 2 − ( R −τ 0 )

=

α 1+

,

k0

(1−β )τ 0

−γ

1+

(47)

.

k0

(1−β )τ 0

Based on demand conservation, we can calculate P1 as



D=

t3

t1

(1 + s(t ))d2 (t )dt = C ((1 − β )τ 0 )



t3

t1

1 dt p(t ) − h

(48)

The integral term can be split into two parts for time periods (t1 , t2 ) and (t2 , t3 ). The constant terms for these two parts from integration could be different, which are defined as C3 and C4 for (t1 , t2 ) and (t2 , t3 ), respectively.



t3

t1

1 dt = p(t ) − h =



1 p (t )

t3  t2 (ln( p(t ) − h ) + C4 ) t + p 1(t ) (ln( p(t ) − h ) + C3 ) t 2

ln(P1 −h )+C4 p (t3 )

−

ln(PR −h )+C4 p (t2+ )

+

1

ln(PR −h )+C3 p (t2− )

−

ln(P1 −h )+C3 p (t1 )

(49)

In order to make the ridesharing payment p(t) a bounded and continuous function, the following equation should hold

ln(PR − h ) + C4 ln(PR − h ) + C3 = . p (t2+ ) p (t2− ) Then we have

(α + γ )ln(PR − h ) = −αC4 − γ C3 Without losing generality, we set C3 = 0, so that

C4 =

(α + γ )ln(PR − h ) . −α

Substitute the above equations into (48), we have the following equation of P1 , PR and k0 .

ln(PR − h ) − ln(P1 − h ) =



D

C ( 1 − β )τ + k0 0

 1

1 α + γ

.

Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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The ridesharing payment at the beginning time t1 and the ending time t3 can be expressed by PR and k0 .





P1 = h + exp



−D

C ( 1 − β )τ 0 + k0

 1

1 α + γ

 + ln(PR − h )

We have the ridesharing payment as a continuous function for t ∈ (t1 , t3 ).

p(t ) =

where

αδt+τ 0 R k0 1+ ( 1 − β )τ 0

· (t − (R − τ 0 )) + PR , for t ∈ (t1 , t3 )

⎧ (P1 − PR )(1 + ⎪ ⎪ ⎪ t1 = R − τ 0 + ⎪ ⎨ α t2 = R − τ 0 , ⎪ ⎪ ⎪ (P − P )(1 + ⎪ ⎩t3 = R − τ 0 + 1 R −γ

( 1 −β ) τ 0 )

(50)

k0

, (51)

k0 ( 1 −β ) τ 0

)

.

It shows that the ridesharing payments should be a piecewise linear function with respect to time t, in order to achieve free-flow traffic for a given constant parking charge. From (43) we can further observe that the ridesharing ratio s(t) is nonlinear, with a decreasing nonlinear part for the early arrivals and an increasing nonlinear part for the late arrivals. This leads to a nonlinear demand rate function for all travelers. Still, the vehicular flow rate is a constant. This indicates that at the peak time, when the ridesharing payments are high, there would be fewer ridesharing passengers in the system. Notice that the nonlinearity in this scheme is different from that of Scheme 1. As shown in (43), the multiplication of the demand rate function and the difference between the linear dynamic ridesharing payments and the pick-up cost for each passenger is a constant, thanks to the constant free-flow travel time. The nonlinearity is introduced by such relation, not by the cost sharing for the parking charges. Even the parking charge is set to zero, the demand rate function in this scheme is still a piecewise nonlinear function. Feasibility The feasibility conditions of Scheme 3 is determined by n, (1 − β )τ 0 , PR − h, D, k0 , α1 + γ1 . First, the ridesharing ratio at all time must be properly bounded.

0 ≤ s(t ) =

( 1 − β )τ 0 − 1 ≤ n, p(t ) − h

which translates to two inequalities )τ 0 i) (1P−β−h ≤ n + 1; 1

Since P1 can be expressed by PR and k0 , such condition in fact regulates the selection of k0 and PR . For a given k0 , the maximum ridesharing payment PR cannot be too low, otherwise there are more than n passengers to be transported in a single vehicle to achieve the system equilibrium. )τ 0 ii) s(R − τ 0 ) = (1P−β−h − 1 ≥ 0; R

This leads to PR ≤ (1 − β )τ 0 + h, which means the peak payment for a passenger cannot be too high, otherwise, at the peak time, nobody would afford to be a passenger. Second, the critical times should be properly sequenced t1 < t2 < t3 ; This is equivalent to P1 > PR . i.e., ln(P1 − h ) > ln(PR − h ) ⇔ k0 < (1 − β )τ 0 . This constraints the selection of k0 . Last, to satisfy the linearization constraint, one needs  to have the minimum ridesharing payment to be lower-bounded by the constraint listed in Appendix B. PR > h + (1 − β )τ 0

h

(1−β )τ 0 +k0

.

Discussion on strict integer ridesharing ratio In Scheme 3, according to (43), if the ridesharing ratio is enforced to be integer values, then p˙ (t ) (the derivative of p(t)) is zero at almost everywhere if it exists, which contradicts (45), where p˙ (t ) = 0. This proves that if s(t) is limited as integer values, no feasible solution can be found in Scheme 3. Remarks on ridesharing payments It is noted that the term p(t ) − h appears frequently in most schemes. It is the difference between the ridesharing payment and the half pick-up trip cost per passenger for the drivers, and can be interpreted as that the ridesharing payment from a passenger to a driver can be decomposed into two parts. The first part is called the ‘base fare’ for the fixed cost for a half of the pick-up trip, which suggests that half of the pick-up cost for each passenger is transferred from the driver to each passenger in term of partial ridesharing payment. In other words, the base fare of the ridesharing payment is always h. The second part is called the ‘variable fare’ representing the shared operational cost, i.e., p(t ) − h. Since the increase of the Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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trip travel times on the bottleneck link (congestion during the commute trip) does not affect the amount of the pick-up cost per passenger, the second part of ridesharing payment may vary in different schemes or different selections of parameters, even the based fare is kept unchanged h. 4. System performance under different ridersharing and parking charge schemes As shown in Section 3, different ridesharing and parking charge schemes leads to different morning commute patterns and hence potentially different system performance. In the following two sections, we are aiming at answering the following questions: (1) can we use parking and ridesharing as viable management tools to enhance system performance? (2) which parking pricing and/or ridesharing schemes are more effective in reducing travelers’ commuting cost and environmental footprint? (3) does there exist any universal guideline to regulate the parking charges and ridesharing payments for bottleneck corridors with different characteristics? and (4) to what extent do the model parameters affect the policy decision making and the resulting system performance? In this section, we will analyze the system performance of those schemes introduced in Section 3. Specifically, we derive the analytical expressions of three system performance indicators, the vehicle miles traveled (VMT), vehicle hours traveled (VHT) and the total cost (TC) for each of the schemes. Such analytical expressions can help answer the questions above in analytical ways. We note that it is hard to compare the explicit analytical results directly in general due to two reasons. First, some solutions are nonlinear so that we may not have explicit expressions, and thus they are solved numerically; Second, the analytical solutions are determined by multiple parameters (such as k0 , p0 , τ 0 , hp , etc.), direct comparison in an analytical way is not straightforward (if not impossible) with multiple dimensions of parameters. Therefore, instead of comparing the explicit analytical results in general, the analytical system performance indicators are presented as functions of critical variables (such as t1 , t2 , t3 , etc., which can be solved either analytically or numerically depending on schemes). We resort to numerical solutions and examples in Section 5 if no explicit expression of the system performance indicators is available. In order to investigate the relation among the parameters of the system, the strategies of parking charges and ridesharing payments and the system performances more clearly, we implement the analysis to numerical examples in Section 5. We first define the three system performance indicators. MV MT MV HT TC

