Coordinated pricing for cars and transit in cities with hypercongestion

Economics of Transportation ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Coordinated pricing for cars and transit in cities with hypercongestion Eric J. Gonzales Department of Civil and Environmental Engineering, University of Massachusetts, Amherst, MA 01003, United States

art ic l e i nf o

a b s t r a c t

Article history: Received 17 June 2014 Received in revised form 12 March 2015 Accepted 18 April 2015

Urban street networks exhibit hypercongested traffic states in which flow and speed drop as vehicles crowd the network. An important feature of aggregate traffic models based on the macroscopic fundamental diagram is that the dynamics of congestion are represented in a physically realistic way. For peak demand in a city in which all travelers use cars, hypercongested states are known to be unstable in equilibrium. In real cities travelers are able to choose whether to use cars or public transit during a peak period. This paper presents a model of the equilibrium for cars and transit, recognizing that hypercongestion often arises when modes are not priced. It is shown that hypercongested traffic can be part of a stable, steady equilibrium state when cars and high-capacity transit are used simultaneously. The paper also shows that coordinated fixed prices for cars and transit can always prevent hypercongested traffic states from developing. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Pricing cars and transit User equilibrium System optimum Hypercongestion Bathtub model Macroscopic fundamental diagram

1. Introduction Traffic in cities worldwide is often characterized by peaked demand that overwhelms the capacity of bottlenecks and street networks. When queues develop at individual bottlenecks or the queues on city streets block intersections, increased vehicle density is associated with reduced vehicle flow and slower traffic speeds. This condition is known as hypercongestion, and it can apply to the traffic state on a single link or to the aggregated traffic state on a network. Arnott et al. (1990b, 1993) are early examples advocating consideration of hypercongestion in the analysis of peak demand at bottlenecks. Hypercongestion on urban networks has gained more recent attention (Arnott, 2013; Fosgerau and Small, 2013). In order to understand traffic in crowded cities, it is necessary to recognize that hypercongested traffic conditions are an inherent part of the dynamics of congestion. In real cities, travelers often have a choice between traveling by car or using an alternative mode, such as public transit. Since the equilibrium distribution of travelers among available modes does not always result in efficient use of urban infrastructure, management strategies are needed, such as pricing. This paper addresses the characteristics of the equilibrium for cars and transit in a city, recognizing that hypercongested traffic states often arise as part of the user equilibrium. A pricing strategy based on coordinating fixed prices for cars and transit is presented that can always prevent the onset of hypercongested states. Most models of multimodal transportation systems describe traffic flows through a bottleneck or along a corridor. On these facilities, increased travel demand causes longer queues, but the flow at the bottleneck remains at a stable capacity. This consistent bottleneck capacity forms the basis for much of the work on optimal congestion

pricing dating back to Vickrey (1969) and developed extensively with deterministic analysis (Hendrickson and Kocur, 1981; Smith, 1984; Daganzo, 1985; Arnott et al., 1990b, 1993) and stochastic models (De Palma et al., 1983; Ben-Akiva et al., 1984, 1986). The bottleneck model has been extended to consider multiple routes (Arnott et al., 1990a) and multiple modes when all users are identical (Tabuchi, 1993; Huang, 2000; Danielis and Marcucci, 2002; Kraus, 2003). Other extensions consider users with heterogeneous sensitivity to schedule delay (Arnott et al., 1988, 1992, 1994) or preferred passage times through the bottleneck that are distributed over a period of time (Gonzales and Daganzo, 2012, 2013). The optimal time-dependent prices for cars that eliminate queuing at a bottleneck have also been shown to eliminate excess queuing in a network to maintain a consistent flow of vehicles as if the capacity were fixed (Geroliminis and Levinson, 2009). Gonzales and Daganzo (2012) show that optimal time-dependent prices of cars and transit at a bottleneck also hold for networks when the transit system operates on dedicated lanes and does not have constrained capacity. Although optimal dynamic pricing strategies allow for traffic conditions to be managed in a detailed way, many cities are unable or unwilling to implement road pricing schemes due to challenges of political acceptability (De Palma et al., 2007). Only a few cities (e.g., Singapore, London, Oslo) have implemented urban road pricing, and fewer still (e.g., Stockholm, Stavanger) have done so with time-dependent prices (Hårsman and Quigley, 2010). Without any pricing intervention, a traveler may choose whether to drive or use transit in whatever way minimizes the cost of his or her own trip. In many cities, this unpriced equilibrium behavior involves hypercongested traffic conditions. The reality is that most cities have not instituted dynamic congestion pricing schemes and have no plans

http://dx.doi.org/10.1016/j.ecotra.2015.04.003 2212-0122/& 2015 Elsevier Ltd. All rights reserved.

Please cite this article as: Gonzales, E.J., Coordinated pricing for cars and transit in cities with hypercongestion. Economics of Transportation (2015), http://dx.doi.org/10.1016/j.ecotra.2015.04.003i

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2

Fig. 1. A typical triangular fundamental diagram represents the traffic states that can exist on a link. The points to the right of C are the hypercongested traffic states (dashed lines). (a) Flow-density relation. (b) Speed-density relation.

1

v

hrs km

traffic states in terms of the fundamental flow-density relation for traffic states on a link. We then consider how hypercongested traffic states emerge as a consequence of queuing at a bottleneck. Finally, we consider the transient nature of hypercongestion when travelers must pass a bottleneck by car and the equilibrium conditions with two modes that allow hypercongestion to exist as a stable steady state. With this understanding of hypercongestion and equilibrium at a single bottleneck, we will be equipped to consider the performance of traffic in networks.

E3 de

ma

nd

E2

1

v

supply

C

E1 qC

q

veh lane·hr

Fig. 2. The static supply curve implied by the fundamental diagram relates travel time (inverse of speed) to flow with a backward sloping branch corresponding to hypercongested states (dashed line). A downward sloping demand curve (thin line) may intersect this curve in three potential equilibrium points (Verhoef, 1999).

to do so, but cities around the world do have transit systems, and policies to subsidize transit fares or increase the cost of driving (e.g., by raising parking prices) are common. If transit fares and car prices are coordinated, then the difference in cost that travelers experience between the two modes can be manipulated to achieve different equilibrium outcomes. This paper addresses the dynamics of traffic states in cities that do not implement optimal time-dependent tolls. Specifically, the focus is on identifying the conditions that lead to steady equilibrium states in which cars and transit are used simultaneously. The coordinated fixed prices that minimize the social cost associated with peak demand are also identified. The paper is organized as follows: Section 2 reviews the dynamics of queuing on a single link, and shows that hypercongestion is a steady equilibrium state only when high-capacity transit operates in a dedicated right of way. Physically realistic models of traffic and transit operations in cities are presented in Section 3. The aggregate city-scale models of traffic and transit are used in Section 4 to extend the equilibrium queuing analysis from the single bottleneck to an urban neighborhood. In addition to identifying the potential for hypercongestion in a steady equilibrium with unpriced cars and transit, a model is introduced to identify coordinated fixed prices for cars and transit that realize steady equilibrium traffic states that are not hypercongested. Section 5 extends the model to the case that transit operates in mixed traffic conditions and is thus subject to the slower speeds of congested traffic. A numerical example is provided to illustrate the cases with transit operations on dedicated and mixed lanes. The results are summarized in Section 6.

2. Hypercongestion and equilibrium at a single bottleneck Before considering the complexities of traffic in urban street networks, it is useful to start with an understanding of traffic flow on a single street, which we will call a link. We define hypercongested

2.1. Hypercongestion on an isolated link Traffic states on a single link can be described in terms of flow, q (veh/lane-hr), density, k (veh/lane-km), and the corresponding speed, v ¼ q=k (km/hr). The traffic states that can be observed on a road are described by a fundamental diagram that expresses flow and speed as functions of density. Empirical investigations of bi-variate relations of traffic variables date back to Greenshields et al. (1935), who hypothesized that the flow-density relation takes the form of an inverted parabola corresponding to a linear speed-flow relation. More recent empirical evidence from observations of traffic in steady states reveals that the fundamental flow-density relation for a link is most nearly triangular as shown in Fig. 1(a) (Cassidy, 1998). The common feature of all theoretical and empirical fundamental diagrams is that flowdensity relation is unimodal with a weakly concave increasing branch (Smulders, 1989; Wu, 2002). The corresponding speed–density relation is always non-increasing, as shown for the triangular fundamental diagram in Fig. 1(b). The triangular fundamental diagram in Fig. 1(a) is for a link with maximum capacity qC (veh/lane-hr), free-flow vehicle speed vf (km/ hr), and jam density kJ (veh/lane-km) when vehicles are stopped in a queue. The left side of the fundamental diagram (k r kC ) is an increasing branch, indicating that flow increases with the vehicle density and traffic moves at the free-flow speed. The right side of the fundamental diagram (k 4 kC ) is a decreasing branch in which both flow and speed decrease with increasing vehicle density. Traffic engineers refer to these right-side traffic states as congested. Economists call these traffic states hypercongested. Hypercongestion has received much attention from engineers and economists over the years, because these traffic states are associated with wasteful delays. Any flow associated with hypercongestion could also be achieved on the left side of the fundamental diagram at a lower vehicle density and higher speed of traffic. The fact that two traffic states are associated with every flow (with exception of the unique maximum capacity point C) poses a challenge for analysts who want to work with a static supply function of the type proposed by Walters (1961). Such static models seek to express travel time (inverse of speed) as a function of flow to be used as a supply curve for static equilibrium analysis. A debate has arisen in the literature over the stability of hypercongested states. The hypercongested branch of the

Please cite this article as: Gonzales, E.J., Coordinated pricing for cars and transit in cities with hypercongestion. Economics of Transportation (2015), http://dx.doi.org/10.1016/j.ecotra.2015.04.003i

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link at time t

A(t)

N (veh)

nðtÞ ¼ AðtÞ  DðtÞ

qA T(t) n(t)

t

tP

3

D(t)

ð1Þ

and the rate of change n(t) is governed by the rates of arrivals and departures.

