On the allocation of city space to multiple transport modes

Transcription

1 Transportation Planning and Technology Vol. 33, No. 8, December 21, On the allocation of city space to multiple transport modes Eric J. Gonzales a *, Nikolas Geroliminis b, Michael J. Cassidy a and Carlos F. Daganzo a a Institute of Transportation Studies, University of California, Berkeley, CA, USA; b School of Architecture, Civil and Environmental Engineering, École Polytechnique Fédérale de Lausanne, Switzerland (Received 4 August 29; final version received 17 September 21) This paper analyzes urban multimodal transportation systems in an aggregated way. To describe the aggregate behavior of traffic in cities, use is made of an idea that is now receiving some attention: the macroscopic fundamental diagram (MFD). We demonstrate through simulation how the MFD can be used to monitor and control a real network, in this case a portion of San Francisco, using readily available input data. We then show how different modes interact on the same network and discuss how these interactions might be incorporated into an MFD for multimodal networks. The work unveils two main results: first, it confirms recent results showing that restricting access to a city s congested areas can improve mobility for all travelers, including those who endure the restrictions; and second, that dedicating street space to collective transport modes can improve accessibility for all modes, even those from which space is taken away. Keywords: congestion; macroscopic fundamental diagram; multimodal traffic; San Francisco Background Cities around the world are growing and motorizing rapidly. This trend is driven by peoples need for improved access to activities such as employment, shopping, education, healthcare, and social events. As more people compete for the limited urban space that is set aside to serve transport, there is an increasing need to understand details of how this space is used and how it can be managed to improve accessibility for everyone. Ultimately, an important goal is to understand what sustainable level of accessibility cities of different structures can achieve. Understanding these accessibility outcomes parametrically for all possible city structures and mixes of transport modes would inform the decision-making process, thereby helping cities achieve their sustainability goals. Following some background discussion on accessibility and on the character of the problem that confronts transport networks with multimodal traffic, the present paper reviews both previous efforts to analyze street networks, and a framework that has more recently been proposed to remedy deficiencies in existing network models. Though this latter framework has thus far been developed for networks that serve only a single transport mode, it provides realistic descriptions of urban congestion. It *Corresponding author. gonzales@berkeley.edu ISSN print/issn online # 21 Taylor & Francis DOI: 1.18/

2 644 E.J. Gonzales et al. does so at an aggregate (macroscopic) level of detail and can be applied with readily available data. The paper demonstrates the robustness of this modeling approach by applying it to the street network of downtown San Francisco (and assuming that the network serves only cars). The paper then describes how the allocation of street space to competing modes affects accessibility outcomes for users. Two curious findings arise. First, the simulation suggests, as conjectured in Daganzo (25, 27), that judiciously restricting access to congested areas will improve mobility for everyone, even those who are restricted; and this is achieved with fewer vehicles on the road. Second, dedicating street space to sustainable modes such as bicycles and buses taking space away from automobiles can improve accessibility for all modes, including automobiles. Accessibility Accessibility is defined as the number of opportunities (employment, shopping, leisure, etc.) that a person can reach with given budgets of time or money. It depends on both the density of opportunities as determined by the city structure, and the speed with which people can move about the city, which can be described as the mobility provided by the transportation system. Greater accessibility is achieved with a denser city structure, which puts opportunities close together, and with increased mobility, which allows people to reach opportunities more quickly. However, urban development and transportation infrastructure compete for the same urban space, so the problem of allocating space to improve accessibility is not trivial. Although denser cities will tend to be more congested, the close proximity of people and opportunities improves accessibility by shortening the distances people must travel. There are two complementary approaches for investigating the effect of city structure and mobility on accessibility. One approach is to look for appropriate city structures given the mobility character provided by their transportation systems for example, when planners study the trade-offs of transit-oriented development. By exploring many city structures for given mixes of modes, the complete space of accessibility outcomes can be explored. A dual approach transportation engineers often use is to consider city structure and transportation demand as given, and look to improve mobility. With this method we can explore the same space of accessibility outcomes because the analysis can be done for any city form. We adopt the engineering approach in this paper. Character of the problem and scope of work Urban transportation systems include multiple modes that compete for the same road space. These modes vary from city to city but may include pedestrians, nonmotorized vehicles, buses, and cars. With different performance characteristics, these modes interfere with one another, and the resulting congestion can restrict mobility for everyone. Effective performance requires careful allocation of the available street space to the different modes. Although the allocation of this space is eminently political, it should be informed by the correct physics. The present paper speaks to these physics.