= pick-up miles traveled by the drivers + corridor miles traveled by the drivers; = pick-up times by the drivers + corridor travel times by the drivers; = individual cost (IC) ∗ number of travelers (under UE).

We then calculate the VMT, VHT and TC for each scheme. The objective is to express these indicators as functions of all parameters, including the parking charges and ridesharing payments if any of them are fixed as constants. It is possible that due to the nonlinearity of the solutions, some of the indicators can only be expressed as functions of variables that are not explicitly expressed as functions of the basic parameters. However, as shown in Section 3, these variables can always be obtained, as long as a specific set of parameter values is given. Therefore the derivations and calculations in this section is critical for the analysis and comparison of the various ridesharing and parking schemes presented earlier. 4.1. The benchmark scheme without ridesharing The benchmark scheme with or without parking is classical and the resulting three performance indicators are simply provided below without further explanation. The case without parking charges:

 



MV MT = h p v p + τ 0 v f D,

 αγ D D, 2 (α + γ ) C   αγ D = hp + τ 0 + D. α+γ C

MV HT = TC

hp + τ 0 +

The case with parking charges:

    = h p + τ 0 D,   αγ D = hp + τ 0 + D. α+γ C

MV MT = h p v p + τ 0 v f D, MV HT TC

We can observe that for both cases, the total travel cost is greater than the VHT, where the difference represents the arrival penalties of all vehicles. For the benchmark scheme, both cases (with or without parking charges) lead to the same VMT, since whether or not the drivers are charged for parking, all drivers have to travel the same distance to reach the destination, which includes the local trips and the corridor trip on the highway. Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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However, the VHT in Case 1 without parking charges is higher than that in Case 2 with optimal parking charges. This shows the optimal parking charges eliminate the deadlock loss in Case 1 and reduce the system travel cost. Total cost of drivers in both cases are the same. These are well known results from the classical morning commute model with or without optimal parking charges. 4.2. Scheme 1 performance indicators In this case, the parking charge is fixed as k0 and shared among the driver and the passengers. The ridesharing payment is fixed as p0 . At UE, the traffic is not in the free-flowing state, and a queue is built up at the bottleneck.

 t3





d2 (t ) (1 + s(t ))h p v p + τ 0 v f dt  t   3 = h pv p (1 + s(t ))d2 (t )dt + τ 0 v f

MV MT =

t1

t1

= h pv pD + τ 0 v f C

t3

t1



t3

t1

d2 (t )dt

1 + τ˙ (t )dt

= h p v p D + τ 0 v f C ( (t3 − t1 ) + τ (t3 ) − τ (t1 ) ) = h p v p D + τ 0 v f C (t3 − t1 ) Due to the ridesharing, the departure time window (t1 , t3 ) is no more than that of the benchmark scheme, which means t3 − t1 ≤ D . Thus the VMT in Scheme 1 is no more than that in the benchmark scheme. C

 t3

d2 (t )[(1 + s(t ))h p + τ (t )]dt  t3  p0 − h t3 = hp d2 (t )[(1 + s(t ))]dt + d2 (t )[(1 + s(t ))]dt 1 − β t1 t1

MV HT =

t1



= 2h +



p0 − h D 1−β p0 −h 1−β

From the feasibility condition, we can have

p0 −h 1−β

≤ τ 0 , then

+ hp ≤ hp + τ 0 < hp + τ 0 +

D αγ 2C α +γ

, which means that

the VHT in Scheme 1 with constant ridesharing payments and parking charges is strictly less than that in both cases of the benchmark scheme. The individual cost for drivers departing at any time is equal under UE. The total cost is calculated via cost of passengers entering the bottleneck link at time t2 .

T C = IC · D = c3 (t2 ) · D



= h(1 + s(t2 )) + βτ



=

m

k0 + p0 + 1 + s(t2 )

h ( 1 − β )τ m + βτ m + p0 + p0 − h

 ·D



k0 1 −β p0 −h

· τm

·D

For the special case of this scheme, where there is no parking charges for the travelers, we can calculate the explicit analytical expression of the VMT, VHT and the total cost. Details can be found in Appendix A. 4.3. Scheme 2 performance indicators In this case, the ridesharing payment is fixed as p(t ) = p0 , and the dynamic parking charges k(t) is imposed to the travelers. )τ 0 D For a time period length of , the number of passengers s(t) is constantly equal to s0 = (1p−β−h − 1, and the drivers (1+s0 )C

0

demand rate d2 (t) is constantly equal to C .

MV MT = (h p v p (1 + s0 ) + τ 0 v f ) · C ·

(1 + s0 )C

From the feasibility condition, τ 0 = (1 + s0 )

MV HT = (h p (1 + s0 ) + τ 0 ) · C ·



D

D

(1 + s0 )C

p0 −h 1−β



= h pv p + ≥

p0 −h . 1−β

= hp +

p0 − h ·v (1 − β ) f

 ·D

So the VMT is no more than that in the benchmark scheme.

p0 − h (1 − β )

 ·D

Similiarly, the VHT is no more than that in the benchmark scheme. Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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The individual cost for travelers departing at any time is equal under the UE. The individual cost of the first driver is γ D c2 (t1 ). As calculated, t1 = R − τ 0 − α +γ · . (1+s0 )C

T C = IC · D = c3 (t1 ) · D



= 0.5(1 + s0 )h p + βτ 0 + p0 + ααγ · +γ =



h ( 1 −β ) τ p0 −h

0

+ βτ 0 + p0 + ααγ · +γ

D p0 −h C ( 1 −β ) τ 0

D p0 −h C ( 1 −β ) τ 0





·D

·D

Clearly, TC is a nonlinear function of h. By fixing other parameters, we can calculate the value of h minimizing TC by letting the gradient with respect to h as zero.