µ

_  DðtÞ _ _ ¼ AðtÞ nðtÞ

t (hrs)

_ ¼q Fig. 3. A queuing diagram for a single bottleneck with arrival rate AðtÞ A exceeding bottleneck capacity μ for t A ð0; t P Þ.

fundamental diagram implies a backward sloping branch for the link's supply curve as shown in Fig. 2. Verhoef (1999) explains that a point on the supply curve crossing demand from above (like E3) is not stable to perturbations in flow or travel time, and attaining a point like E2 would require inflows to be unequal from outflows on the link. The distinction between flows arriving on the link and departing from the link is critical for modeling the inherent dynamics of traffic congestion or hypercongestion, and this point has been reiterated in subsequent studies (May et al., 2000; Small and Chu, 2003; Verhoef, 2003). Hypercongested traffic states cannot develop as a consequence of flows entering the upstream end of a homogeneous link. Only point E1 could actually develop as vehicles flow onto the link. If the demand curve were to increase so that it does not intersect the supply curve at a feasible point, a queue would develop upstream of the link as vehicles wait to enter at rate qC. Walters (1961) aptly called the points on the hypercongested branch of the supply curve “the bottleneck case,” because these traffic states can only occur when a bottleneck restricts the departing flow of vehicles from the link. 2.2. Hypercongestion and equilibrium at a bottleneck with only cars No street exists in complete isolation, so analysis of a link without connections to other parts of the network is not realistic. The simplest case in which hypercongested traffic states can arise is in the queue that builds up on a link upstream of a bottleneck. Suppose that the link described in Fig. 1 is restricted at the downstream end by a bottleneck with capacity μ. If the arriving demand on the link is less than μ, the flow of arriving vehicles can depart without obstruction, and the traffic state will be associated with the point on the left side of the fundamental diagram with the corresponding flow. If the arriving flow qA (associated with point A in Fig. 1(a)) exceeds μ, the link will not be in a stationary state, because more vehicles enter the link than can depart from it. According to kinematic wave theory (Lighthill and Whitham, 1955; Richards, 1956), a queue will develop, and the traffic state within the queue will be the hypercongested state with flow qB equal to the bottleneck capacity μ; point B in Fig. 1(a). The imbalance between arriving and departing vehicle flows will cause the queue to grow until the high rate of arriving vehicles subsides. Thus, a static link supply curve of the type illustrated in Fig. 2 is not suitable for describing the dynamics of queuing congestion. A useful way to visualize and analyze the dynamics of congestion at a bottleneck is with a queuing diagram that shows the cumulative count of vehicle arrivals by time t, A(t), and the cumulative count of vehicle departures by time t, D(t). Fig. 3 shows the queuing diagram for vehicles arriving at rate qA for a period of time t A ð0; t P Þ. We will denote the time derivative by an overdot, so _ ¼ q for t A ð0; t P Þ, and the slope of D(t) is the slope of A(t) is AðtÞ A _ DðtÞ ¼ μ whenever a queue is present. The vertical distance between the arrival and departure curves represents the number of delayed vehicles that are waiting on the

ð2Þ

If the queue follows a first-in first-out discipline (i.e., vehicles do not overtake one another), the horizontal distance between A(t) and D(t) represents the delay experienced by each vehicle. Since the bottleneck capacity is constant, a driver planning to arrive at the bottleneck at time t can anticipate the delay that he or she will experience, T(t), as a consequence of n(t) TðtÞ ¼

nðtÞ

μ

:

ð3Þ

The evolution of the queue over time is represented in the queuing diagram in Fig. 3. The increasing delays represented by T(t) correspond to the increasing lengths of the queue through which drivers will travel at the hypercongested state B. Since the queue length is transient, so is the nature of the hypercongested state for this peak demand. The conventional bottleneck model for departure time choice builds on this queuing formulation to consider the preferences that users have for when they wish to reach their destinations and hence when they wish to depart the bottleneck (Vickrey, 1969; Arnott et al., 1990b). If users wish to depart the bottleneck at a rate that exceeds μ, they cannot all pass on time. Some travelers will be early or late. The resulting user equilibrium is described by the cumulative arrival curve that allows no user to unilaterally change his or her arrival time at the bottleneck in order to reduce the combined cost of queuing and schedule delay for his or her own trip. A unique equilibrium has been proven to exist for a broad range of wished bottleneck passage time distributions and schedule penalty functions (Smith, 1984; Daganzo, 1985). It is common to consider a piecewise linear schedule penalty function with cost e equivalent minutes of queued travel time per minute of earliness and L equivalent minutes of queued time per minute of lateness.1 For N travelers in a peak period with values e and _ ¼ μ=ð1  eÞ for the first L, the equilibrium arrival curve has slope AðtÞ _ ¼ μ=ð1 þ LÞ for the remaining Ne=ðe þ LÞ travelers and slope AðtÞ NL=ðeþ LÞ travelers (Hendrickson and Kocur, 1981; Fargier, 1981). The equilibrium with departure time choice, as described by the bottleneck model, is a dynamic process. The equilibrium arrival curve implies the growth and dissipation of the queue depending on the relative values that users place on being early or late. The resulting equilibrium queues are never stationary when all users travel by car through the bottleneck, so the hypercongested traffic state is associated with the transient traffic conditions on the link resulting from the imbalance of arrivals and departures. 2.3. Hypercongestion and equilibrium at a bottleneck with cars and transit In many urban environments, travelers can choose whether to travel by car or to use an alternative mode, such as public transit. The total demand for travel can be broken down into cumulative arrivals by car, Ac(t), and cumulative arrivals on transit, At(t), such that AðtÞ ¼ Ac ðtÞ þ At ðtÞ:

ð4Þ

The arrival rates are given by the time derivative of (4), so the 1 These costs are often formulated in terms of cost per travel time α, cost per time early β, and cost per time late γ (Arnott et al., 1990b). From these values, e ¼ β=α and L ¼ γ =α.

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cost of driving would exceed transit and all users would be diverted to transit, thus causing the queue to shorten. If the queue were to get shorter, the cost of transit would exceed driving and all users would choose to drive, causing the queue to lengthen. When cars and transit are used simultaneously, the delay and _ ¼ 0. accumulation of vehicles stabilize at Tcrit and ncrit, so nðtÞ _ _ Evaluating (2) with A_ c ðtÞ in place of AðtÞ and μ in place of DðtÞ, because a queue is present, this condition implies that A_ c ðtÞ ¼ μ at

A(t)

N (veh) qA

Ac(t) µ Dc(t) Tcrit µ

ncrit t1

tP

t (hrs)

Fig. 4. A queuing diagram for a bottleneck with arrival rate exceeding bottleneck capacity μ, and an alternative transit mode with capacity exceeding qA  μ.

arrival rate for each mode must satisfy _ ¼ A_ c ðtÞ þ A_ t ðtÞ: AðtÞ

ð5Þ

We suppose that each user chooses the mode that minimizes the cost of his or her own trip. This will lead to a Wardrop (1952) equilibrium condition in which users choose the mode with the lowest cost or use both modes simultaneously only if the costs are equal. We will consider a transit service that operates on a reliable schedule and has high enough capacity that passengers are always able to board the next vehicle. Such a system allows each user to anticipate the time to access the transit stop or station, wait for the vehicle, and ride within the vehicle to reach his or her destination. As a result, the cost of using transit is associated with a fixed generalized cost, zt (hours). The generalized cost of a free-flow trip by car is zc0 (hours), which represents the free flow travel time and fixed costs associated with making a car trip. Since delays vary over time depending on the length of the queue, the generalized cost of travel by car including queuing delay is zc ðtÞ ¼ zc0 þ TðtÞ:

ð6Þ

If zt o zc0 , then transit is always the less costly mode, and all trips will be made by transit. This may be the case in cities in which parking and registration fees are very expensive compared to the cost of using an efficient transit system or for travelers who cannot afford to pay for a car. The more interesting case is when zt 4 zc0 . Users will choose to drive at any time that zc ðtÞ o zt . By substituting (6) into the inequality, and solving for T, we can see that users will prefer to drive at all times satisfying: TðtÞ o zt  zc0 :

ð7Þ

This condition implies a critical delay, T crit ¼ zt  zc0 , below which all users drive and at which both modes may be used simultaneously. Note that this critical delay depends only on the difference between the generalized costs for free-flow trips by car and transit, and it is not a function of time t. The condition (7) can be expressed in terms of the accumulation of delayed vehicles on the link by substituting (3) for T(t) and solving for n(t): nðtÞ o μðzt  zc0 Þ:

all times that cars and transit are used simultaneously. Based on constraint (5) and the arrival rates associated with condition (8), the equilibrium arrival rates for cars and transit must be ( _ AðtÞ if nðtÞ o ncrit A_ c ðtÞ ¼ ð9Þ μ otherwise _  A_ c ðtÞ: A_ t ðtÞ ¼ AðtÞ

ð10Þ

These conditions imply that only cars are used in the beginning of the rush while queues are developing. Once the accumulation of delayed vehicles reaches ncrit, the arrival rate for cars matches the bottleneck capacity and the excess demand is diverted to transit.2 For t A ðt 1 ; t P Þ, the accumulation of delayed vehicles remains constant at ncrit. The hypercongested traffic in the link's queue during this period is in a steady equilibrium state, because it can be sustained as long as the demand exceeds the bottleneck capacity. There have been a number of studies that consider mode choice with the trip scheduling decision (Tabuchi, 1993; Huang, 2000; Danielis and Marcucci, 2002; Kraus, 2003). Gonzales and Daganzo (2012) show that high capacity transit constrains the queuing delay at the bottleneck not to exceed Tcrit. In the beginning and end of the rush, delays are short, so only cars are used and the equilibrium arrival curve has the same slopes as for early and late travelers in the singlemode bottleneck equilibrium. In the middle of the rush, transit allows travelers to reach their destinations on time, and travelers use both modes simultaneously with A_ c ðtÞ ¼ μ in order to maintain ncrit as described above. Although transit vehicles may travel slower than cars, this is reflected in the value of zt. Transit riders may have to start their trips a little earlier in order to reach their destination on time, but the equilibrium is assumed to be a condition in which each user has perfect information about the costs of travel by the available modes and can choose the timing and transportation mode accordingly. The problem with using a static supply curve for hypercongested states as shown in Fig. 2 is that these states can only exist when a queue develops on a link. Then, these hypercongested states only become stable if the arrival rate of cars drops to the bottleneck capacity (A_ c ðtÞ ¼ μ) after the queue develops. This steady state is unlikely to arise if all users must travel by car through a bottleneck, as explained in Section 2.2. However, the presence of an alternative transit system with sufficiently high capacity bounds the queuing delays at Tcrit and allows for a period of steady state hypercongestion in the equilibrium for t A ðt 1 ; t P Þ, as shown in Fig. 4. The remainder of this paper is primarily concerned with the characteristics of this steady state in the middle of the rush, which can be achieved with travelers simultaneously using cars and transit.

ð8Þ

Similarly, this condition implies a critical accumulation of delayed vehicles, ncrit ¼ μðzt  zc0 Þ, below which all users drive and at which both modes are used simultaneously. Consider again the bottleneck with arriving demand as shown in Fig. 3, except that now an alternative transit mode is available to users. The queuing diagram for this two-mode case is shown in Fig. 4. At the beginning of the peak period, the accumulation of _ vehicles is less than ncrit, so all users drive, and A_ c ðtÞ ¼ AðtÞ. Once the accumulation of delayed vehicles reaches ncrit at time t1, the queue length must stabilize. If the queue were to get longer, the

3. Modeling city-wide traffic and transit Link-based models may be combined to represent a whole network of streets with explicit representation of bottlenecks and queues on each link of the network. For large networks, this 2 If the transit system capacity exceeds A_ c ðtÞ  μ, the system has sufficiently high capacity that transit trips can be served with constant generalized cost zt; i.e., without additional delays. If transit capacity is constrained, there will be queuing on transit as described in Gonzales and Daganzo (2013).