3 Transportation Planning and Technology 645 Previous work Traffic in real cities is complex, with many modes sharing streets, and congestion evolving as demand patterns changeover the course of a day. Existing literature on the physics of urban mobility (see below) can be divided generally into city-scale (macroscopic) efforts and street-scale (microscopic) works. City-scale investigations have thus far looked only at the behavior of one mode without considering the dynamics of traffic congestion. Studies of multiple modes, on the other hand, have only been made at the street-level scale for unrealistic time-independent scenarios. Thus, there is a need for macroscopic models that are physically realistic, timedependent, and multi-modal. Single mode at the city scale Early models set out to determine how much car traffic a city s network can serve. Smeed (1966) modeled the maximum number of vehicles that can enter the central area of a town without jamming the streets based on the town s area, the fraction of this area devoted to roads, and street capacity in units of vehicles per time per width of road. Comparing this theoretical capacity of street networks with different structures (ring network, radial-arc network, and radial network) with data collected from several towns, the work suggests that capacity can be roughly predicted from the area and structure of a city s street network. A further contribution was the proposal of a linear relation between average speed and total network flow (Thomson 1967). Wardrop (1968) expanded this model to relate average traffic speed in a central urban area to average network flow based on road width, density of signal-controlled intersections, and the proportion of the signal cycle devoted to green-time. Recognizing the need for a simple macroscopic model, Zahavi (1972a) conjectured that the ratio between the length of road per unit area and the spacemean speed in a city, denoted a, should be proportional to the observed traffic intensity in the city. This a-value can vary between different neighborhoods, and Zahavi (1972b) argued that a can be used as a performance indicator. Although these models look at traffic on the city scale, they are limited because they assume that the speed-flow relationship is monotonically increasing. Thus the models do not apply to severely congested networks, when both speed and flow are low. Godfrey (1969) examined the relation between average speed and the concentration of vehicles and vehicle-miles traveled in an urban network, and proposed what appears to be the first city-scale speed-flow relation that is meaningful for the full range of speeds. The two-fluid model (TFM), later introduced by Herman and Prigogine (1979), and further verified in Herman and Ardekani (1984), also leads to well-behaved macroscopic relationships, but this model attempts to do many other things and therefore contains unnecessary restrictions. More recently, Daganzo (25, 27) has proposed a less restricted form of this flow-density relation, which with some additional assumptions, leads to a dynamic theory of urban traffic that reproduces the phenomenon of gridlock. Not all work at the macroscopic scale has focused on cars. Modelers have also looked at how to design city-scale public transport systems. Holroyd (1967) and Wirasinghe et al. (1977), among others, have explored how to systematically design a bus system in an idealized city by minimizing the costs incurred by users and

4 646 E.J. Gonzales et al. operators. The methods in these references can also be applied to other urban systems, including freight, at the aggregate city scale (Daganzo 1992). However, to our knowledge this has thus far been done only for one mode in steady state conditions. Multimodal work at the street scale Studies exploring how multiple modes share road space in cities appear to have been limited to the microscopic street scale. Sparks and May (1971), Dahlgren (1998), Daganzo and Cassidy (28), and others have studied how different modes use freeways, recognizing that if different modes serve different numbers of passengers, then analyses should not view all vehicles the same. But these works are limited to small scale systems. Systematic analyses of multimodal systems on urban streets, such as Radwan and Benevelli (1983) and Black et al. (1992), are also limited in scale. In addition, these studies ignore fluctuations and spillover effects from queues, which play a major role in the development of traffic congestion in cities. To our knowledge the only studies that include these effects are micro-simulations such as Currie et al. (24), which argues the need for city-scale planning studies that account for the full economic, social, and environmental impacts of multiple modes. Unfortunately, micro-simulation studies cannot systematically cover the space of possible inputs, and therefore an understanding of multimodal traffic on the cityscale remains elusive. Analysis framework for city-scale modeling Micro-simulation models require mountains of hard-to-get data, such as timedependent origindestination tables. In addition, the detailed behavior of congested networks cannot be predicted due to their chaotic nature (Daganzo 1998). In view of these limitations, we shall focus on aggregate models that involve less detail, easily obtained inputs, and observable outputs. Looking at traffic on a road system in an aggregate way is much like looking at a tub of water a reservoir. Knowledge of details concerning each individual drop of water is not required to understand the water s general behavior in the reservoir. Certain characteristics of the basin, and the volume of water in it, provide enough information to determine how quickly it will drain. For traffic, certain knowledge of the road network in a city or neighborhood, and the accumulation of traffic on those streets, is enough information to predict how quickly vehicles and people move to their destinations. The difference between a water reservoir and the proposed traffic reservoir is that if the latter contains too many cars, the output of the system will be low. A theory of urban traffic dynamics along these lines has been formulated in Daganzo (25, 27) and further verified in Geroliminis and Daganzo (27). In the above-cited theory, the size of the traffic reservoir depends on a city s structure and may also be affected by the design of the network and by the city s traffic management policies. Adding streets to the network, for example, increases the size and changes the character of the reservoir. Experiments show that for a given set of trips, reservoir models of traffic can reliably predict measures like the time and kilometers traveled by vehicles and people by mode and time of day, without requiring information about origin-destination pairs (Geroliminis and Daganzo 27). This is valuable information because these