∂ TC | = ∂ h h=hopt

p 0 ( 1 −β ) τ 0 ( p0 −h )2

− ααγ +γ

D 1 C ( 1 −β ) τ 0

⇒ hopt = p0 − (1 − β )τ 0



=0 C D



p0 α1 + γ1



4.4. Scheme 3 performance indicators In this case, the ridesharing payment p(t) is time-varying, while the parking charges are fixed as k(t ) = k0 . The vehicular traffic is free-flowing. For the time period (t1 , t3 ), the drivers demand rate d2 (t) is constantly equal to C . Then we have

MV MT =

 t3  t1



h p v p (1 + s(t )) + τ 0 v f · C dt

= h p v p D + τ 0 v f C (t3 − t1 ) MV HT = h p D + τ 0C (t3 − t1 ) Similar to Scheme 1, due to ridesharing, the departure time window (t1 , t3 ) is no more than that of the benchmark scheme, which means t3 − t1 ≤ D . Thus the VMT in Scheme 3 is no more than that in the benchmark scheme. C The individual cost for travelers departing at any time is equal under the UE. The individual cost of a passenger who encounters no arrival penalty is c3 (t2 ), which equals the individual cost of any traveler.



T C = IC · D = c3 (t2 ) · D =



h ( 1 − β )τ 0 k0 + βτ 0 + · (PR − h ) + PR D PR − h ( 1 − β )τ 0

As TC is a nonlinear function of h in this Scheme 3 as well, by fixing other parameters, we can calculate the value of h minimizing TC by letting the gradient with respect to h as zero.

∂ TC | = ∂ h h=hopt

PR (1−β )τ 0 (PR −h )2

− (1−kβ0 )τ 0 = 0

⇒ hopt = PR − (1 − β )τ 0



PR k0

5. Numerical examples Not all of the expressions we have derived earlier for the three performance indicators of different ridesharing and parking schemes can be evaluated explicitly because some of the variables involved do not have explicit solutions. In this section, we present numerical examples to show the system performance and the dynamics of demand flow, parking charges and ridesharing ratios and payments for all schemes with given parameter values. We also conduct sensitivity analysis to show how the system performance indicators change with changes in parameter values. Specifically, in the numerical examples, the parameters of the morning commute model are given as follows. Free-flow travel time is initially set as τ 0 = 10 min, β = 0.75, total demand D = 60 0 0, bottleneck capacity C = 600 veh/min, α = 0.5, γ = 1.5, preferred arrival time R = 25 min, maximum passengers per vehicle n = 4, The travel speed on local roads for pick-up trips is v p = 30 mph, and the free-flow travel speed on the highway with a single bottleneck is v f = 60 mph. All time cost can be converted into monetary cost with the value of time 1 dollar/min. We initially set the pick-up time per passenger h p = 1 min, and thus h = 0.5 min. Note that τ 0 or hp can be chosen freely as long as the other parameters satisfy the feasibility conditions. Here both the value of time and pick-up time parameter values are set in such a way to simplify computations. Since it is the relative, not the absolute commuting costs that we are interested in our analysis, the use of these high/low values does not affect our conclusions. But more realistic values can be used should they be known in a specific corridor. Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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R. Ma, H.M. Zhang / Transportation Research Part B 000 (2017) 1–30 Table 2 System performance for Benchmark Scheme and six cases of Scheme 1. Case No.

k0 ($)

p0 − h ($)

C1

VMT (mile)

VHT (min)

TC ($)