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approach lends itself to simulation, because the complexity of interactions between links quickly becomes intractable analytically. Aggregate models of urban transportation systems provide a simplified representation of traffic in a network by relating aggregate values, such as the total number of vehicles or passengers circulating in a network, with average flows, densities, and speeds in the network. Recent advances in modeling urban networks at the aggregate or macroscopic scale provide simple representations of traffic and transit operations, which allow the formulation of analytical equilibrium models to evaluate city-wide pricing and traffic control strategies. First, an aggregated model of car traffic in urban networks is presented based on recent work with the Macroscopic Fundamental Diagram (MFD) and the bathtub model. This model represents both free flowing and hypercongested traffic conditions and is used to develop a cost function for driving that is consistent with the physics of urban traffic. Then, an aggregated model of city-wide transit operations is presented, which relates the optimal headway of transit service and the resulting cost per passenger to the demand for transit service. This model is important, because transit service can be adapted to the number of users, and this has implications for the efficiency of the transit system and the generalized cost for the average transit trip. In the following section, the models for cars and transit will be connected to identify the equilibrium of car and transit use.

3.1. Traffic states in urban street networks There are many approaches to modeling traffic in urban networks. Many modeling approaches seek to explicitly represent the detailed interconnections between links in a network. Simulation models, for example, require that each link be represented based on its geometric and operational characteristics and that the timedependent demands for travel from origins to destinations are explicitly represented (Dowling et al., 2004). Traffic assignment models are used to predict how each trip will be routed through the network, taking into account the effects of all other trips. Even a static traffic assignment model requires extensive data on network infrastructure and demand patterns. Dynamic traffic assignment models not only require extensive data inputs, but often involve computationally intensive solution algorithms. There are advantages to detailed modeling approaches in that they can be

q(

)

5

used to study the detailed effects of small changes in the network or impacts of spatially-detailed policies. The drawback is that the data requirements and computational complexity of detailed models hinder the ability to systematically study network-wide policies and traffic phenomena. What aggregate models lack in detail, they make up for in parsimony (Daganzo et al., 2012). The advantage of aggregate traffic models is that they focus on relationships between network-wide values, such as total accumulation of vehicles, average traffic speeds, and average flows on the network (Herman and Prigogine, 1979; Ardekani and Herman, 1987; Williams et al., 1987; Daganzo, 2007). These models are appealing to engineers and economists, because they present analytically tractable relationships between variables that can be measured or estimated relatively easily. A consistent relationship between average vehicle flow, q (veh/lane-hr), and average vehicle density, k (veh/lane-km), in a network has come to be known in the transportation engineering field as a Macroscopic Fundamental Diagram (MFD), which is a property of the network that can be predicted theoretically based on the free flow vehicle speed, jam density, saturation flow per lane, block lengths, and signal timings (Daganzo and Geroliminis, 2008). Recent studies have begun investigating the effect of transit operations on the shape of the MFD itself (Zheng and Geroliminis, 2013). The existence of an MFD for a network is supported by empirical evidence from real-world measurements in Yokohama, Japan (Geroliminis and Daganzo, 2008), Toulouse, France (Buisson and Ladier, 2009), and Minneapolis, USA (Geroliminis and Sun, 2011), and from simulations of San Francisco, USA (Geroliminis and Daganzo, 2007), Amsterdam, Netherlands (Ji et al., 2010), and Nairobi, Kenya (Gonzales et al., 2011). Fig. 5 shows a comparison of the MFD for Yokohama, San Francisco, and Nairobi. In all cases, the MFD represents flow as a concave function of density. The downward sloping side of the MFD (to the right of the peak) represents the hypercongested traffic states in the network. Theoretically, the shape of the MFD depends on characteristics of the network including the block lengths, intersection controls, and link-level fundamental diagrams (Daganzo and Geroliminis, 2008). This accounts for some of the differences in free flow speed and maximum network capacity among the MFDs in Fig. 5. Additionally, Geroliminis and Daganzo (2008) show empirically from measurements in Yokohama that the average trip length tends to be constant over time. This means that the MFD also

0.15

0.125 Yokohama, Japan

0.1 0.075

San Francisco, USA

0.05 0.025

Nairobi, Kenya

0 0

0.02

0.04

0.06

0.08

0.1

k(

)

Fig. 5. Comparison of empirical Macroscopic Fundamental Diagrams (MFD) from measurements in Yokohama, Japan (Geroliminis and Daganzo, 2008), simulations of San Francisco, USA (Geroliminis and Daganzo, 2007), and Nairobi, Kenya (Gonzales et al., 2011).

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defines a consistent relationship between the total number of vehicles in the network and the rate that vehicles exit the network. Vehicles exit either by driving outside the edge of the network or reaching their destination and parking within. The relationship between total exit rate and vehicle accumulation has been distinguished from the MFD, which relates average flow to average density, as the Network Exit Function (NEF) (Gonzales and Daganzo, 2012). Although the MFD presents a flow-density relation for networks that appears to be very similar to the fundamental diagram for a single link (Fig. 1(a)), traffic in networks differs from traffic on a link in a very important way. At a single bottleneck, the capacity for vehicles to depart the bottleneck is always μ, no matter how long the queue grows. In a network, queues of vehicles on one link back up to block flows on other links. As queues develop in a network, the increasing accumulation of vehicles changes the rate that vehicles can exit the network, so an urban street network has state dependent capacity defined by the NEF. A hypothetical MFD for a network is described by a concave function Q(k) as shown in Fig. 6(a). This function may be piecewise linear for an analytically approximated MFD (Daganzo and Geroliminis, 2008), or it may be defined by a curve fitted to measured data (Daganzo et al., 2012). The corresponding NEF is defined by scaling Q(k) by the total length of the network l (lanekm) and the average length of a trip d (km) l n FðnÞ ¼ Q ð11Þ d l

maintaining, insuring, fueling, and parking the vehicle.4 The generalized cost of a trip completed by car follows a similar structure to (6), now expressed as a function of the vehicle accumulation in the network

where k has been replaced by n=l and the flow (veh/lane-hr) is scaled to the exit rate from the network (veh/hr) by l=d (Daganzo, 2007). As shown in Fig. 6(b), the NEF is a scaled version of the MFD, so the concave flow function Q(k) will always correspond to a concave exit function F(n). The NEF is useful for modeling traffic in cities, because the rate that trips are completed and depart the network depends only on n.3 This forms the basis of what Arnott (2013) calls a bathtub model in which network traffic can be viewed as an aggregate quantity, and the rate that the network “drains” depends only on how full it is. The NEF exhibits properties of real urban networks in that increasing vehicle accumulations result in increasing exit flows up to a maximum flow associated with nC and exit rate μC (see Fig. 6 (b)). To the right of C, hypercongested states occur as further increases in vehicle accumulations result in reduced flows, exit rates, and traffic speeds. A state of complete gridlock or traffic jam occurs when the accumulation reaches nJ and FðnJ Þ ¼ 0; point J in Fig. 6. The traffic speed associated with density k is graphically represented in Fig. 6(a) as the slope from the origin to the corresponding traffic state on the MFD; vðkÞ ¼ Q ðkÞ=k. The corresponding slope on the NEF (Fig. 6(b)) is FðnÞ=n. By substituting (11) for F(n), and kl for n, it follows that FðnÞ=n ¼ vðkÞ=d. Thus, the slope to a traffic state on the NEF represents the inverse of the travel time to complete a trip of length d in the network. This travel time can be expressed in terms of the vehicle accumulation as

_ ¼ A_ c ðtÞ  FðnðtÞÞ nðtÞ

TðnÞ ¼

n : FðnÞ

zc ðnÞ ¼ zc0 þ

3 Traffic states on the NEF correspond to steady state conditions. When demand changes suddenly, the duration of the transition is comparable to the duration of a trip (Daganzo, 2007), so the approximation is good if steady conditions are maintained for a rush that is much longer than a trip time (Gonzales and Daganzo, 2012) and if traffic is uniformly dispersed on the network (Geroliminis and Sun, 2011).

ð13Þ

where zc0 represents the fixed costs of the trip expressed in units of travel time. This cost function may also vary with time to the extent that the vehicle accumulations vary over time as described by n(t). 3.2. Traffic dynamics in urban street networks With the MFD and NEF describing the traffic states that can occur in a network, queuing diagrams can be employed to track the dynamics of network traffic states. The traffic conditions in a rush period can be determined as a consequence of the number of vehicles entering the network over time and the resulting departures from the network as described by the NEF. The cumulative arrivals of cars into the network are described by Ac(t), which accounts for vehicles driving into the network at the edges or starting their trip from a parking space within. The rate of _ c ðtÞ ¼ FðnðtÞÞ. Unlike departures from the network at time t is D the single bottleneck case in which the queue always discharges at rate μ, the departure rate from a network is a function of the traffic state variable n(t). The dynamics of traffic in a network are governed by the following ordinary differential equation: ð14Þ

which follows from (2). Solving this ODE is not trivial, because F(n) may take many possible forms. The complete D(t) can be constructed numerically using Euler's method to track n(t) based on A (t) and an initial accumulation of vehicles, nð0Þ. Queuing diagrams are shown for three example arrival curves in Fig. 7 for traffic in a network with a NEF as shown in Fig. 6(b). In all cases, the initial accumulation of vehicles is assumed to be nð0Þ ¼ 0. The arriving demand rate during a peak period t A ð0; t P Þ is λ, and if all users travel by car, this implies A_ c ðtÞ ¼ λ during that period.5 The cases are as follows:

 Case 1: λ r μC (Fig. 7(a)) – The accumulation of vehicles



ð12Þ

The generalized cost of completing a trip in the network by car includes the travel time T(n) and other fixed costs associated with each trip including the expenses associated with owning,

n FðnÞ



increases as traffic enters the network until the accumulation _ c ðtÞ. satisfies FðnðtÞÞ ¼ λ. This is equivalent to saying A_ c ðtÞ ¼ D _ ¼ 0, so the traffic Following (2), this condition implies that nðtÞ has reached a steady state. This steady traffic state will be on the left side of the NEF, because the accumulation stabilizes as soon as the condition is met. There would be no way for additional vehicles to crowd the network and induce a hypercongested traffic state. The steady traffic state remains until the arrival rate drops at time tP. Case 2: λ 4 μC , nðt P Þ o nJ (Fig. 7(b)) – As the accumulation of vehicles increases, the departure increases to μC and then decreases as the rate of vehicle arrivals exceeds the rate of departures, casting the network into hypercongested conditions (shown by the dashed line). If the rate of vehicle arrivals drops below the departure rate before n(t) reaches nJ, then D_ c ðt P Þ 40, and the network can recover as the accumulation of vehicles diminishes. Case 3: λ 4 μC , nðt P Þ Z nJ (Fig. 7(c)) – The accumulation of vehicles increases as in Case 2. The difference is that with high

4 Details about how car costs relate to trip parameters, such as d, are available in Gonzales (2011). 5 For the network, we use λ to denote aggregate vehicles per hour as represented on the NEF, whereas q is left to denote veh/lane-h as represented on the MFD.