5 Transportation Planning and Technology 647 measures are key determinants of the sustainability of urban transport. Based on the character of the reservoir, model outputs can be correlated with indicators of interest including: accessibility levels, the costs to users and service providers, and externalities such as emissions and resource consumption. The reservoir model can also be extended to encompass more than one travel mode. For example, the reservoir might be partitioned into sub-reservoirs containing different modes. In the following section, we demonstrate that the San Francisco downtown area, despite its irregularities, exhibits well-defined macroscopic relations among its traffic variables when traffic is treated as a single mode. Restricting access for a single mode: a case study of San Francisco As explained in Daganzo (25, 27) and Geroliminis and Daganzo (27), two key requirements to describing reservoir dynamics are: (1) a well-defined macroscopic fundamental diagram (MFD), i.e. the relationship between the number of cars on the neighborhood street network (the reservoir accumulation) and the average flow on the network links; and (2) a fixed ratio between the average flow on the network links and the number of vehicles exiting the network either by driving out of the neighborhood or by reaching their destinations within it the reservoir outflow. This relationship between accumulation and outflow arising from (1) and (2) is just a scaled up version of the MFD arising from (1). Both relationships are expected to be similar in shape to the so-called fundamental diagram of traffic flow for a single road link (e.g. Lighthill and Witham 1955), but the new relationships describe traffic behavior over an entire network. As in the case of a single link, we expect these MFDs to be unimodal and non-negative. At low accumulations the outflow should be low because there are few vehicles in the reservoir, so few can exit. Likewise, at very high accumulations the outflow should be low because traffic congestion impedes vehicles moving toward their destinations. There should be a sweet-spot in between at which outflow is greatest. This is the capacity of the system, and it is at this point that the network produces the greatest mobility. To test these conjectures we ran 4-hour simulations of San Francisco s central business district with different origindestination tables. In our simulations, the reservoir was taken to be the entire network of streets shown in Figure 1. Estimated demand patterns were scaled up to simulate a full range of vehicle accumulations: from empty streets to complete gridlock. Remarkably, despite the chaotic behavior at the local link level, there was order at the macro-level. In every simulation run, the global outflow vs. accumulation scatterplot fell on a well-defined curve. And more importantly, the set of all runs produced points that grouped along the same curve, as shown in Figure 2 (this figure shows the different demand scenarios by means of four different symbols). The spatial distributions of demand for two of the four simulated scenarios are presented in Figure 3. Notice that in the scenario labeled 1, trips are predominantly inbound, with 8% originating outside the CBD, while Scenario 2 has 7% of trips generated internally and heading outbound. Our findings mean that despite the very different distributions of trip origins and destinations, the outflows can be predicted by monitoring the accumulation of vehicles in the network without knowing the demand patterns. Since the system s performance is found to be consistent for a broad range of demand patterns, a neighborhood s MFD can be used to make traffic control and

6 648 E.J. Gonzales et al. Low accumulation High accumulation Figure 1. Views of the San Francisco network under low and high accumulations. White dots represent vehicles, and black sections show vacant road space. For our animation, see Outflow (veh/hr) 1,5, 1,2, 9, 6, 3, Accumulation (veh) Figure 2. Outflow vs. accumulation for the San Francisco CBD network. Different symbols correspond to simulation runs with different demand patterns.