Benchmark, no parking charge Benchmark, parking charges

– –

– –

– –

630 0 0 630 0 0

77250 660 0 0

88500 88500

Scheme Scheme Scheme Scheme Scheme Scheme

1 2 1 2 2.5 60

1 1 2 2 2.5 0.1

−1.9653 −1.5698 −1.0156 −0.2444 1.0079 10.047

25156 25084 43950 43489 51126 N/A

30 0 0 0 30 0 0 0 540 0 0 540 0 0 660 0 0 N/A

69205 71581 80906 85533 96046 N/A

1, 1, 1, 1, 1, 1,

Case Case Case Case Case Case

1 2 3 4 5 6

5.1. Commuting pattern and system performance under Scheme 1 Since we do not have explicit expressions of given parameters for all system performance indicators, these indicators need to calculated case by case. We here select some representative cases with different settings of the constant ridesharing payment and parking charges, as listed in Table 2. From the feasibility condition p0 − h ≤ (1 − β )τ 0 , we know p0 ≤ 0.25 ∗ 10 + 0.5 = 3. For each pair of selected p0 and k0 , the solution C1 to the constraints can be found via a numerical search procedure. 5.1.1. Detailed results and comparisons Fig. 2 shows details of the patterns of travel times τ (t), ridesharing ratios s(t) and demand rates for all travelers d(t) and vehicular demand rate d2 (t) for these six cases. The horizontal axes are time t. In all cases, it is observed that both the travel times and the ridesharing ratios are increasing in the early arrivals (left wings) and decreasing in the late arrivals (right wings). For each wing, the trend is nonlinear. The less the parking charges and the ridesharing payment are, the less significant are the nonlinearity. It is very obvious that in Case 5, the travel time function is an increasing concave function on its left wing, and a decreasing convex function on its right wing. While for Cases 1 to 4, although it is not that obvious, the analytical solutions in Section 3 do show the nonlinearity in general. Such nonlinear pattern of the travel times suggests that the queuing congestion is built up quicker than the linear cases (with only solo drivers). The demand rates are no longer step functions in the linear cases. Among these cases, Fig. 2(e) clearly show that for Case 5, significant high demand rates are observed in the early morning, which quickly build up the queues. The ridesharing ratios share the same pattern with the corresponding travel time functions in all cases. It can be observed that the vehicular demand rates d2 (t) in Cases 1 to 5 share similar patterns. Most of the vehicles depart in the early arrival time period (left wings). As the parking charges and ridesharing payment increase, the nonlinearity of the vehicular demand rates are more prominent and easier to see. For each case, the vehicular demand rate on the left wing is a decreasing function, and the demand rates on the right wing are much lower than that of the left wing, which means that the starting time instant is the most busy time instant for the ridesharing drivers of the entire departure time window. However, such a statement is not true for the demand rates of all travelers, including passengers and drivers. In fact, the piecewise monotonicity observed in d2 (t) does not hold for d (t ) = (1 + s(t ))d2 (t ), since the ridesharing ratio s(t) is not constant, and the monotonicity of s(t) is in fact inverted from the corresponding vehicular demand rate d2 (t), as observed in the figures. Specifically, in Cases 1, 2 and 3, where k0 and p0 − h are not high, the travelers’ demand rates are increasing functions of time, which is opposite from their corresponding vehicular demand rates. In Cases 4 and 5 with higher k0 and p0 − h, the trend inherited from d2 (t) dominates d(t). Other than the first five cases, Case 6 does not have a feasible solution. In Fig. 2(f) we show the resulting s(t) is exceedingly large in the range of 24 to 24.2, compared with the maximum ridesharing ratio of 4. Case 6 suggests that the ridesharing payment cannot be set too low, otherwise there would be too few drivers and too many passengers in the ridesharing system. The genuine nonlinearity brought by the ridesharing travelers were not considered in the traditional ADL models. Such genuinely nonlinear functions in the ridesharing schemes differ from the linear results not only on the shapes (piecewise convex or concave), but also on their critical departure times of the first vehicle, the last vehicle, and the vehicle with no arrival penalty. Here we compare the results of the ridesharing Scheme 1 Case 5 and the traditional ADL model results. Due to the difficulties of directly comparing analytical functions without explicit forms in general, we present the differences among the following three cases with the same total demand. (1) Genuinely nonlinear travel time and demand rates of Scheme 1, Case 5 with time-varying ridesharing ratios; (2) Piecewise linear travel time and piecewise constant demand rate of the non-ridesharing benchmark scheme; (3) Piecewise linear travel time and piecewise constant demand rate with a scale-down total demand per the average ridesharing ratio of Scheme 1, Case 5 (The average ridesharing ratio is s = 0.1977). As presented in Fig. 3, during the early departure times, the travel time of the ridesharing scheme increases nonlinearly and quicker than that of no ridesharing and fixed ridesharing schemes. This is due to the nonlinear demand rates for ridesharing scheme 1, of which the maximum demand rate occur at the earliest departure time, and drops significantly and nonlinearly in the early departure times. During the late departure times, the reduction of travel time is also quicker in the Please cite this article as: R. Ma, H.M. 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Fig. 2. Dynamics of travel time, ridesharing ratio and demand rates for Scheme 1.

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Fig. 3. Travel time and demand rates in results for ridesharing, no ridesharing and fixed ridesharing schemes.

Fig. 4. Percental errors of travel times using ADL model without time-varying ridesharing.

ridesharing scheme. By comparing the travel times of no ridesharing and fixed ridesharing schemes with that of Scheme 1, Fig. 4 shows that the percentage differences of travel times between no ridesharing and Scheme 1 are from -13.79% to 14.8%. The travel times calculated by the fixed ridesharing ratio (as the average of that in Scheme 1) is systematically lower. The percentage differences between fixed ridesharing and Scheme 1 are from −22.17% to 2.32%. Table 3 summarizes the critical characteristics of results, including critical departure times, maximum values of travel time and demand rates, the error ranges of travel times, VMT and VHT for ridesharing, no ridesharing and fixed ridesharing schemes. The critical departure times are different among these schemes. The first departure time of Scheme 1 is the latest, and the latest departure time of Scheme 1 is the earliest among these three schemes. In general, the dynamic ridesharing scheme has the shortest departure time window, and the fixed ridesharing has the longest one. The results in Table 3 also suggest that if the fixed ridesharing ratio scheme is used as an approximate of the actual dynamic ridesharing scheme with nonlinear results, the VMT would be slightly overestimated, and the VHT would be underestimated. Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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21

Table 3 Critical characteristics of results for ridesharing, no ridesharing and fixed ridesharing schemes. Scheme

t1

t2

t3

τ (t2 )

max of d2 (t)

Error range of τ (t) Comparing to Scheme 1

VMT (mile)

VHT (min)

Scheme 1, Case 5 Benchmark, no ridesharing Fixed ridesharing ratio s = 0.1977

8.984

10.231

17.005

14.766

6285

–

51126

660 0 0

7.5

11.25

17.5

13.75

1200

−13.79% ∼ 14.8%

630 0 0

77250

8.738

11.869

17.087

13.130

1200

−22.17% ∼ 2.42%

51596

60939

Fig. 5. Change of the system performance indicators with respect to h.

5.1.2. System performances Table 2 shows the system performances of the benchmark scheme with no ridesharing, and Scheme 1 with six selected cases with different k0 and p0 values. For each case except Case 6, the VMT, VHT and TC are calculated, respectively. First, it is observed that the VMT of all cases of Scheme 1 are less than both cases of the benchmark scheme. For the VHT, the worst case (Case 5) of Scheme 1 has the same VHT as the benchmark scheme with parking charges, and it is still better than that of benchmark scheme with no parking charge. These observations indicate that ridesharing can reduce VMT and VHT even only constant parking charge and ridesharing payments are applied. For TC, except the worst case (Case 5), all other cases in Scheme 1 have less TC than that of the benchmark scheme, which shows that generally ridesharing can reduce the system total cost effectively. Comparing Cases 1 and 2 in Scheme 1, we can observe that the VMT is reduced slightly with a higher constant parking charge in Case 2, the VHT are identical in both cases, and the TC of Case 2 is higher than that of Case 1. Similar observations can be found from Cases 3 and 4. This suggests that for a fixed ridesharing payment, the increase of constant parking charges do not change the total time spent by all vehicles in the system, while the total distance that all vehicle traveled decreases due to higher parking charges, and the total cost (including all time and monetary costs) would increase due to a heavier burden of shared parking charges to all travelers. This also confirms the analytical expression of VHT, where the VHT is not a function of k0 . For the VMT, however, the analytical expression shows that it is a function of departure time window t3 − t1 , which is further a function of k0 implicitly. Case 5 shows another parameter selection, where both of k0 and p0 − h are chosen higher than the first four cases. As a result, all of the three system performance indicators are higher than those β )τ β )τ of Cases 1 to 4. In Case 6, the feasibility condition 0.5 = (1−1+ ≤ (1−1+ ≤ p0 − h is not satisfied, and thus it leads to an n n unrealistic solution with excessively high ridesharing ratios. In order to achieve better system performance, it is desired to have lower VMT, VHT and TC. If the parking charges and the ridesharing payments are both fixed as constants, the results in Table 2 suggests that it is better to have lower variable fare p0 − h for a given constant parking charge, since the VMT, VHT and TC all increase from Cases 1 and 2 to Cases 3 and 4, respectively. While for a given variable fare, higher constant parking charges can reduce the VMT but increase the TC, as seen from the increasing k0 from Cases 1 and 3 to Cases 2 and 4, respectively. Starting from Case 4, we further change the pick-up cost h. It is found that with the same k0 and p0 − h, changing h alone does not alter the resulting commuting pattern, which suggests that the constant parking charges and the variable fare can define a solution regardless what the base fare is, i.e., the value of the pick-up cost per passenger. As shown in Fig. 5, for a given corridor free-flow travel time τ 0 = 10, the system performance indicators, i.e., the VMT, VHT and TC, increase as h increases with constant slopes. Fig. 6 show how the system performance indicators change, if the pick-up cost is fixed as h = 1 and we let free-flow travel times of the corridor vary. The free flow travel times τ 0 are from 9 min to 12 min. While τ 0 increases, both the VMT and TC increase, while the VHT remains a constant. The major difference for the VMT and TC from Fig. 5 is that the changes 0