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q(

µ(

)

C

qC

7

)

C

µC

E

E Q(k)

v(k )

F(n)

1 T(n )

J

J k

k

k

n

k

n

n

n (veh)

Fig. 6. The MFD and NEF for a hypothetical network. Traffic states to the right of point C represent hypercongested traffic conditions in the network (dashed lines). (a) Macroscopic fundamental diagram (MFD). (b) Network exit function (NEF).

N (veh)

A (t) D (t)

t

t (hrs)

A (t)

N (veh)

N (veh) A (t) D (t) n(t ) < n

C n D (t)

J C

C t

t (hrs)

t

t

t (hrs)

_ Fig. 7. The queuing diagram for traffic in a network shows that departure rate, DðtÞ, is state-dependent and deteriorates in hypercongested conditions (dashed lines). (a) Case 1: λA o μC , (b) Case 2: λA 4μC , nðt P Þ o nJ and (c) Case 3: λA 4 μC , nðt P Þ Z nJ .

enough demand there will be a time t J r t P when nðt J Þ ¼ nJ . The consequence is that the network reaches a state of complete gridlock at time tJ after which no vehicle is able to exit the network, even after the arriving demand diminishes. The shape of the queuing diagram for the network is different from the queuing diagram for a single bottleneck with fixed capacity in Fig. 3. The reduced exit rate from the network when traffic conditions become hypercongested contributes to larger total delays experienced by travelers (i.e., the area between Ac(t) and Dc(t)). If the demand is high enough, the network can reach a state of complete gridlock in which all vehicles come to a complete stop, and this is the only stable hypercongested state in an unpriced network where all users travel by car. Of course, conditions of total gridlock as illustrated in Fig. 7(c) are not observed in reality, because travelers are not willing to tolerate infinitely long travel times. In reality, the rate that vehicles arrive into the network is not independent of conditions in the network, because drivers can make choices about when to travel. Although an elegant equilibrium solution exists for the trip scheduling problem at a single bottleneck with a wide range of preferred passage time distributions and schedule penalty functions (Smith, 1984; Daganzo,

1985), the equilibrium for drivers in a hypercongested city has been recognized as an intractable problem due to the complex relationship between dynamic traffic conditions and travel times in the network (Arnott, 2013). The challenge is that a hypercongested network is only in a steady state if A_ c ðtÞ ¼ FðnðtÞÞ or the traffic reaches a state of complete gridlock. Equilibrium queues must grow and recede at rates related to when each user will depart the network, but this depends on the evolution of the vehicle accumulation after the start of his or her trip.6 The availability of transit as an alternative to driving can provide a bound for the delays that can develop, and this will change the nature of the user equilibrium. We first consider a more realistic model for the average generalized cost of transit. In the following section, the model for traffic dynamics in a street

6 It has been shown that system optimal dynamic prices can incentivize drivers to arrive at a rate that sustains a specific traffic state during a rush period (Geroliminis and Levinson, 2009; Gonzales and Daganzo, 2012; Daganzo et al., 2012). This result is convenient, because a sustained exit rate allows a network to behave like a fixed-capacity bottleneck, so optimal pricing for bottlenecks can also apply to networks.

Please cite this article as: Gonzales, E.J., Coordinated pricing for cars and transit in cities with hypercongestion. Economics of Transportation (2015), http://dx.doi.org/10.1016/j.ecotra.2015.04.003i

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8

network will be paired with the transit cost function in order to identify the equilibrium with both cars and transit.

where the first term is the time required to travel a unit distance at free-flow speed vf, and the second term is the loss time for vehicle stops based on the loss time per stop τs and stop spacing s.

3.3. Transit in urban networks 4. Equilibrium in a network with transit in dedicated lanes A simple analysis of traffic in networks with competing transit service may treat the transit operations as a fixed design. If the route network and headways of service are planned exogenously, then the average cost to complete a trip of length d in the city may be represented by a fixed cost zt as presented in the single bottleneck case (Section 2.3). However, a responsive transit agency is able to adjust the amount of transit service provided to match the demands for the service. Aggregate models of transit operations in cities provide analytical approximations for the minimum generalized cost to completing a trip by transit, including travel time and agency costs, when the headway of service is optimized for the rate of transit demand. Simple models for the performance of transit on grid networks date back to Holroyd (1967) and have been used recently to compare the performance of radial, grid, and hybrid network structures (Daganzo, 2010). The costs for users and operating agencies can be associated with the performance of a transit system based on a small number of design variables (e.g., route spacing, stop spacing, and service headway) and the number of people using the system. For the present paper, suppose that the layout of routes and stops in a network is fixed but the service headway on the routes, H, can be optimized based on the demand. The generalized cost of a trip by transit depends on a number of characteristics of the physical transit operations and the costs associated with capital and operations. We are interested in how the generalized cost of a transit trip depends on the demand for transit, λt (trips/hr). The average cost of a transit trip, expressed in units of travel time is sffiffiffiffiffiffiffiffiffi d zt0 lt zt1 z t ðλ t Þ ¼ τ a þ þ þ 2 ð15Þ vt λ t vt λt where τa is the time for a passenger to access a transit stop or station (hr), d is the length of a trip (km), vt (km/hr) is the average speed of transit including stops for passengers, zt0 is the total generalized cost of maintaining infrastructure per hour of operation, lt is the total directional route length of the transit system, and zt1 is the generalized cost per vehicle hour of operation. Endogenously determined within this model is the optimal headway of transit service, which is assumed to be the same on each line within the network. See Appendix A for a more complete description of the components of travel time and cost and the derivation of zt ðλt Þ. An important characteristic of the average generalized cost of a transit trip, zt ðλt Þ, is that it is a decreasing function of the demand λt, because transit exhibits economies of scale. As the demand for transit increases, the optimal headway decreases, because the agency provides more service to reduce the time that passengers spend waiting. Furthermore, the fixed costs for the transit agency zt0 are divided among the users, so increased demand reduces the share attributed to each trip. The speed of transit is also an important component of the generalized cost of a transit trip. When transit operates on a dedicated right of way (e.g., dedicated bus lanes) the speed of transit is not impacted by traffic conditions on the network, and vt may be considered to be a fixed value. A reasonable estimate of the average transit speed in dedicated lanes is  vt ¼

1 τs þ vf s

1 ð16Þ

Equipped with models of network traffic and transit, we are now prepared to return to an analysis of equilibrium traffic states. We shall consider a city in which a large number of users wish to travel at rate λ during a peak period, t A ð0; t P Þ, and who otherwise have identical preferences (e.g., trip length, value of time, preference for car versus transit, etc.). A transit system operates in dedicated lanes so that the speed of transit, vt, is independent of the accumulation of vehicles in the network, n(t), and the transit vehicles have sufficient capacity so that passengers are always able to board the first vehicle that arrives at a stop. We will first consider the equilibrium that results when there is no pricing interference with the modes. Then, a method for implementing optimal coordinated fixed prices for cars and transit will be presented. Finally, a numerical example will be presented to illustrate the results. 4.1. Unpriced cars and transit We first consider the case of unpriced modes in which each traveler experiences the generalized cost of the mode that he or she chooses. That is to say that there is no additional toll or fare charged above the generalized cost of the driving or using transit and no subsidy provided to allow a traveler to experience a lower cost. In equilibrium, travelers are expected to use the mode with the lowest cost at each time during the rush (Wardrop, 1952). In addition to the monetary and travel time costs, travelers may experience additional schedule costs if they must exit the network before or after their preferred time. Gonzales and Daganzo (2012) show that a transit system that is not capacity-constrained allows passengers to exit the network at their desired time. Therefore, when cars and transit are used simultaneously, the costs of a trip by each mode are as described by (13) for the car cost, zc(n), and (15) for the transit cost zt ðλt Þ. The equilibrium for cars and transit in a network is identified by following a similar line of logic as used for a single bottleneck in Section 2.3. We start by comparing the generalized cost of using each mode at the beginning of the rush when nð0Þ ¼ 0. Using (13), we can approximate the cost of a car trip in the network at this initial state by evaluating the following limit: lim zc ðnÞ ¼ zc0 6 þ lim

n-0

n-0

n d ¼ zc0 þ FðnÞ vf

ð17Þ

where vf is the average speed of a free-flow trip in the network, and d=vf is the free-flow travel time for a trip of length d. The car cost function, zc(n), is a monotonically increasing function with limit limn-nJ zc ðnÞ ¼ 1, because F(n) is concave and Fð0Þ ¼ 0.7 Therefore, zc(n) will always be greater than the limit expressed in (17). Since transit exhibits economies of scale, the least possible cost of a transit trips is zt ðλÞ when all trips are served by transit. This is the appropriate cost of transit to compare against the initial cost of driving, because all users will choose transit if 7 The proof follows directly from the definition of concavity: if F(n) is concave and Fð0Þ ¼ 0, then Fðαn þ ð1 αÞ0Þ Z αFðnÞ þ ð1  αÞFð0Þ for all α A ð0; 1Þ. It follows that FðαnÞ=αnZ FðnÞ=n or, equivalently, αn=FðαnÞ r n=FðnÞ for all 8 α A ð0; 1Þ, which means the travel time is not less for n than at any accumulation between 0 and n. As a result zc(n) is monotonically increasing, because it is composed of the travel time and an additive constant cost.