7 Transportation Planning and Technology 649 Demand Scenario 1 (predominantly inbound) Demand Scenario 2 (predominantly outbound) Distribution of trip origins # Trips per Node # Trips per Node Y X Y X Distribution of trip destinations # Trips per Node Y X Figure 3. Visual representations of the spatial distribution of trip origins and destinations for two of the demand scenarios simulated to construct Figure 1: inbound demand from outside entering the San Francisco CBD (left), and outbound demand from within exiting the CBD (right). policy decisions that are robust in the face of unpredictable variations in the distribution of demand. Note that the outflow in Figure 2 consistently drops toward zero after the accumulation is allowed to exceed a reproducible value. To increase outflows, we might try to discourage vehicles from entering a neighborhood that is already crowded beyond the sweet-spot for example, by re-timing the signals or with congestion pricing. In a second simulation of Scenario 1, vehicle entries into the reservoir were restricted by reducing the green-time allocated to the inbound directions on the peripheral traffic signals. This was done in the simplest way without adaptation as if the signals were pre-timed. The goal was preventing the neighborhood accumulation from growing past the sweet-spot. As shown in Figure 4, the result was that more trips had reached their destination with control than without control. The plot for the original uncontrolled case also shows that the trip completion rate increases at first as the accumulation rises, and then drops about half way through the simulation as the accumulation surpasses the sweet-spot and traffic congestion prevents vehicles from reaching their destinations. In the signal-controlled case, the higher trip completion rate is sustained throughout the simulation. And quite importantly, this is done with fewer vehicles on the road. # Trips per Node Y X

8 65 E.J. Gonzales et al. 75, No Control 75, Control Cumulative outflow (veh) 6, 45, 3, 15, Time (min) Cumulative outflow (veh) 6, 45, 3, 15, Time (min) Figure 4. Cumulative outflow vs. time for uncontrolled (left) and controlled (right) systems. For an animation, see Alternative strategies to control congestion that could be conveniently studied with the MFD include: license plate rationing, mileage-based and/or cordon-based congestion pricing, as well as parking restrictions and pricing. However, to be truly comprehensive one must be able to account for all the modes that may be competing for scarce resources. Reservoir models, including sub-reservoirs for different modes, may offer a means for designing suitable multimodal policies. We next present a first step in this direction. Multiple modes: empirical evidence Here we consider a single reservoir serving two generic modes. The two modes can be either mixed or separated on parallel lanes; see Figure 5. Modes should be separated when they are very different for example, as in the obvious case of pedestrians on sidewalks but there could be cases when modes should be separated even if they are very similar, for example, as in the case of high occupancy vehicles (HOVs). In view of this, we now present some results for HOV lanes (very similar modes) and their implications for the general case. A surprising result of research on freeway HOV lanes is that total car flow through a bottleneck can actually increase when HOVs are separated from general traffic. Previous simulations have shown that bottleneck discharge rates after an HOV lane is activated can be greater than when all vehicles travel together, even when the HOV lane is underutilized (Menendez and Daganzo 26). These simulations suggested that the separation of modes reduced the frequency of disruptive lane-changing maneuvers, and thus smoothed and increased the bottleneck flow. Hence, the phenomenon has been called the smoothing effect. This phenomenon has been confirmed with real freeway data (Cassidy et al. 26), and shown in Figure 6. The cumulative vehicle counts are plotted on oblique coordinates, subtracting a background rate to visually emphasize changes in vehicle flow (see Zwilinger 23). Cassidy et al. (26) show that the capacity of the bottleneck before the HOV restriction goes into effect is 696 vph (the flow in Figure 6b between 14:45:3 and 14:57 hrs). Note that although the flow of vehicles in the HOV lane drops from 183 vph to 143 vph, the total discharge flow of all lanes including the HOV lane remains constant at 6996 vph.

