A NEW METHOD TO ESTIMATE VALUE OF TIME FOR HIGH-OCCUPANCY-TOLL LANE OPERATION

Transcription

1 A NEW METHOD TO ESTIMATE VALUE OF TIME FOR HIGH-OCCUPANCY-TOLL LANE OPERATION Xuting Wang Department of Civil and Environmental Engineering Institute of Transportation Studies University of California, Irvine Irvine, CA, Wenlong Jin (corresponding author) Department of Civil and Environmental Engineering California Institute for Telecommunications and Information Technology Institute of Transportation Studies University of California, Irvine Irvine, CA, -00 Phone: -- Word Count: words + figure(s) + table(s) = words Submitted to 0 TRB annual meeting November, 0

2 Wang and Jin 0 ABSTRACT Road pricing has been advocated for decades as an efficient way of reducing congestion since the 0s. In the U.S., there are several types of managed lanes, and the high-occupancy-toll (HOT) lane is an application of road pricing on freeway. At the same time, value of time (VOT) is a key factor when deploying congestion pricing. In this study, we propose one new approach to estimate drivers VOT and determine the real-time tolling strategy for HOT lane simultaneously when lane drop occurs downstream of freeway segment. There are two goals of operating HOT lanes, one is to maximize the freeway s throughput, and another one is to maintain the free flow speed. The traffic state variables used in this paper is the queue length on HOT lane and turning in-flux of paid single-occupancy-vehicles (SOVs). The traffic dynamics is described by a point queue model. When the HOT lane is operated at optimal, we can use feedback control algorithm to estimate drivers VOT; and the pricing rate for the HOT lane is calculated by the product of estimated VOT and travel time difference of two types of lanes. Simulations results are provided to validate our approach. Keywords: High-occupancy-toll lane; Point queue model; Dynamic congestion pricing; Value of time; Feedback system

3 Wang and Jin INTRODUCTION In recent years, traffic congestion has become more severe in large cities and on highways and it affects economic competitiveness, driving safety, and air quality. According to the European UNITE project, the total social cost of congestion, on average, is equal to approximately % of GDP in Western Europe (). Chang and Xiang () stated that the accident frequency on both freeways and arterial roads tends to increase with the congestion levels. During peak hours, the average annual delay in 0 for an auto commuter in Los Angeles was about 0 hours, and the average congestion cost was dollars (). Furthermore, CO emission for vehicles with a stop-and-go driving pattern would be higher than those with constant speed (). In the U.S., one type of managed lane called high-occupancy-vehicle (HOV) lane is widely used on freeways to reduce congestion. This lane is reserved for cars with a minimum of two or three occupants. Many studies have concerned the low utilization of HOV lanes. Kwon and Varaiya () collected speed and flow measurements from more than 00 stations during peak hours of weekdays in 00, and found % of HOV lane flow rates are below 00 vehicles per hour per lane. In this case, it is necessary to find a strategy to improve the freeway performance. Congestion pricing has received more and more attention both in the field of economics and transportation since the work by Pigou () and Knight (). Both static and dynamic pricing are widely studied. Sorensen et al. () indicated that many strategies (such as raising fuel price) would provide short-term relief, but only pricing strategies could manage congestion in the long run. High-occupancy-toll (HOT) lane is the application of congestion pricing on HOV lane. In HOT lane operation, we can charge for those single-occupancy vehicles (SOVs) if they want to use HOT lanes to reduce travel time. HOT lanes have been regarded as an improvement of HOV lanes in terms of better lane utilization. And it is pointed out that HOT lanes work relatively well in reducing congestion under different levels of initial delay and HOV lane usage () The first HOT lane was implemented on SR- in California, which began operation in. Different tolling rates are set based on time of day and congestion level. The second HOT lane was operated on I- near San Diego in. This HOT lane applied a distance-based dynamic pricing scheme for SOVs between $0.0 and $.00 to maintain level of service B on HOT lane, and the rate changes every six minutes. By the end of 0, 0 pricing projects have covered about 00 miles of priced lanes, and of them are HOT lane projects (0). Details can be found in reports of I- in Minnesota, I- in Florida, and I- in Colorado. Some literatures have studied HOT lane operations. Based on survey data collected from I- and Dallas North Tollway, Li and Govind () developed a model to evaluate tolling strategies. Zhang et al. () adopted a piecewise feedback control model, based on the different speeds on HOT and general purpose (GP) lane, to calculate the optimal flow ratio for HOT lane. Then the toll rate is estimated backward based on discrete choice model. Yin and Lou () proposed two methods, including feedback and reactive self-learning approach, to determine the dynamic pricing rate regarding the arrival flow rate on GP and managed lanes. Based on the self-learning approach proposed in (), Lou et al. () applied multi-lane hybrid traffic flow model () to consider the impacts of the lane-changing behaviors. Laval et al. () proposed a system optimal (SO) toll for real-time arrivals under inelastic demand, but the solution is not well defined when flow rates reach capacity. When making a pricing scheme, we need to take value of time (VOT) into account. In economics, VOT represents the opportunity cost of the time that a traveler spends on trips. It is the