Please cite this article as: Gonzales, E.J., Coordinated pricing for cars and transit in cities with hypercongestion. Economics of Transportation (2015), http://dx.doi.org/10.1016/j.ecotra.2015.04.003i

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zt ðλÞ r limn-0 zc ðnÞ. This case is not particularly interesting, because all travelers would choose transit and cars would never be competitive. We will now consider the more interesting case that limn-0 zc ðnÞ r zt ðλÞ, in which drivers at the beginning of the rush choose only to drive, and A_ c ð0Þ ¼ λ. From this initial condition, vehicles will begin to accumulate in the network as described in Section 3.2. The objective is to identify the critical accumulation of vehicles, ncrit, at which the cost of driving is equal to cost of using transit, and travelers are willing to use both modes. For the same reasons as in the bottleneck case, the accumulation of vehicles cannot be changing while both modes are used, because this would give one mode a cost advantage over the other. _ ¼ 0 on (14), it follows that A_ c ðtÞ ¼ Fðncrit Þ. Thus, By imposing nðtÞ (9) and (10) can be rewritten for the network cases as ( λ if nðtÞ o ncrit _ ð18Þ A c ðtÞ ¼ Fðncrit Þ otherwise A_ t ðtÞ ¼ λ  A_ c ðtÞ:

ð19Þ

These conditions imply that the arrival rate for transit when both modes are used depends on these accumulation of vehicles in the system

λt ðncrit Þ ¼ λ  Fðncrit Þ

ð20Þ

An unpriced equilibrium in which both modes are used is achieved at the first accumulation that reaches the value ncrit such that zc ðncrit Þ ¼ zt ðλ Fðncrit ÞÞ:

ð21Þ

In order to identify the critical accumulation, we will define the transit demand more generally as λt ðnÞ ¼ λ  FðnÞ so that the cost functions for cars and transit can be compared across all possible vehicle accumulations. Example cost functions and the corresponding queuing diagram are shown in Fig. 8 for cars and transit reaching an equilibrium at ncrit ¼ nE at time t1. Recall that the car cost function, zc(n), is a monotonically increasing function with limit limn-nJ zc ðnÞ ¼ 1. The transit cost function depends on the magnitude of λ relative to the maximum exit flow that the network can support, μC . If λ 4 μC , λt ðnÞ is positive valued for all possible vehicle accumulations. Since F(n) is concave and unimodal with a maximum value at nC, it also follows from (20) that λt ðnÞ is convex and unimodal with a minimum value at nC. We know that zt ðλt Þ is monotonically decreasing, so the shape of λt ðnÞ implies that zt is a continuous unimodal function of n with maximum value at nC. The minimum cost of transit is zt ðλÞ for n ¼ f0; nJ g, at which FðnÞ ¼ 0. Since limn-0 zc ðnÞ r zt ðλÞ o 1, the transit cost zt ðλt ðnÞÞ must cross zc(n) at least once from above at a stable equilibrium point, shown by E in Fig. 8(a). Depending on the shapes of the cost functions, there may be multiple vehicle accumulations that satisfy (21). Of these potential

z( )

9

values, the left-most (smallest) value of ncrit will be the stable equilibrium that is achieved, because the cost of transit is greater car at n¼0 so the transit cost must cross the car cost from above. Small perturbations will affect the relative costs of the modes to return the system to this point. If the total demand is low enough that λ o μC , it is possible for the network to serve all trips by car as shown in Fig. 7(a). As a result, there are a range of vehicle accumulations for which FðnÞ 4 λ that will never be achieved as a steady state traffic condition; i.e., n A ðnλ1 ; nλ2 Þ are infeasible, where nλ1 is the point left of C at which Fðnλ1 Þ ¼ λ, and nλ2 is the point right of C at which Fðnλ1 Þ ¼ λ. The cost of transit based on (20) is undefined in this interval, because the traffic states cannot be sustained if the demand λ is lower than the exit rate from the network. The cost of a transit trip remains equal to zt ðλÞ at n ¼ f0; nJ g, as explained above. The limit of the transit cost as ncrit approaches the boundaries of this interval are lim zt ðλt ðnÞÞ ¼ lim zt ðλt ðnÞÞ ¼ 1:

n-nλ1

If the equilibrium point E lies at an accumulation nE o nλ1 , a stable equilibrium with transit will be reached. If the equilibrium point E lies at an accumulation nE 4 nλ2 , the network will reach a stable traffic state on the right side of the NEF with all trips served by cars. Whether or not both modes are used, depends on the specific cost functions for cars and transit. Identifying the stable bimodal equilibrium, if it exists, is important for two reasons. First, the equilibrium value of ncrit is the greatest number of vehicles that are expected to circulate in the city, because transit becomes competitive at the equilibrium and serves the additional arriving demand. Second, the total cost for all users of the network can be approximated by the equilibrium condition when the rush is long compared to the period of time when the network is loading ð0; t 1 Þ. A more detailed estimate of the equilibrium A(t) and D(t) curves at the beginning and end of the rush when travelers experience schedule delay may be achieved by using a numerical method such as the one proposed in Geroliminis and Levinson (2009) or Arnott (2013), but there is no exact analytical solution for this equilibrium problem in networks. The total cost per hour for all users during the steady equilibrium period is ZðnE Þ ¼ λzc ðnE Þ, because all users of cars and transit experience the same equilibrium cost. 4.2. Coordinated fixed prices for cars and transit The equilibrium state defined by ncrit can be manipulated by changing the relative cost of driving versus riding public transit using pricing. A price pc can be applied to each car trip so drivers experience a cost of zc ðnÞ þ pc . Likewise, a price pt can be applied to

A(t)

N (veh)

z (n)

ð22Þ

n-nλ2

A (t) n

F(n )

E z( ) z +

d v

E z ( (n))

n

C n

D (t)

F(n )

E n =n

n

n (veh)

t

t

t (hrs)

Fig. 8. Equilibrium with cars and transit is achieved at the value of ncrit ¼ nE where the cost per trip is the same for both modes. Cost curves shown for the case that λ 4μC . Hypercongested traffic conditions are indicated by the dashed curve. (a) Generalized cost per trip. (b) Queuing diagram.

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10

µ (veh/hr) 35,000

C

µC = 33,168

30,000

25,000

20,000

F(n)

15,000

10,000

5,000 nC = 8217 0 5,000

0

10,000

15,000

20,000

25,000

n (veh) Fig. 9. Network exit function for Yokohama, Japan (source of data points: Geroliminis and Daganzo, 2008).

transit so that users experience a cost of zt ðλt Þ þ pt . Prices may be positive if they represent an extra charge to increase the costs of travel and negative if they represent a subsidy to reduce the cost of travel. Here prices are expressed in units of equivalent units of travel time. With prices, the equilibrium condition from (21) must hold at the critical accumulation of vehicles that makes the userexperienced costs including prices the same for both modes zc ðncrit Þ þ pc ¼ zt ðλ  Fðncrit ÞÞ þ pt

ð23Þ

This condition can be rewritten in terms of the difference of prices, Δp ¼ pc  pt

Δpðncrit Þ ¼ zt ðλ  Fðncrit ÞÞ  zc ðncrit Þ: which

depends

on

ncrit.

Defining

ð24Þ (24)

more

generally

as

ΔpðnÞ ¼ zt ðλt ðnÞÞ  zc ðnÞ allows us to compare coordinated prices

across all feasible vehicle accumulations. Although there is flexibility in the specific prices that are charged to car and transit users, it is important that these prices are coordinated to maintain ΔpðnÞ. We will call ΔpðnÞ the coordinated prices, because it defines the difference between pc and pt, which is the relevant coordination of prices that is required to achieve a steady state equilibrium at n. We will consider coordinated prices that do not change over time, which are most easily implemented by applying fixed prices to one or both modes. The choice of Δp allows a city to choose the equilibrium point nE to achieve objectives such as preventing hypercongested traffic conditions from occurring in the network. A reasonable objective is to minimize the total generalized cost of transportation for all travelers. Although optimal time-dependent prices for cars and transit have been identified in Gonzales and Daganzo (2012, 2013), fixed coordinated prices are much easier to implement with existing technologies and political constraints. The total cost per hour during the middle of the rush is the sum of the costs for all car users and all transit users ZðnÞ ¼ FðnÞzc ðnÞ þ ðλ  FðnÞÞzt ðλ  FðnÞÞ:

ð25Þ

The first term is the product of the rate that trips are served by car and the generalized cost per car trip. The second term is the product of the remaining rate of trips served by transit and the total cost per transit user. Since the units of zc and zt are in equivalent hours of passenger travel time per trip, the units of Z are equivalent hours of travel time per hour of rush period (hrs/ hr). This cost could be converted to monetary values, by multiplying it by the average value of time (e.g., dollars per hour). To minimize the total transportation cost for cars and transit with fixed prices, the optimal vehicle accumulation is   nn ¼ arg min ZðnÞ : n A ½0; nJ ; FðnÞ r λ : ð26Þ n

The first constraint requires that the vehicle accumulation be nonnegative and not exceed the jam density. The second constraint ensures that when λ o μC only feasible accumulations are considered; i.e., n2 = ðnλ1 ; nλ2 Þ. A closed-form analytical expression for nn can only be derived for some specific functional forms of F(n), but it is easily solved with a computer for a general F(n) and mode costs. Based on the characteristics of F(n), the optimal vehicle accumulation nn is never a hypercongested traffic state. For any potential value of ncrit in a hypercongested state, there is an uncongested state with the same network exit flow and lower vehicle accumulation. For example, for every unpriced equilibrium accumulation in the hypercongested range of traffic states (nE 4nC ), there is an accumulation n0E o nC such that Fðn0E Þ ¼ FðnE Þ, so zc ðn0E Þ o zc ðnE Þ. The cost of transit is the same in both cases, because λ Fðn0E Þ ¼ λ  FðnE Þ. Therefore, the total cost Zðn0E Þ oZðnE Þ is an improved total cost that can be achieved with coordinated fixed prices. This also provides an upper bound for the value of the optimized total cost, because Zðnn Þ r Zðn0E Þ. The extent to which a further reduction in total cost can be achieved by adjusting ncrit from n0E depends on the shapes of the cost functions for cars and transit. The coordinated prices to move an unpriced equilibrium from the hypercongested region to the uncongested region will always be positive, Δpðn0E Þ 4 0, because transit is too costly for users to choose at the lower vehicle accumulations based only on the generalized costs of the modes. A positive coordinated price

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implies that pc 4 pt . This can be accomplished by charging a toll to drivers, subsidizing transit, or implementing some combination of both policies. Transit subsidies are common in many cities, and a fare-free system may present a practical lower bound for the price of transit. There may be political barriers to charging a toll to drive on streets that have previously been free, so pc could be indirectly increased through policies like pricing parking.

4.3. Numerical example We now present a numerical example to illustrate the characteristics of this equilibrium and the effect that the choice of ncrit has on the performance and costs of the system. The example is based on the well-defined NEF of Yokohama, Japan, that was measured using taxi data (Geroliminis and Daganzo, 2008) and later fitted with a polynomial function (Daganzo et al., 2012). The network covers an area of roughly 100 km2, the network length is l ¼157 lane-km, and the average length of a vehicle trip within the network is d ¼2.3 km. Fig. 9 shows observations up to an accumulation of approximately 14,000 vehicles, which is associated with an average traffic speed of 4.53 km/hr. The accumulation reached the same limit on both days that data were available, suggesting that this is the traffic state beyond which travelers are unwilling to choose car over other modes. For this example, we are primarily interested in identifying efficient equilibrium traffic states that can be achieved to the left of C. These traffic states can be reliably estimated by a curve fitted to the data. To provide a complete illustration, we extend F(n) beyond the measurements all the way to a gridlocked state. The shape of F(n) is constrained to be concave and the maximum vehicle accumulation should be associated with a feasible jam density, such as kJ ¼160 veh/lane-km, which implies that nJ ¼25,000 veh. The shape of the hypercongested branch of the NEF could be piecewise linear according to analytical approximation methods (Daganzo and Geroliminis, 2008), but simulated data suggests some curvature (see Fig. 5). We use the following fitted function: ( FðnÞ ¼

2:28  10  8 n3  8:62  10  4 n2 þ 9:58n

for n A ½0; 13; 000

 9:50  10  5 n2 þ 1:20n þ 2:94  104

for n A ð13; 000; 25; 000

ð27Þ This network can serve a maximum exit rate of μC ¼ 33; 168 veh=hr when the vehicle accumulation is nC ¼8,271 veh. In Fig. 9, F(n) is denoted by a black curve over the range of observations (n r 14; 000) and by a gray curve where the shape is assumed for the sake of illustration (n 4 14; 000). Using values originally derived and compiled in Gonzales (2011) and valuing time at $20 per hour, the following cost parameters are used to model hypothetical car and transit costs in the network: zc0 ¼ 0:0297 hr=trip, vt ¼ 14.2 km/hr,8 zt0 ¼ 0 for simplicity, zt1 ¼ 3:9 hr=veh  hr, and τa ¼ 0:139 hr=trip. With these coefficients the lowest cost that a transit can approach with very high demand is limλt -1 zt ðλt Þ ¼ τa þ d=vt ¼ 0:30 hr, and this will always be more costly than a free-flow car trip, because limn-0 zc ðnÞ ¼ zc0 þ d=vf ¼ 0:13 hr. We consider the effect that λ and n have on the mode costs and coordinated prices by comparing several levels of λ over the range of possible ncrit. Each combination of λ and n is associated with an equilibrium with cars and transit that would be achieved with ΔpðnÞ calculated from (24). The resulting total cost is given by (25). 8 The transit speed is given by (16), based on vf ¼ 22 km/hr from the slope of the NEF at n¼ 0. The loss time per stop is assumed to be τs ¼ 0:0125 hr=stop, and the stop spacing is s ¼0.5 km/stop.