9 Transportation Planning and Technology 651 Figure 5. Road space in a multimodal reservoir can be (a) shared by mixed traffic or (b) separated by modes. Space can be dedicated to specific modes such as special-use lanes for (c) bicycles or (d) high-occupancy vehicles. Photographs courtesy of: (a) Wendy Tau, (b) Cláudia Almeida, (c) ITS Berkeley, and (d) Valerie Reneé. These results suggest that the MFD (Q) for each of the two identical modes, expressed in terms of cars per lane kilometer (k) and cars per lane hour (q), could perhaps be of the form qq(k) if the two modes are segregated, and qbq(k) for some bb1 if they are not (Cassidy et al. 28). The constant b measures the benefit of separation; the smaller the better. For the data shown in Figure 6, b.93. This kind of knowledge would allow us to predict the quantitative aggregate effects of different multimodal policies as we did in Figure 2 for a single mode. Of course, predicting the qualitative impact of some policies is much easier, as we discuss below. Multiple modes: qualitative impact of road space allocation decisions Even if cases arise in which lanes carry the same maximum flow whether traffic is mixed or separated, space for efficient modes should be reserved taking into account the spatio-temporal differences in demand and road geometry. With regard to space, a fixed number of lanes can be reserved everywhere in a network, or the amount of dedicated road space can be varied from place to place. With regard to time, the number of lanes reserved can be the same at all times or vary with time of day. Variable controls could be based on historical information or by monitoring and incorporating real-time measurements. These spatio-temporal decisions are important because they allow for customized and targeted control strategies. If decisions are made incorrectly, e.g. ignoring the smoothing effect, space could be wasted.

10 652 E.J. Gonzales et al. (a) Bottleneck location 69 ft 37 ft X 1 X 2 X 3 Median (HOV) lane Shoulder lane (b) Tennyson Rd. over-crossing Pedestrian over-crossing SR-92 over-crossing (c) Oblique count (all lanes) Oblique count (HOV lane) t = 14:47 Flow disruption t = 14: vph Capacity drop t = 14:45:3 X 1 X 2 & X vph 696 vph 684 vph t = 14:57 t = 15:5 14:4 14:45 14:5 14:55 15: 15:5 15:1 216 vph t = 14:47 Time (measured at X 3 ) 183 vph t = 14: vph t = 15:5 143 vph 14:4 14:45 14:5 14:55 15: 15:5 15:1 Time (measured at X 3 ) Figure 6. Oblique cumulative plots compare (a) vehicle flow for a freeway bottleneck before and after the activation of the HOV lane at 15:. Plots compare (b) flows on all lanes and (c) with the HOV lane alone. Source: Cassidy et al. (26).

11 Transportation Planning and Technology 653 A key issue in partitioning road space is that if a dedicated lane is introduced into a system, and the lane is underutilized, the queue of traffic in the remaining general purpose lanes will lengthen to occupy more space on the roadway. However, if the dedicated lane is only slightly underutilized and the individual lanes have the same capacity to carry traffic flow in both scenarios (i.e. no smoothing occurs), then the queue will not lengthen significantly. In this case, the total vehicle-hours of travel (including delay) are nearly the same in both scenarios. By dedicating a lane to efficient modes, all of this fixed delay is transferred to low occupancy vehicles. Thus, dedicated lanes reduce the total number of person-hours of delay and increase accessibility. The greater the vehicle occupancy, the greater the benefit, so clearly the separation of modes is good if the dedicated lane is well utilized and if the mode given preference carries many more occupants. The benefits can be amplified if separating the modes induces smoothing. There are even some situations where dedicating road space can be good even if the reserved space is severely underutilized. The lengthening of a queue may not be problematic if there is uncongested roadway upstream for the queue to grow. One example of this is a radial road leading into a city center during the morning commute. A problem arises in this case only if the queue blocks upstream intersections with significant exit flows. If those intersections do not have significant demand, as is likely in the morning, the queue-lengthening effect will be small (Daganzo and Cassidy 27). By allowing efficient modes to bypass this queue, even underutilized reserved lanes can reduce the total person-hours of delay, and therefore increase accessibility without significantly increasing the vehicle-hours of delay. Separating modes can be bad, however, if there is no room upstream for the growing queue. This could happen on congested ring roads where queues could fill the entire road space and reduce flow. Furthermore, the queues could spill over onto adjacent streets reducing flows there as well (see Figure 7). Another bad candidate Dedicated space Queue spillover Restricted flow Figure 7. Separating modes can be undesirable for cases like a congested ring road if queues fill the remaining road space (shown in dark), restricting flow and increasing delay on adjacent streets.