m

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Fig. 6. Change of the system performance indicators with respect to τ 0 .

Fig. 7. Dynamics of optimal parking charges, ridesharing ratios and demand rates for Scheme 2.

of VMT and TC are both nonlinear with respect to the changes of free-flow travel time. The dash lines in Fig. 6 show the increments of VMT and TC, respectively, by increasing the values of τ 0 by 1. It is observed that the increments of both VMT and TC are not constants over different values of τ 0 , which suggests that VMT and TC are both nonlinear with respect to τ 0 . The explanation is that when τ 0 increases, the overall travel distances increase, while the departure time window shrinks to accommodate such changes. The shrank departure time window introduces higher congestion level, which further increases the travel cost besides the free flow travel time. The VHT remains constant thanks to a higher ridesharing ratio, which is also a result from the shrank departure time window. 5.2. Commuting pattern and system performance under Scheme 2 We first fix τ 0 = 10 min and h p = 1 min (h = 0.5). According to the feasibility condition, given such parameters, the minimum parking charge is calculated as p0 = (1 − β )τ 0 /(1 + n ) + h = 1, while the maximum one should be p0 = (1 − β )τ 0 + h = 3. Fig. 7 shows the resulting flow, cost, and ridesharing patterns for this scheme with different values of the constant ridesharing payments p0 , from the minimum p0 = 1 as Case 1 in Fig. 7a, the medium p0 = 2 as Case 2 in Fig. 7b, to the maximum p0 = 3 as Case 3 in Fig. 7c, respectively. It is observed that the departure time window, as well as the duration of the dynamic parking charges, is expanded as p0 increases. Since the vehicular demand in all cases are constantly 600 vph as the same as the flow capacity of the bottleneck, expanded departure time window means higher vehicle demand in total. In this scheme, a higher p0 leads to a lower ridesharing ratio. With the minimum p0 , the ridesharing ratio is constantly 4, which is the maximum number of passengers per vehicle; while with the maximum p0 , the ridesharing ratio goes down to zero, and the result reduces to the benchmark scheme with only solo drivers. Clearly the dynamic parking charges with minimum ridesharing payments concentrate the travelers’ demand. The optimal dynamic parking charges in all three cases are piecewise linear functions. The highest parking charges in all three cases occur at time 15 min with the same value of 3.75. Since Case 1 enjoys the least time duration for parking charges, the total monetary cost for all travelers as an entity is minimized, compared with the other two cases. Similar to the traditional bottleneck tolling, the parking charges eliminate the queuing delays in the ridesharing system in all three cases. Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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Fig. 8. Change of the system performance indicators respect to τ 0 .

Fig. 9. Patterns of ridesharing payment, ratio and demand rate for Scheme 3.

By fixing the variable fare p0 − h = 2, and the free-flow travel time τ 0 = 10, the VMT, VHT and TC are all observed as linearly increasing functions with respect to hp . The linear trends are the same as shown in Fig. 5. This is consistent with the analytical expression in Section 4, where the VMT and VHT are both linear functions of h and p0 − h, and the TC is a linear function of p0 and thus a linear function of hp for a given p0 − h. Fig. 8 shows the change of the system performance indicators over varying τ 0 . The variable fare p0 − h is fixed to be 2 as well, and the based fare hp is fixed to be 1. The VMT and VHT are both constants with respect to τ 0 , which is not possible in the benchmark scheme without ridesharing, since ridesharing adjusts the vehicular demand and the departure time window for various τ 0 , and thus makes VMT and VHT invariant to τ 0 . Such observations are consistent with the analytical expression in Section 4, where the VMT and VHT are not functions of τ 0 . On the other hand, TC is observed as an increasing nonlinear function with respect to τ 0 with an increasing derivative (the dash line in Fig. 8). 5.3. Commuting pattern and system performance under Scheme 3 Similar to Scheme 2, we first fix τ 0 = 10 and h p = 1 (h = 0.5). As discussed in Section 4, for Scheme 3, a constant parking charge of k0 is imposed to all vehicles, while the ridesharing payments are time-varying in order to eliminate the queuing delays at the bottleneck. For a given constant parking charge k0 , the parameter PR = p(t2 ), the minimum parking charges within the entire time period, is bounded according to the feasibility conditions. Fig. 9 shows the resulting patterns of ridesharing payment, ratio and demand rate for Scheme 3 with k0 = 2 and different selections of parameter PR , from the minimum PR = 1.65 as Case 1 in Fig. 9a, the medium PR = 2.33 as Case 2 in Fig. 9b, to the maximum PR = 3 as Case 3 in Fig. 9c, respectively. Similar to Scheme 2, it is observed that the departure time window is expanded as PR increases. However, the commuting patterns in Scheme 3 is fundenmentally different from that of Scheme 2, since the ridesharing ratios s(t) within the departure time window is no longer a constant. In fact, s(t) is a piecewise genuinely nonlinear function for each case. The increase of PR does not only expand the departure time window, but also increases the ridesharing payment for all times Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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Fig. 10. Change of the system performance indicators respect to τ 0 .