11

Fig. 10 shows the cost per trip and total system cost for all feasible n and the coordinated price ΔpðnÞ that makes travelers willing to use cars and transit in order to sustain the accumulation of n vehicles as a steady state in equilibrium. The unpriced equilibrium is shown for each value of λ with an open dot. The equilibrium resulting from coordinating prices to minimize the total cost is shown with a filled dot. A half-filled dot is used when these equilibrium states coincide. All curves are shown in black over the range where the NEF conforms to observed values and in gray over the range where the shape of the NEF is assumed. For values of λ A f35; 000; 45; 000; 55; 000; 65; 000g 4 μC , the generalized cost of a transit trip is defined for the full range of n A ½0; nJ . The unpriced equilibrium is at the intersection of zc(n) with zt ðλt ðnÞÞ, which is also where the coordinated prices are ΔpðnÞ ¼ 0. For the NEF and cost functions used in this example, the unpriced equilibrium states are all at hypercongested states. These types of functions are typical of wealthier cities where the slower speed of transit is a major competitive disadvantage against car. The optimal equilibrium with coordinated fixed prices is always at a traffic state left of nC corresponding to the minimum of Z(n). As the total demand increases, the priced optimum moves to less congested traffic states (to the left), because increasing transit ridership makes the transit system more efficient and competitive. For λ A f15; 000; 25; 000g o μC , the demand is low enough that all trips can be served by car without reaching hypercongested states, and this is associated with a lower cost than any of the alternatives that involve transit. There are values of n that are infeasible, so the unpriced equilibrium is associated with everyone traveling by car at an accumulation of nλ1 , as described in Section 4.2. For the NEF and cost functions used in this example, the unpriced and optimized equilibrium coincide. A comparison of the numerical values for the unpriced user equilibrium and the optimized coordinated prices is presented in Table 1. Numerical values of ncrit are shown for the unpriced equilibrium, denoted by nE, and with optimal coordinated prices, denoted by nn. Note, for example that for every value of λ, nE Z nn and nn o nC ¼ 8217 veh=hr. Table 1 also presents the rate of car use, Fðncrit Þ, and transit use, λt ðncrit Þ, in the equilibrium in order to provide a sense of the mode share in the steady state. The social benefit of implementing a coordinated pricing policy can be estimated by comparing the total cost of the system optimum and unpriced user equilibrium. In Table 1, the percent change in total cost is expressed relative to the total cost of the unpriced user equilibrium, ZE. In the low demand cases, where both equilibrium states coincide, there is no need to implement pricing. However pricing cars and transit provides large benefits for congestion reduction when λ 4 μC . With increasing λ , the benefit diminishes along with the optimal difference of coordinated prices. This is because there are many more transit users in both the unpriced and priced cases as total demand increases. With more transit users, the cost per user is lower, and both the unpriced equilibrium and the system optimum are associated with less congested traffic states. This implies that the densest cities will not have the worst traffic congestion, because there is enough demand to support efficient alternative transit systems. Table 1 includes one other parameter of the equilibrium state, which is relevant to the dynamics of reaching the stable equilibrium during a rush period. The time t1 indicates the elapsed time from the start of the rush until the stable equilibrium with cars and transit is reached (see Fig. 8(b)). The value of t1 is useful to compare against the duration of a rush period to make a judgement about whether such an equilibrium state is likely to be reached. For example, it takes t 1 ¼ 3:34 hr for the accumulation of vehicles to reach the stable unpriced equilibrium state with λ ¼35,000 trips/hr, so a steady-state

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12

z (hrs/trip)

Z (hrs/hr)

1.50

5

r

s/h rip 0t ,00 5 = 6 ,000 55 00 ,0 45 000 , 35 00 ,0 25 0 ,00 15

1.25 4 = 15,0

1.00

3

00 t

zt( t(n))

ri p

0.75

s /h

25, 000 45,000 35,000 55,000 65,000

2

r

0.50 0.25

1

zc(n)

0

0 0

5,000

10,000

15,000

nC

20,000

25,000

0

5,000

n (veh)

10,000

nC

15,000

20,000

25,000

n (veh)

(hrs) 0.60 0.45

000

,00 5 0 5,

0

0

,00

45

00

25,000

35,

65

0.15

= 15,000 trips/hr

0.30

0

nC

-0.15 0

5,000

10,000

15,000

20,000

25,000

n (veh) Fig. 10. Costs and pricing for all vehicle accumulations that can achieve a steady equilibrium state with coordinated prices. Filled dots represent optimal coordinated fixed prices, open dots represent unpriced equilibrium, and half-filled dots are where the optimum and unpriced equilibrium coincide. (a) Average generalized costs, zc, zt. (b) Total generalized cost per hour, Z. (c) Coordinated prices, Δp. Table 1 Unpriced equilibrium and optimal coordinated fixed prices for dedicated transit lanes. Parameter

Units

Unpriced user equilibrium nE veh FðnE Þ trips/hr λt ðnE Þ trips/hr ΔpðnE Þ hrs ZðnE Þ hrs/hr t1 hrs Optimal coordinated fixed prices veh nn trips/hr Fðnn Þ n trips/hr λt ðn Þ hrs Δpðnn Þ hrs/hr Zðnn Þ t1 hrs Benefit of coordinated pricing hrs/hr Zðnn Þ ZðnE Þ %

Total demand, λ (trips/h) 15,000

25,000

35,000

45,000

55,000

65,000

1863 15,000 0

3749 25,000 0

2308

4491

14,217 27,259 7741 0 19,292 3.34

12,939 29,031 15,969 0 21,389 0.80

12,264 29,895 25,105 0 24,195 0.46

11,852 30,416 34,584 0 27,256 0.32

1863 15,000 0

3749 25,000 0

2308

4491

5323 30,008 4,992 0.41 9271 0.43

4882 28,878 16,122 0.28 13,386 0.19

4734 28,453 26,547 0.24 17,154 0.13

4650 28,200 36,800 0.22 20,784 0.10

0 0

0 0

 10,021  51.9

 8003  37.4

 7040  29.1

 6472  23.7

analysis is probably not appropriate for a typical morning or evening rush that lasts only a few hours. The duration of t1 declines quickly as total demand increases, and the values are

even lower when coordinated prices are implemented. For a typical rush period that may last a few hours, the steady state period is likely to be long compared to the time it takes for the

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network to build up the accumulation of vehicles to reach the stable equilibrium.

13

In wealthy cities, however, transit is usually subsidized. To analyze such cases, we must consider pricing. 5.2. Coordinated prices for cars and transit

5. Equilibrium in a network with transit in mixed traffic When transit operates in mixed traffic, the principles for equilibrium and pricing remain unchanged. However, transit operations are affected by traffic congestion on the network. The cost function for car trips remains as in (13). The cost function for transit must be revised to reflect the fact that vehicles can only travel between stops at the speed of the traffic in the rest of the network. In mixed traffic, the speed of transit depends on the average speed of traffic as defined by the traffic state on the NEF. Recalling from (12) that the travel time to complete a trip of length d by car in a network with an accumulation of n vehicles is n=FðnÞ, the average traffic speed associated with n is vðnÞ ¼ FðnÞd=n. Substituting v(n) into (16) in place of vf, the average transit speed for operations in mixed traffic is   n τs  1 þ vt ðnÞ ¼ : ð28Þ FðnÞd s Naturally, the speed of transit slows as the average speed of traffic in the network slows. The cost function for transit is revised by replacing vt with vt(n) in (15). Thus, the transit trip cost function depends on λt and F(n) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     n τs zt0 lt zt1 n τs þ þ þ þ2 zt ðλt ; FðnÞÞ ¼ τa þ d ð29Þ FðnÞd s λt λt FðnÞd s The transit demand, λt ðnÞ, depends on the accumulation of vehicles according to (20) in equilibrium when both cars and transit are used. The competitive advantage of transit is reduced in mixed traffic because the congested conditions that slow traffic also slows transit. 5.1. Unpriced cars and transit The equilibrium conditions when cars and transit are unpriced and operating in mixed traffic can be assessed following the same line of reasoning used to assess transit in dedicated lanes in Section 4.1. Users are expected to choose the mode with the lowest cost, and cars and transit are only used simultaneously if the generalized cost of travel is the same for cars and transit. The economies of scale for transit again imply that the lowest possible cost for a transit trip is achieved when all trips are served by transit. If zt ðλÞ rlimn-0 zc ðnÞ, then all trips will be served by transit, because cars are too costly even in free-flow traffic conditions. The result is the same for mixed or dedicated lanes, because the equilibrium involves transit use without any traffic on the streets. The more interesting case is when the cost of transit is always greater than a free flow car trip; i.e., limn-0 zt ðλt ðnÞ; FðnÞÞ 4 limn-0 zc ðnÞ. This is where the results for mixed traffic differ from the case of dedicated lanes. The slowing effect of traffic undermines the ability of transit to compete with cars. In fact, if transit in mixed traffic is not competitive with a free-flow car trip, the cost of a transit trip will be greater than the cost of a car trip at all values of n. A proof for this statement is provided in Appendix B. As a result, no stable equilibrium with transit can exist, and the system can only be represented with the dynamics of car traffic presented in Section 3.2. This result seems to counter the fact that we see transit in mixed traffic being used in real cities with hypercongestion. In real cities, the diversity of travelers means that there will always be at least some people (and perhaps many people) who find cars too costly even in free-flow conditions and who are willing to take transit even if traffic condition slows the system.