12 654 E.J. Gonzales et al. would be outbound radial roads emanating from a city center. The spillover queues on these facilities can develop into gridlock, negatively affecting many vehicles. The growth of queues is of course diminished, and sometimes can even be reversed if one separates modes that experience significant smoothing. Recall that the smoothing effect was observed when dedicating freeway lanes to HOVs, even though the two separated modes are composed of vehicles with the same characteristics, such as size and acceleration. This suggests that the smoothing effect would be much more pronounced when separating modes that are more distinct, such as buses and cars. (However, although it should be smaller, the exact magnitude of b associated with the mixing of buses and cars is not currently known.) Separating the latter two modes would reduce disruptive vehicular interactions since cars would not be delayed by large, slow, and frequently stopping buses. Likewise, buses would not be delayed by queues of cars blocking bus stops and intersections. Since b can be quite small, there could be cases where, by separating the modes on urban streets, vehicles of all types should be better off. Equally, it may also be worthwhile to dedicate space to buses even if separation makes cars worse off because buses carry more people. Complications arise in real cities because modes can only be effectively separated if the road is sufficiently wide. When making allocation plans, there will typically be some links that are too narrow to accommodate separation schemes, such as near bottlenecks. Queues from these bottlenecks often spillover onto wider streets in the network, however, and on these wider links, separation of modes is possible (see Figure 8). While the capacities of the network bottlenecks do not change in terms of vehicle flows by applying this form of space allocation, the passenger-carrying capacity of these locations can significantly improve. Total person-hours of delay can thus be reduced by providing a congestion bypass lane for buses on these wider streets. Bus lanes and HOV lanes can therefore be deployed in much the same way. The more accentuated smoothing effect expected for buses means that bus lanes may even be deployed in congested ring roads or congested city centers. A nice feature of city street networks is that they are dense, and while an entire freeway cannot be devoted to efficient modes such as HOVs, an entire city street may be devoted to buses if parallel streets can carry the other modes. This is most effective if the bus lines in a city can be designed to fully utilize the dedicated space. An example of this is Oxford Street in London, which is devoted to 18 bus lines and is so fully utilized that it tends (a) (b) Figure 8. It may not be possible to separate modes on some congested streets, but queues spill over onto wider streets (a), and delay can still be reduced by providing congestion bypass (e.g. a bus-only lane) where space is available (b).

13 Transportation Planning and Technology 655 at times to be congested with buses. In light of the usefulness of reservoir models, methods to systematically analyze all the above issues now seem within our grasp. Conclusions and future work This paper has analyzed urban multimodal transportation systems in an aggregated way. To describe the aggregate behavior of traffic in cities, use has been made of an idea that is now receiving some attention: the MFD. We demonstrated through simulation how the MFD can be used to monitor and control a real network, in this case a portion of San Francisco, using readily available input data. We then showed how different modes interact on the same network and discussed how these interactions might be incorporated into an MFD for multimodal networks. The work has identified two main results: first, it confirms recent results showing that restricting access to a city s congested areas can improve mobility for all travelers, including those who endure the restrictions; and second, that dedicating street space to collective transport modes can improve accessibility for all modes, even those from which space is taken away. Although we have shown how reservoir models can be used to model approximately some large-scale networks that serve single travel modes, and shown how this approach can be extended to model multimodal networks, the ideas presented in response to the latter issue pertain only to networks with two modes and have not yet been embedded in a dynamic quantitative model. The main reason for this is because the necessary empirical fieldwork has not yet been carried out. Further empirical work on the behavior of more distinct traffic modes (e.g. buses and cars) is needed. These advances should help decision-makers, engineers, and planners understand how to allocate urban space to the various modes, and to design sustainable transportation systems that improve accessibility. Not only should the allocation of existing road space be understood, but so too should the consequences of devoting different total amounts of space to transportation, and the ultimate impacts of city infrastructure on accessibility. Acknowledgements This work is supported by the University of California, Berkeley Center for Future Urban Transport (a Volvo Research and Educational Foundations International Center of Excellence). References Black, J.A., Lim, P.N., and Kim, G.H., A traffic model for the optimal allocation of arterial road space: a case study of Seoul s first experimental bus lane. Transportation Planning and Technology, 16 (3), Cassidy, M.J., Jang, K., and Daganzo, C.F., 28. The smoothing effect of carpool lanes on freeway bottlenecks. Berkeley, CA: Institute of Transportation Studies, University of California, UCB-ITS-VWP Cassidy, M.J., et al., 26. Empirical reassessment of traffic operations: freeway bottlenecks and the case for HOV lanes. Berkeley, CA: Institute of Transportation Studies, University of California, UCB-ITS-RR-26-6.

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