and lowers the curves of the ridesharing ratios. In Case 1, PR is selected to be minimum, and the ridesharing payment of the first and the last departing travelers have the full occupancy in the shared vehicle, i.e, s(t1 ) = s(t3 ) = 4. While in Case 3, the maximum PR leads to zero ridesharing ratio for the travelers departing at time t2 = 15, which means the high ridesharing payment prevents travelers to become passengers at the peak time, so that the travel demand of passengers is moved to the early or late arrival time periods. The vehicular demand in all cases are still constantly 600 vph as the same as the flow capacity of the bottleneck, which suggests that the dynamic ridesharing payment eliminates the vehicular queuing delays in the ridesharing system in all three cases. According to the feasibility condition, the maximum k0 is (1 − β )τ 0 = 2.5. Thus, the parking charge is set as k0 = 2 < 2.5. The free-flow travel time is set as τ 0 = 10, and the parameter PR is its maximum value as h + (1 − β )τ 0 . All of the VMT, VHT and TC linearly increase with respect to hp , which have the same trends as in Fig. 5. Fig. 10 shows the change of the system performance indicators over varying τ 0 . It is observed that both of the VMT and VHT are increasing functions with respect to τ 0 . In fact, the derivatives of both indicators over τ 0 are increasing. It is suggested that the departure time window t3 − t1 is also an increasing function of τ 0 . On the other hand, TC is observed as a linear increasing function of τ 0 , which is consistent with the analytical expression of TC, where PR − h happens to cancel out the τ 0 in the denominator. 5.4. System performance indicators for all schemes with varying hp /τ 0 Figs. 11–13 compare the system performance indicators for all schemes with different hp /τ 0 , with a fixed τ 0 = 10. Cases in the benchmark scheme without and with the optimal parking charges are included as ‘0: Solo drivers, k(t ) = 0’ and ‘0:Solo drivers, optimal k(t)’, respectively for VHT. For VMT and TC, these two cases in the benchmark scheme have identical results and are thus combined, respectively. For Scheme 1, we fix k0 = 2 and p0 = 2. For Scheme 2, two cases with different constant ridesharing payments are included. For Scheme 3, the maximum payment PR is set as 2.5, and two cases with different parking charges are included. We also note that the feasible regions of hp in these case could vary. Fig. 11 shows a general trend that the VMT is decreasing for an increasing ratio of hp /τ 0 for all ridesharing cases. Such trend suggests that for a given policy of ridesharing payment and/or parking charge, the longer the ridesharing pick-up trips, the less VMT the system would generate. This is a result of increasing ridesharing ratios. For instance, for Scheme 1 with p0 = 2 and k0 = 2, with an increased pick-up trip, more ridesharing travelers choose to become passengers, as they benefit from paying the same to the drivers for longer pick-up trips; on the other hand, the increasing ridesharing ratios also benefit the drivers by increasing their incomes from more passengers. As a result, the equilibrium at a longer pick-up trip eventually lead to more ‘compact’ ridesharing vehicles with higher ridesharing ratios. We also note that such decreasing functions are not linear, which are consistent with the analytical discussion in Section 4. By comparing these numerical cases, we have the following detailed observations. Two cases of Scheme 2 have the highest VMT, and higher p0 leads to higher VMT for Scheme 2. One case of Scheme 3 has the least VMT with k0 = 1, while the other case of Scheme 3 has a higher VMT with a higher k0 = 2.5. Scheme 1 has a bit less VMT, comparing with the second case of Scheme 3. These observations suggest that Scheme 3 with constant parking charge and the corresponding optimal dynamic ridesharing payments can achieve lower VMT for the system than the constant ridesharing payments with optimal dynamic parking charges, while constant parking charge and ridesharing payments can achieve the same level of VMT comparing with Scheme 3. Fig. 12 shows the VHT of all schemes with increasing hp /τ 0 . The VHT for each ridesharing case has a decreasing trend with an increasing hp /τ 0 . Similar to the VMT, cases of Scheme 3 achieve the lowest VHTs, while Scheme 2 with a higher constant ridesharing payment p0 = 3 leads to the highest VHT, comparing with other ridesharing cases. We note that the VHT of Scheme 1 is always the same with that of Scheme 2 with the same p0 = 2, which is consistent with the analytical Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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Fig. 11. VMT of all schemes with varying hp /τ 0 .

Fig. 12. VHT of all schemes with varying hp /τ 0 .

results in Section 4. Observations from Figs. 11 and 12 imply that under the optimal dynamic parking charges (ridesharing payments), the fixed ridesharing payment (parking charges) should be as low as possible, so that more travelers are encouraged to become ridesharing passengers, and thus lower VMT and VHT are expected. We observe more complex trends of TC with increasing hp /τ 0 in Fig. 13. All ridesharing cases show convex trends with increasing hp /τ 0 for TC. More specifically, for Schemes 1 and 2, the TC are first decreasing and then increasing, while within Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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Fig. 13. Total cost of all schemes with varying hp /τ 0 .

the feasible regions, the TC for cases in Scheme 3 are increasing. From the analytical results in Section 4, it is calculated that the optimal h minimizing the TC for Scheme 2 is hopt = 0.174 for p0 = 2 and hopt = 0.764 for p0 = 3, respectively. The observation from Fig. 13 confirmed the results. For Scheme 3,hopt = −1.45 for k0 = 1 and hopt = 0 for k0 = 2.5, respectively, which are out of the feasible regions of h. Therefore, the TC functions for Scheme 3 are increasing. An interesting observation is found that the TC of ridesharing Scheme 2 can be higher than that of solo driver benchmark, when the ratio of pick-up trip time over bottleneck link free-flow travel time hp /τ 0 is high. It shows that when p0 = 2, if h p /τ 0 = 0.3, or equivalently h p = 3 min, the TC is higher than that of solo driver benchmark; when p0 = 3, if hp /τ 0 ≥ 0.44, or equivalently hp ≥ 4.4 min, the TC is higher than that of solo driver benchmark. The increment of TC for Scheme 3 is less sensitive to the increasing ratio, and the TC for cases of Scheme 3 are the lowest compared to other schemes. Although there is no explicit expression of hopt for Scheme 1, in the numerical result we can see that when h p /τ 0 = 0.14, or equivalently h p = 1.4 min, the TC for Scheme 1 reaches its minimum. From above numerical results, it is found that parking charges and ridesharing payments are viable management tools to enhance system performance. By simultaneously setting the parking pricing and ridesharing payments properly, travelers’ mode and departure-time choices can be shifted for significant reduced VMT, VHT and TC. Regulations that enforce price floors and caps are needed to the parking prices and the ridesharing payments, so that realistic and feasible ridesharing can be conducted by general passenger vehicles. The numerical results suggest that Scheme 3 (constant parking charge with optimal dynamic ridesharing payments) can achieve best system performance in terms of VMT, VHT and TC with a relative low parking charge constant. It also shows that if the ratio of pick-up trip cost over free-flow bottleneck link trip cost is higher than certain criteria value, Scheme 1 (constant ridesharing payment and optimal dynamic parking charges) has less VMT and VHT, but yields higher TC than solo driver benchmark. Note that the numerical results do not cover all possible parameter settings, and there may not be a universal strategy to manage the ridesharing in bottleneck corridors with different characteristics.