Coordinated prices for cars and transit when the modes operate in mixed traffic are defined in the same way as presented for dedicated lanes in Section 4.2. Users will be indifferent between cars and transit if the generalized cost including prices is the same for both modes, and this is the condition required for a stable equilibrium with travelers using cars and transit simultaneously. In order to achieve this condition at a vehicle accumulation of n, the prices for cars and transit must differ by ΔpðnÞ as defined in (24), with the only difference being that the speed of transit also depends on n. Any traffic state to the left of C, n onC , can be attained as a stable equilibrium with the appropriate price ΔpðnÞ. A proof for this statement is provided in Appendix C. The existence of the priced equilibrium is important, because it means that the availability of transit with a sufficient subsidy can allow cities that would otherwise be susceptible to unstable hypercongested conditions to operate with stable equilibrium states during busy periods of the day. The optimal coordination of prices minimizes the total cost per hour during the middle of the rush when both cars and transit are used simultaneously. The optimal vehicle accumulation that minimizes the total cost is still given by (26), with the only difference being that the total cost function is defined by ZðnÞ ¼ FðnÞzc ðnÞ þ ðλ  FðnÞÞzt ðλ  FðnÞ; FðnÞÞ

ð30Þ

because the vehicle accumulation determines the transit demand and the transit speed. The optimal accumulation, nn, will never be in a hypercongested traffic state, because the same flow of cars can be served at lower cost to the left of C. Since any accumulation to the left of C can be achieved with fixed coordinated prices, coordinated fixed prices can always be used to achieve an efficient stable equilibrium with cars and transit during a rush period with λ 4 μC . 5.3. Numerical example The numerical example based on Yokohama, Japan, is now used to illustrate the effect of operating transit in mixed traffic. Similar plots of the average cost of a trip by car, zc, and transit, zt, as well as the the total cost per hour, Z, are shown in Fig. 11. With the cost parameters used in this example, limn-0 zt ðλt ðnÞ; FðnÞÞ r limn-0 zc ðnÞ, so a stable unpriced equilibrium with both cars and transit does not exist, as explained in Section 5.1. There is no value of λ or n at which ΔpðnÞ ¼ 0. Only when the total demand is low enough that all trips can be completed by car (i.e., λ A f15; 000; 25; 000g o μC ) does a stable unpriced equilibrium exist with all travelers using cars (Fig. 11(a)). When coordinated prices can be implemented, a stable equilibrium with cars and transit appears for every case that λ 4 μC . As described in Section 5.2 and shown in Fig. 11(b), the priced equilibrium that minimizes the total cost is always to the left of C, meaning that optimally coordinated prices will not result in hypercongested conditions. The coordinated prices that are required to achieve equilibrium at each accumulation, ΔpðnÞ, are shown in Fig. 11(c). Transit will always need a subsidy in order to be used, but this subsidy provides a constraint on traffic congestion that prevents the costs of congestion from growing without bound. Table 2 provides numerical values associated with the optimal coordinated fixed prices for transit operating in mixed traffic. The unpriced equilibrium states are not reported, because they are either the same as the optimal case for λ o μC or the equilibrium

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14

z (hrs/trip)

Z (10,000 hrs/hr)

3.0

1.5

0 15,00

4

25,0 00

5,0 00 tri ps 55 /hr ,00 0 45 ,00 0 35, 000

2.0

=6

2.5

25,000

= 15,000 trips/hr

5

3

zt( t(n)) 35,000

2

1.0

00

00 55,0 45,0

00 65,0

0.5

1

zc(n)

0

0 0

5,000

10,000

nC

15,000

20,000

25,000

5,000

0

n (veh)

10,000

nC

15,000

20,000

25,000

n (veh)

(hrs)

1.00 0.75

25,000

= 15,000 trips/hr

1.25

35,000

0.50 65,000

45,000 55,000

0.25 0 0

5,000

10,000

nC

15,000

20,000

25,000

n (veh)

Fig. 11. Costs and pricing change depending on the choice n. The system optimum is shown with a filled dot. (a) Average generalized costs, zc, zt. (b) Total cost per hour, Z. (c) Coordinated prices, Δp.

Table 2 Optimal coordinated fixed prices for mixed traffic. Parameter

nn Fðnn Þ λt ðnn Þ Δpðnn Þ Zðnn Þ t1 Benefit of dedicated lanes Zðnn ÞDed  Zðnn ÞMix

Units

veh trips/hr trips/hr hrs hrs/hr hrs hrs/hr %

Total demand, λ (trips/hr) 15,000

25,000

35,000

45,000

55,000

65,000

1863 15,000 0

3749 25,000 0

2308

4491

5538 30,490 4510 0.57 9944 0.47

4326 27,158 17,158 0.36 14,880 0.16

3637 24,537 30,463 0.31 19,193 0.09

3100 22,095 42,905 0.28 23,241 0.06

0 0

0 0

 1494  10.0

 2038  10.6

 2457  10.6

does not exist. The same parameter values are provided as in Table 1: optimal accumulation of vehicles nn, the corresponding trips per hour served by car Fðnn Þ and by transit λt ðnn Þ, the coordinated prices for cars and transit that achieve an efficient equilibrium Δpðnn Þ, the total cost per hour of equilibrium Zðnn Þ, and the time t1 for the system to reach the equilibrium accumulation of vehicles. Values from Table 2 can be compared directly against values from Table 1. It is possible to estimate the social benefit of implementing dedicated lanes for transit during the rush by comparing the total cost with optimal pricing of dedicated lanes

 673  6.8

to the total cost of optimal pricing with mixed traffic. The difference is reported in Table 2 as Zðnn ÞDed  Zðnn ÞMix , where the subscripts distinguish total cost for dedicated lanes and mixed traffic operations. The total cost associated with the system optimum for dedicated lanes is unchanged when λ o μC , because the network has sufficient capacity to serve low demand without transit. For cases that λ 4 μC , the total cost associated with the system optimum is always greater with mixed traffic than with dedicated lanes, because the slower traffic speeds make travel for all road users more costly. Dedicated lanes allow lower subsidies (smaller

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Δp) to achieve equilibrium conditions with lower total social costs. As total demand λ increases, the value of operating transit in dedicated lanes increases, reaching about 10% reduction in total social cost relative to operating transit in mixed traffic conditions. Whether or not a city can dedicate lanes to transit, coordinated pricing provides a way to bound the cost of travel by car by preventing traffic conditions from becoming hypercongested.

6. Conclusion Urban street networks are known to exhibit hypercongested traffic conditions when too many vehicles crowd the infrastructure. Static models are problematic for representing traffic conditions in hypercongested networks, because they do not recognize the dynamics of traffic congestion that make some traffic states inherently transient. However, physically realistic models that relate the aggregate rate that trips are served by a network to the accumulation of vehicles within the network provide a basis for modeling congestion and hypercongestion in networks. This paper has made use of aggregate traffic models to show that when high-capacity transit service is available to travelers in a city, steady equilibrium states can arise in which cars and transit are used simultaneously. Recognizing that stable steady traffic states, including hypercongested states, can emerge from the dynamics of a peak period is valuable for identifying unpriced equilibrium states and simple pricing strategies to minimize the generalized costs of a rush period. In the case that the cost of serving all trips by transit is lower than the cost of a free-flow car trip (i.e., zt ðλÞ o zc0 þ d=vf ), the stable equilibrium state is for all travelers to use transit. Cars will never be competitive in such a city. More often, we are interested in cities where the generalized cost of a transit trip exceeds the cost of a free-flow car trip. In this case travelers have an incentive to travel by car until traffic congestion increases to the point that the cost of a car trip is equal to the cost of a transit trip, and users are willing to use both modes simultaneously. Three findings are presented in this paper for networks in which cars and transit operate without pricing interventions, and the cost of transit exceeds the cost of a free-flow car trip 1. When the peak demand is less than the maximum exit rate from the network (i.e., λ o μC ), the network can serve all trips by car in a traffic state that is not hypercongested. 2. When the peak demand exceeds the maximum exit rate from the network (i.e., λ 4 μC ) and transit operates in dedicated lanes, there is always at least one stable unpriced equilibrium state. Depending on the shape of the NEF and cost functions for cars and transit, the stable equilibrium traffic state that is reached without pricing may be a hypercongested traffic state. 3. When peak demand exceeds the maximum exit rate from the network and transit operates in mixed traffic, there is no stable unpriced equilibrium. Traffic congestion will always slow transit, keeping it less competitive than cars. A reasonable goal for a network manager would be to prevent accumulations of vehicles on a network from rising into hypercongested states. Traffic states on the increasing branch of the MFD and NEF can always serve the same flow with lower generalized costs. The paper shows that whether transit is operated in dedicated lanes or in mixed traffic, fixed coordinated prices for cars and transit always exist to achieve a stable equilibrium state without hypercongestion. The important feature of the coordinated prices is that the difference in cost that users experience when traveling by car or by transit must be controlled to incentivize travelers to use cars and transit in an efficient way.

15

An efficient pricing strategy may be to pick the coordinated prices that minimize the total generalized cost for the transportation system. In cities with mixed traffic, the implementation of coordinated prices makes the difference between being able to bound the costs of traffic congestion or not. Although traffic congestion is inherently a dynamic phenomenon, and these dynamics are important, the existence of stable traffic states is useful. Since the stable state with travelers using cars and transit simultaneously can be sustained as long as the demand remains high, the total costs associated with the peak period can be approximated by focusing on the steady state. The time it takes for the accumulation of vehicles to build up the critical accumulation for steady equilibrium gets shorter as λ increases. In the numerical example, pricing makes this loading time less than 0.2 hr (12 min) for demands 35% greater than the network's maximum exit rate. This is a short time compared to a typical rush period. Although working out the detailed user equilibrium travel patterns at the beginning and end of the rush remains a research challenge, errors associated with estimating the beginning and end conditions are small compared to the total cost of the transportation system when the peak period is long compared to the loading time. The models presented in this study provide a way to conduct a systematic analysis of traffic congestion in cities around the world. The models account for the performance of the street network as described by the MFD, the characteristics of trips such as the distance and demand rate, and the costs associated with each mode. From this information, it is possible to estimate the magnitude of social benefit from implementing a coordinated pricing scheme or operating transit in dedicated lanes. By recognizing that traffic conditions in networks can and often do become hypercongested, the analysis and design of pricing schemes are consistent with the behavior of the real networks to which they will be applied.