4 Wang and Jin 0 amount that a traveler would be willing to pay in order to save time, or accept as compensation for lost time. Vickrey () mentioned that variation of VOT is important when deciding the pricing strategy. In the last fifteen years, some economists have done some studies in VOT with real data. Lam and Small () estimated VOT based on the survey data and loop detector data on SR-. The VOT is obtained when maximum log-likelihood achieved. Brownstone et al. () collected revealed preference data from drivers, loop detector and toll data on I-, and unweighted maximum likelihood estimation is carried out for the mode choice. Then, the VOT for each respondent is calculated using the estimated parameters. Yin and Lou () used a logit model to formulate route choice in a simulation for a segment of freeway with HOT and GP lanes. But unlike the papers listed above, they applied Kalman filtering technique, an estimation method in control theory, to calibrate drivers VOT. However, there are some deficiencies in literature. Zhang et al. () used the speed difference to update the pricing strategy, but their method cannot capture the traffic dynamics, such as the queue evolution. In the step-wise function, multiple parameters are needed, but these parameters have no economic meaning. Later, Yin and Lou () proposed two methods to calculate dynamic pricing rate. In their feedback approach, the error term is the difference between desired occupancy and measured occupancy. The tolling rate on HOT lane can be stated as: p(t + ) = p(t) + K (O HOT (t) O HOT ) () where O HOT (t) is the measured occupancy on HOT lane at time step t, O HOT is the desired occu- pancy, and is usually equal or slightly less than critical occupancy; and K is a regulator parameter. 0 This method can be regarded as an application of ALINEA strategy (0), which is widely used in ramp metering. Different K values are tested in the simulation, but the performance is inferior to the second method, in terms of queue length and throughput. And the self-learning approach applied a logit model to show the lane group choice. With the flow rates collected on GP, HOV and HOT lanes, the parameters in the model can be estimated recursively by Kalman filtering tech- nique, and the VOT is obtained by the ratio of the first two parameters. For the traffic part, they applied the point queue concept () to capture traffic dynamics. Then, the optimal tolling rate can be determined. More computation time is needed for this method, because in each iteration, they need to update parameters, such as Kalman gain and error covariance matrix. In this paper, we propose one new approach to determine dynamic pricing for HOT lane, 0 which can be regard as an in-between approach of two methods inyin and Lou (). The major contribution of this paper is the estimation of VOT, which can be easily understood, and involves less computation. We design a feedback approach to estimate VOT, with the queue length on HOT lane and the difference between the optimal and actual turning in-flux serving as the error term. An integral controller is used to control the system. The rest of this paper is organized as follows. In section, the problem statement is pre- sented. In section, we will first describe the traffic dynamics using a point queue model in both continuous and discrete form, and calculate the travel time on two types of lanes. And an inte- gral controller will be used to estimate drivers VOT when HOT lane is operated at optimal(both free-flow speed on HOT lane and maximum throughput of the freeway are reached), then the pric- 0 ing rate is calculated recursively based on discrete choice model. Simulation results are shown in section. Finally, we have conclusion and future work in section.

5 Wang and Jin 0 PROBLEM STATEMENT When congestion arises on freeways, people will experience longer travel time. As mentioned in literature, in most cases, HOV lanes are not used efficiently; at the same time, some SOV drivers have the incentive to pay to reduce the delay on regular lanes. But if too many drivers change lanes, vehicles on HOT lanes cannot reach free flow speed. So, it is necessary to analyze the traffic dynamics on both lanes to provide an effective control strategy. This study starts from a simple scenario with two classes of lanes on the freeway without any ramps. The layout is shown in Figure. In this freeway segment, an instantaneous lane drop occurs on GP lane. In the upstream of the drop, there are two GP lanes and one HOT lane in this freeway segment with the same free flow speed; the capacity of HOT and HOV lane is C per hour per lane, and the capacity for GP lane is C per hour per lane. HOV HOT GP GP FIGURE : Layout of a freeway with HOT and GP lanes 0 There are two sets of loop detectors on the freeway. The first is placed before the toll reader to record the approaching flow rate on HOV and GP lanes, and the second is placed behind the reader to detect the flow rates on HOT and GP lanes. In this study, we have three types of vehicles. The first type is the HOV, which has two or more passengers. The second type is the regular SOV, and it uses GP lane all the time. The last type is the paid SOV, which has an incentive to pay to use HOT lane in order to reduce delay. In Figure, q (t) represents the flow-rate of HOVs; q (t) means the flow-rate on GP lane before the toll reader, which is the sum of two types of SOVs; q (t) represents the flow-rate for the paid SOVs. So, the flow rates after the toll-reader on GP lane and HOT lane should be q (t) - q (t) and q (t) + q (t), respectively. There are several assumptions in this study. First, we initially do not know the true VOT, but we are able to use our framework to estimate it. Second, HOVs do not need to pay when using HOT lane, while SOVs need to pay to get access. Third, we assume all HOVs will stay on managed lanes, and all the lane changes for SOVs to HOT lane happen before the toll-reader. Fourth, for SOV drivers, their travel cost is the sum of travel time cost and the real-time HOT lane pricing rate. When the HOT lane is operated efficiently, vehicles on HOT lane should move at free flow