6. Conclusions and future research 6.1. Conclusions and discussions This paper studied the dynamic ridesharing problem on a highway with a single bottleneck together with parking. In the studied ridesharing problem, a traveler is either a driver or a passenger. Given the piecewise linear arrival penalty function, dynamic user equilibrium of travelers’ departure-time and mode choices are modeled. Specifically, we let the travel costs of any ridesharing traveler with any departure time equal to each other, no matter they are passengers or drivers. The analysis Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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of the morning commute problem on the corridor with a single bottleneck with ridesharing merits much more discussion, since in these ridesharing schemes, system performances are district from the traditional morning commute problem. In this paper, we studied three schemes with distinct parking and ridesharing policies, including 1) The ridesharing scheme with constant parking charges and ridesharing payments; 2) The ridesharing scheme with constant ridesharing payments and dynamic parking charges to achieve the free-flowing traffic; 3) The ridesharing scheme with constant parking charges and dynamic ridesharing payments to achieve the free-flowing traffic. Besides those schemes, the traditional morning commute problem (i.e., solo drivers only) with and without the parking charges are listed as well for comparison purposes. By deriving the analytical formulation of the traffic dynamics, the choice behaviors and the system performance indicators for each scheme, it was found that the dynamics and the system performance indicators varied over different schemes; even in the same scheme, the selections of certain parameters may significantly influence the system performance. One of the major findings was that nonlinear departure rates and travel time functions can be generated in Schemes 1 and 3, which was not observed in traditional single bottleneck models without ridesharing or with only fixed ridesharing ratios. Such nonlinearity is generated from the cost sharing behavior. More specifically, since the parking charges are shared among all travelers in a commuting vehicle, the shared parking cost would carry the number of travelers per vehicle in the denominator and thus brings genuine nonlinearity. Solving such nonlinear ordinary differential equations further involves the calculation of the travel time functions, as well as the ridesharing ratios, so that the genuine nonlinearity also appear in the expressions of travel time or ridesharing ratios. For instance, in Scheme 3, although the parking charges are piecewise linear, the resulting ridesharing ratios are genuinely nonlinear. By comparing the benchmark scheme (only with solo drivers) and the ridesharing schemes with different parameter selections, the results showed that dynamic ridesharing payments and parking charges can serve as effective management tools to mitigate the traffic congestion and reduce vehicle-miles-traveled (VMT) and vehicle-hours-traveled (VHT) in the ridesharing schemes. However, in terms of total travel costs (TC) of all travelers, dynamic parking charges with constant ridesharing payments may help little or even increase the total cost if the pickup trip costs are higher than certain criteria; while properly selected constant parking charges with corresponding optimal dynamic ridesharing payments can achieve the best system performance among all schemes in terms of VMT, VHT and TC, by encouraging more travelers to participate ridesharing programs as passengers and eliminating queuing congestion. The numerical results, together with the analytical formulations, have shown that the strategy in Scheme 3 with a lower constant parking charge performs the best, with the least VMT, VHT and TC. However, it was also noted that the feasible region of the pickup cost in such strategy is the most restricted. If the ratio of single pickup trip time over the bottleneck free-flow travel time hp /τ 0 is higher than 0.2, such strategy in Scheme 3 cannot work due to infeasibility. The results also showed that the constant parking charges with constant ridesharing payments may bring limited benefits, when the pickup trips are relatively short. In fact, Scheme 1 has the worst performance in terms of total travel cost among all ridesharing schemes when the ratio hp /τ 0 is below 0.1. Such observations suggest that in order to properly manage the ridesharing system, an integrated strategy with both ridesharing payments and parking charges should consider the relative length of the pickup trips with respect to free-flow travel time on the bottleneck link. Different from emerging implementation of dynamic parking charges, the results from numerical results suggest that the shared parking charges should not vary over time-of-day, while the ridesharing payments should be time-varying to encourage ridesharing services at different levels for different time-of-day to spread out the demand peak.

6.2. Future research In this paper, the equilibrium solutions are derived from equilibrium conditions with parameters such as average pick-up time h and the parameter for modified bottleneck trip cost of passengers. There may be potentially very interesting topics on calibrating or validating such parameter selections, and/or refining the cost structure for the generalized cost of ridesharing travelers. More realistic parameters and cost structure can be integrated into the proposed formulation in the future study, while current methodology of deriving the equilibrium solutions should still apply fundamentally. This paper assumes homogeneous travelers, i.e., that all travelers join the ridesharing program and share the same valueof-time. In practice, not all drivers may choose to participate ridesharing programs, and they tend to continue solo driving. That is, given the ridesharing market as described in this paper, a driver can still choose to be out of the market, i.e., to be a solo driver. This is, however, another layer of choice behavior which can be modeled prior to the propose framework. It merits more investigations on the equilibrium of solo-driving and ridesharing simultaneously with the equilibrium of ridesharing modes and departure time choices with the introduction of High-Occupancy-Vehicle (HOV) lanes, or High-Occupancy-Tolls (HOT). We assumed in this study that the ridesharing ratios are continuous variables, and all ridesharing travelers have exactly the same generalized cost under equilibrium. In the future research, such assumption can be extended with discrete/integer ridesharing ratios and generalized costs with bounded rationality user equilibrium, where the generalized costs of ridesharing travelers may differ within certain bounds. Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002

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Another future research topic is to expand with the heterogeneity of ridesharing travelers on the ownership of cars, among many other heterogeneity properties, such as preferences of early/late arrivals over queuing delays, and staggered preferred arrival times. The authors are currently investigating the morning commute problem with a single bottleneck and with heterogeneous ridesharing travelers. Current research on a corridor with a single bottleneck can also be extended to general networks with multiple origindestinations. This would be pursued in future research as well. Acknowledgment This work is based on research supported by the National Science Foundation under Grant CPS-1544835. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation. Appendix A. Ridesharing with no parking charge and fixed ridesharing payments Pick-up cost for drivers at time t is 2h(1 + s(t )). The payment is fixed as p0 . The derivative of the arrival penalty function αδt+τ 0 R . Define t1 , t2 and t3 so that the first vehicle departs at time t1 , the last vehicle departs at time t3 , and the peak time (vehicles departs at this time would have zero arrival penalty) is t2 . Drivers cost: c2 (t ) = 2h(1 + s(t )) + τ (t ) + F (t + τ (t )) − s(t ) p0 . Passengers cost: c3 (t ) = h(1 + s(t ))βτ (t ) + F (t + τ (t )) + p0 . Under equilibrium, drivers and passengers share the same cost, c2 (t ) = c3 (t )⇒ (1 + s(t ))( p0 − h ) = (1 − β )τ (t ). For t < t1 , the vehicle departure rate is zero, d2 (t ) = 0. ∂F ∂τ =

0 For t1 ≤ t ≤ t2 , let d2 (t ) = d > C . τ (t ) = τ 0 + q(t+τ ) = τ 0 +

C

d−C (t C

− t1 ).