Appendix A. Transit cost functions Aggregate models for transit can be developed based on some basic operating characteristics of the transit system. Examples include representation of a simple grid network dating back to Holroyd (1967). More recently, models of hybrid grid-radial networks (Daganzo, 2010) and ring-radial networks (Badia et al., 2014) have been developed using similar methods. Here, we present a general model for the generalized cost of operating a transit system, including the costs experienced by users and the operating costs incurred by the agency. Suppose that the route network and location of stops are fixed. The total length of directional route distance is lt, which may be arranged in any geometry (e.g., radial, grid, hybrid, etc.). Transit stops have an average spacing of s. For example, a square city with edge length D and grid network with route spacing S has total directional route length lt ¼ 4D2 =S (Daganzo, 2010). The transit agency can choose the service headway between vehicles, H, on each line in response to the passenger demand. An efficient transit system will operate with the headway that minimizes the sum of travel time for all passengers and operating cost for the agency, expressed in equivalent units of travel time. There are three components that constitute the travel time that users experience when traveling by transit traveling in dedicated lanes: 1. Access time, τa, is required to get from an origin to the transit system and then to get from the system to the final destination. The value of τa depends on the distance that must be traveled

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16

and the speed of the access mode, which could be walking, bicycle, or some other mode. For a fixed network, τa is constant. 2. Waiting time is associated with the service headway, H. For trips on a grid or radial network with one transfer, the waiting time may be approximately a full headway on average (i.e., half a headway at the first stop and half a headway at the transfer stop). In other network structures, the waiting time may be another multiple of H. 3. In-vehicle travel time is associated with the distance that must be traveled in the transit vehicle to complete the trip. For a grid transit network operating on a grid street network, the average length of a trip by car and by transit will both be d. Other transit network geometries may introduce circuity that increases the average distance of travel on transit by a multiplicative factor. The transit vehicles travel in dedicated lanes at the network free-flow speed, vf, between stops. A loss time of τs is associated with each stop due to deceleration, dwell time, and acceleration. The result is the average speed for transit, including stops for passengers is   1 τs  1 vt ¼ þ ðA:1Þ vf s where the first term is the travel time per distance while moving between stops, and the second term represents the loss time per distance due to stops. The resulting in-vehicle travel time is d=vt .

The agency costs for operating a transit system accrue with guideway distance, vehicle distance traveled, and vehicle hours of operation. For simplicity, these costs can be represented by two parameters: 1. A fixed hourly cost, zt0, is associated with maintaining infrastructure and keeping up a fleet of transit vehicles even when they are not in use. 2. A variable cost per vehicle hour of operation, zt1, is multiplied by the total number of transit vehicles in operation, m, to represent the total vehicle operating cost per hour. The agency cost per trip is the result of dividing the total cost per hour represented by zt0 and zt1 m by the hourly transit demand, λt. The resulting total cost of transit per passenger is zt ðλt Þ ¼ t a þH þ

d zt0 þ zt1 m þ vt λt

ðA:2Þ

when dollar costs are represented in terms of equivalent passenger travel time. The total number of transit vehicles in operation is related to the average headway by lt m¼ vt H

ðA:3Þ

for any network structure. For a square city with edge length D and grid network with route spacing S, the number of transit vehicles simultaneously in service is then m ¼ 4D2 =SHvt (Daganzo, 2010). By substituting (A.3) into (A.2), the cost per user can be defined in terms of the headway. It is possible to express the headway endogenously within the transit model by supposing that the agency always chooses the headway that minimizes zt ðλt Þ. Since zt is a convex function of H, the optimal headway, H n ðλt Þ, can be determined by solving dzt =dH ¼ 0. The result is sffiffiffiffiffiffiffiffiffi lt zt1 ; H ðλt Þ ¼ vt λt n

ðA:4Þ

and (A.2) becomes

sffiffiffiffiffiffiffiffiffi d zt0 lt zt1 zt ðλt Þ ¼ τa þ þ þ 2 : vt λt vt λt

ðA:5Þ

If agency costs are passed on to users in the form of fares, then zt ðλt Þ represents the cost that users experience when choosing to complete a trip by transit. This function exhibits the well-known economies of scale for transit systems. As more and more trips are served by transit, the cost per trip decreases approaching the sum of access time and in-vehicle travel time.

Appendix B. Nonexistence of stable unpriced equilibrium for mixed traffic A proof is provided here for the following statement: if a transit trip is more costly than a car trip as n approaches 0, the cost of a transit trip will always be more costly than a car trip for transit operating in mixed traffic. The condition that a free-flow car trip is less costly than serving all trips on transit in free-flowing traffic conditions can be expressed in terms of the limits of each cost as n approaches 0 lim zc ðnÞ o lim zt ðλt ðnÞ; FðnÞÞ

n-0

n-0

ðB:1Þ

By substituting (13) for zc(n) and (29) for zt ðλt ðnÞ; FðnÞÞ into (B.1), the inequality becomes sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    ffi n n τs zt0 lt zt1 n τs þ þ lim zc0 þ o lim τa þ d þ þ2 : F n n-0 FðnÞd s n-0 λt ðnÞ λt ðnÞ FðnÞd s ðB:2Þ Recognizing that limn-0 n=FðnÞ ¼ d=vf from (17) and the fact that λt ð0Þ ¼ λ, the inequality can be rewritten and simplified as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     d 1 τs zt0 lt zt1 1 τs þ þ2 ðB:3Þ þ þ zc0 þ o τa þ d vf vf s λ λ vf s dτs zt0 zc0  τa  o þ2 s λ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   lt zt1 1 τs þ λ vf s

ðB:4Þ

To prove that the cost of a transit trip will always exceed the cost of a car trip, we must verify that zc ðnÞ o zt ðλt ðnÞ; FðnÞÞ for all values of n A ð0; nJ . This inequality can be expressed with the same substitutions used to construct (B.2), this time without the limits, and then simplified as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    ffi n n τs zt0 lt zt1 n τs zc0 þ o τa þ d þ þ þ þ2 ðB:5Þ FðnÞ FðnÞd s λt ðnÞ λt ðnÞ FðnÞd s zc0  τa 

dτ s zt0 o þ2 s λt ðnÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi lt zt1 n τs þ λt ðnÞ FðnÞd s

ðB:6Þ

The left side of (B.6) is the same as the left side of (B.4). The inequality will be true if the right side of (B.6) is not less than the right side of (B.4). In other words, the cost of a transit trip always exceeds the cost of car trip if sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    ffi zt0 lt zt1 1 τs zt0 lt zt1 n τs þ r þ2 þ2 þ : ðB:7Þ λ λ vf s λt ðnÞ λt ðnÞ FðnÞd s The value of transit demand can never exceed λ, because F(n) is positive-valued. Therefore, λt ðnÞ ¼ λ  FðnÞ r λ, which implies that 1

λt ðnÞ

1 Z :

λ

ðB:8Þ

The value of n=FðnÞ is monotonically increasing, because F(n) is a concave function passing through Fð0Þ (see explanation in Section 4.1 for why zc(n) is monotonically increasing), so

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E.J. Gonzales / Economics of Transportation ∎ (∎∎∎∎) ∎∎∎–∎∎∎

n n d Z lim ¼ : FðnÞ n-0FðnÞ vf

ðB:9Þ

F

It follows from (B.8) and (B.9) that the condition (B.7) must be true, because each term on the right must be greater than or equal to the corresponding term on the left. Together, (B.4) and (B.7) imply that (B.6) must be true, which confirms that zt ðλt ðnÞ; FðnÞ 4 zc ðnÞ for all n A ½0; nJ .

k

Appendix C. Existence of stable priced equilibrium for mixed traffic

network exit function (NEF) defining exit rate in terms of n

L l lt m N n ncrit

A proof is provided here for the following statement: if a transit trip in a city with λ 4 μC ¼ FðnC Þ is more costly than a car trip at ncrit ¼ 0, any traffic state ncrit o nC can be achieved as a stable equilibrium with cars and transit by implementing the corresponding coordinated prices Δpðncrit Þ. The proof is based on comparing the slope of the average cost functions for cars and transit relative to n for values of n onC . Taking the derivative of (13), the slope of zc(n) is   dzc ðnÞ d n ¼ : ðC:1Þ dn dn FðnÞ For transit, we substitute λt ðnÞ ¼ λ  FðnÞ into (29) and taking the derivative, the slope of zt(n) is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    !   dzt ðnÞ d n d zt0 d lt zt1 n τs ¼ þ 2 þ þ dn dn FðnÞ dn λ FðnÞ dn λ  FðnÞ FðnÞd s ðC:2Þ We know that F(n) is a concave function with maximum value at nC, and Fð0Þ ¼ 0. Therefore, F(n) is positive-valued and monotonically increasing for n A ð0; nC Þ, so 1=ðλ FðnÞÞ is positive valued and monotonically increasing in the same interval. It follows that the second term of (C.2) is positive. We also know that n=FðnÞ is monotonically increasing for F(n) with the stated conditions (see explanation in Section 4.1 for why zc(n) is monotonically increasing). The square root of a monotonically increasing function is still monotonically increasing, and the product of two monotonically increasing functions must also be monotonically increasing. Therefore, the third term of (C.2) must also be positive. Since the second and third terms of (C.2) are positive, and the first term of (C.2) is the same as (C.1), it follows that dzt ðnÞ=dn 4 dzc ðnÞ=dn for all n o nC . If prices are coordinated for cars and transit with difference ΔpðnÞ, users will experience the same cost for both modes when the accumulation of traffic is n. For values of n o nC , this will be a point where the average cost curve for transit including pricing crosses the average cost curve for cars including pricing from above, because fixed prices do not change the slope of the cost curves. Such a point will be a stable equilibrium, a lesser accumulation of vehicles will incentivize car use and a greater accumulation of vehicles will incentivize transit use, leading the system back to the stable accumulation n. Thus, the statement is proven. Appendix D. List of variables A (t) D cd e

cumulative arrivals to a bottleneck/network by time t; subscripts c and t indicate cars and transit (t) umulative departures from a bottleneck/network by time t; subscript c indicates cars average length of a trip (km) cost of earliness in units of equivalent travel time (n)

17

q Q

s T Tcrit t tP t1 v vf vt Z zc zc0 zt zt0 zt1

λ λt μ

τa τs

vehicle density (veh/lane-km); subscript indicates specific traffic state cost of lateness in units of equivalent travel time length of the street network (lane-km) directional length of routes in transit network (km) transit fleet size total number of travelers in the peak period accumulation of vehicles at a bottleneck or in a network; subscript indicates traffic state (veh) critical accumulation of vehicles at which cars and transit are equally attractive to users (veh) vehicle flow (veh/lane-hr); subscript indicates specific traffic state (k) macroscopic fundamental diagram (MFD) defining flow in terms of k transit stop spacing (km) travel delay at a bottleneck or travel time in a network (hr) critical delay or travel time at which cars and transit are equally attractive to users (hr) time (hr) duration of peak period (hr) time at which cars and transit become equally attractive to users (hr) vehicle speed (km/hr) free flow speed (km/hr) average speed of transit, including stops for passengers (km/hr) total generalized cost per hour of the peak period in units of travel time (hrs/hr) average generalized cost of a car trip, in units of travel time (hr) genearlized cost of a free flow car trip (h) average generalized cost of a transit trip, in units of travel time (hr) generalized cost of transit infrastructure per hour of operation (hr) generalized cost per hour of transit vehicle operation (hr) rate of total travel demand (trips/hr) rate of transit demand (trips/hr) exit rate from the bottleneck or network (veh/h) or (trips/ hr) access time for a traveler to reach a transit stop or station (hr) loss and dwell time per transit stop (hr)

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