Passengers that depart at different time share the same cost in the equilibrium, c˙ 3 (t ) = 0. ⇒ d−C ) C



h (1−β ) p0 −h

+β



d−C C

− α (1 +

= 0, h ( 1 −β ) +β p −h d = h(1−β0 ) C, +β −α p −h

(A.1)

0

β) which implies that h(p1−−h + β > α. 0

For t2 ≤ t ≤ t3 , let d2 (t ) = dˆ < C . τ (t ) = τ 0 + 1 (q(t2 + τ 0 ) − (C − dˆ)(t − t2 )) = τ 0 + C

1 C

 (d − C )(t2 − t1 ) − (C − dˆ)(t − t2 ) .

The last vehicle would experience no congestion, τ (t3 ) = τ 0 .

⇒ (d − C )(t2 − t1 ) = (C − dˆ)(t3 − t2 ) Passengers that depart at different time share the same cost in the equilibrium, c˙ 3 (t ) = 0. ⇒ − C −dˆ ) C

= 0,

dˆ =

 h ( 1 −β )



p0 −h h ( 1 −β ) + p0 −h

+β



h (1−β ) p0 −h

+β



(A.2) C −dˆ C

+ γ (1 −



C,  β +γ

(A.3)

At time t = t2 , the departing drivers and passengers experience no arrival penalties, i.e., they arrive exactly on the preferred arrival time R, which is

t2 + τ (t2 ) = t2 + τ 0 +

d −C

(t2 − t1 ) = R

 β For t1 ≤ t ≤ t2 , 1 + s(t ) = p1−−h τ 0 + d−C ( t − t1 ) . C 0

 ˆ β For t2 ≤ t ≤ t3 , 1 + s(t ) = p1−−h τ 0 + d−C (t2 − t1 ) − C−C d (t − t2 ) . C C

(A.4)

0

Denote t2  t2 − t1 , t3  t3 − t2 . According to (A.2),

d −C · t2 = C − dˆ

t3 =

α ( h(p10−−hβ ) + β + γ ) γ ( h(p10−−hβ ) + β − α )

Demand conservation



D=

t2

t1

d (1 + s(t ))dt +



t3

t2

· t2

dˆ(1 + s(t ))dt

(A.5)

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29

Then we have

D=

1 −β p0 −h 0

( 1 −β ) α 2( p0 −h )(β −α ) 2

τ · t2 +

· ( t2 ) + 2

1 −β p0 −h

τ m · t3 +

(1−β )(−γ ) 2( p0 −h )(β +γ )

· ( t3 )

2

(A.6)

= b t2 + a( t2 ) where

τm =τ0 +

d−C C

)τ 0 a = (1p−β−h · 0

b =

 1 −β  p0 −h

(t2 − t1 ) = τ 0 + β −α α (t2 − t1 ) = τ 0 +   h (1−β ) 1+



α(

+β +γ )

p0 −h

γ ( hp(1−−hβ ) +β −α )

 

0

α

·

α β −α t2

h (1−β ) + p0 −h

·

β −α

1 2

A positive solution of t2 is t2 = So that

α

+

h (1−β ) + p0 −h

−b+

β −α

h (1−β ) + p0 −h

β +γ

γ

+

−γ

β) 2( hp(1−−h +β +γ ) 0

α

h (1−β ) + p0 −h

β −α

(

h (1−β ) + p0 −h

γ

β +γ

2



(A.7)

)

√

b2 +4aD . 2a

t1 = R − τ m − t2 t2 = R − τ m

(A.8)

t3 = R − τ m + t3 The system performance indicators, namely VMT, VHT and TC, are calculated below, respectively. During time periods [t1 , t2 ) and (t2 , t3 ), the vehicular demand rates d2 (t) are d and dˆ, respectively.

 MV MT =

t3



 

t1



d2 (t ) 2(1 + s(t ))hv p + τ 0 v f dt

= 2 hv p d

t2

t1

1 + s(t )dt + dˆ



t3

t2



1 + s(t )dt

+ τ 0 v f (d t2 + dˆ t3 )

= 2hv p D + τ 0 v f (d t2 + dˆ t3 )  MV HT =

t3

t1

d2 (t )[2(1 + s(t ))h + τ (t )]dt

= 2h +

 = 2h +

p0 − h 1−β

 

t3

t1

d2 (t )(1 + s(t ))dt

p0 − h D 1−β

The individual cost for drivers departing at any time is equal under the UE. t2 is the departure-time of the passengers arriving at the destination exactly on time, but experiencing the longest travel time τ m , i.e., t2 = R − τ m .

T C = IC · D = c3 (t2 ) · D = (βτ m + p0 ) · D Appendix B. Linearization of p(t) in Scheme 3 We recap the original nonlinear ODE (44) of p(t) in Scheme 3,

h ( 1 − β )τ Or



0



1 p(t ) − h



  ∂ F (t + τ 0 ) k0 + + 1+ p˙ (t ) = 0. ∂τ ( 1 − β )τ 0

−h(1 − β )τ 0 k0 +1+ ( p(t ) − h )2 ( 1 − β )τ 0



( p(t ) − h ) = −

∂ F (t + τ 0 ) . ∂τ

(B.1)

(B.2)

0 k Denote H = h(1 − β )τ 0 , K = 1 + (1−β0)τ 0 , G = − ∂ F (∂t+τ τ ) and x(t ) = p(t ) − h, then





−H

(x(t ))

2

+K

dx(t ) = G. dt

(B.3)

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Using the method of separation of variables, we have the integration form as



x−2 dx + Kx = Gt + C,

−H

(B.4)

or

x+

H/K = (G/K )t + C. x

(B.5)

where C is a constant.  Given the property of such type of function, when x → +∞, x → (G/K )t + C; and if x >> H/K , then x(t ) ≈ (G/K )t + C.



That is, if min p(t ) > h + (1 − β )τ t

p(t ) ≈ −

0

h

( 1 − β )τ 0 + k0

, then

∂ F (t + τ 0 ) 1 . ∂τ 1 + (1−kβ0 )τ 0

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Please cite this article as: R. Ma, H.M. Zhang, The morning commute problem with ridesharing and dynamic parking charges, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.07.002