6 Wang and Jin speed, and the throughput on freeway should be maximized. A real-time pricing strategy for HOT lane is applied to achieve these goals. At the same time, in order to get the pricing strategy, we need to estimate VOT. In this study, we try to solve these two problems simultaneously. In the next section, we will provide a block diagram to show the framework of this study, and explain how to track traffic dynamics, estimate drivers VOT and set the tolling rate in detail. METHODOLOGY When implementing congestion pricing strategies, we need to take drivers VOT into account. In literature, VOT has been collected through surveys, or estimated from data of freeways with toll lanes(such as CA-, I- in California). In this study, we show that drivers VOT can be gradually learned through HOT lane operation, with the application of control theory. In the simulation, the information available is the arriving flow-rate on and and the capacity of both HOV and GP lanes. We need to estimate drivers VOT, and calculate the turning flux of SOVs from GP lane to HOT lane, queuing time on both GP and HOT lanes, and the pricing rate in a certain time period. In this part, we will show our approach to estimate drivers VOT in a feedback system, track traffic dynamics, and provide a real-time pricing strategy for HOT lane operation. Block Diagram Description In this section, we will show how traffic flow theory and control theory are coordinated. The block diagram in Figure is used to illustrate the relation between these components. From the block diagram, we can see this is a multiple-input and multiple-output (MIMO) system. The inputs of the system are optimal turning in-flux for SOVs (i.e., C q (t)) and optimal queue length for HOT lane (it should be 0 here); and the outputs are actual turning in-flux of SOVs (i.e., q (t)) and the queue lengths on both HOT and GP lanes (i.e., λ (t), λ (t)). For the controller (marked as K I in the diagram), one integral (I) controller is used to reveal the true value of VOT. The error signal is denoted by u(t) at time step t. The first error term is the difference between the actual in-flux turning from GP lane and optimal turning in-flux. And the queue length on the HOT lane represents another error term. And there are three control blocks in the diagram. The first block is used to get the pricing rate, and the detail discussion is provided in pricing strategy part later. The second block describes the dynamics of the system, and it has two subsystems: route choice model and traffic flow model. We use discrete choice model to model SOV drivers lane group choice; and the traffic dynamics is captured by a point queue model. For the third block, the inputs are the queue lengths on two types of lanes, and with queuing models, the output will be the travel time difference on HOT and GP lanes (i.e., w(t)).

7 Wang and Jin + - FIGURE : Block diagram of our approach 0 In order to simulate the diagram, we need some initial and boundary conditions. The first initial condition is the initial assumption of VOT (i.e.,v(0)). At the same time, in this case, we assume there is no queue downstream at the beginning of the simulation, so the initial queue lengths on HOT and GP lanes are zero (i.e., λ (0)=λ (0)=0). And the boundary conditions are in-fluxes of HOV lane and GP lanes before the toll reader (i.e., q (t) and q (t)). And in each iteration, the solution should be v(t), λ (t), λ (t), and the intermediate solution is q (t). Point Queue Model In point queue models, all drivers follow first-in-first-out(fifo) rule, and the travel time is composed of the free flow travel time and the queuing time. For HOT and GP lanes in a homogeneous road segment, the free flow travel time is the same, the only difference is the queuing time. And in this section, we will introduce the point queue model proposed by Jin (), in both continuous and discrete form to capture the queue evolution. Continuous Form We consider a point facility, which can be regarded as a limit of a link with a storage capacity of Λ. δ(t) is the origin demand, σ(t) is the the destination supply, d(t) and s(t) are a link s demand and supply respectively. λ(t) is the size of point queue, and it is used to model the dynamics of a queue.

8 Wang and Jin FIGURE : An illustration of point queue model Given demand and supply function, we can have a point queue model: d λ(t) = min{δ(t),s(t)} min{σ(t),d(t)} () dt which means, the rate of change in queue length is the difference between the upstream influx and downstream out-flux. When the storage is infinite, we have s(t) =, and the rate of change is expressed by: d λ(t) = δ(t) min{σ(t),d(t)} () dt For a very small time step t, we can have an approximate form of the point queue demand d(t) = δ(t) + λ(t) t, so the queue dynamics is: d dt λ(t) = max{δ(t) σ(t), λ(t) } () t So, for HOT and GP lanes in this study, the point queue model becomes: Discrete Form d dt λ (t) = max{q (t) + q (t) C, λ (t) } t (a) d dt λ (t) = max{q (t) q (t) C, λ (t) } t (b) For the continuous point queue models, we use λ(t+ t) λ(t) t form, the queue evolution on both lanes is: to replace dt d λ(t), and make it a discrete λ (t + t) = max{0,(q (t) + q (t) C ) t + λ (t)} λ (t + t) = max{0,(q (t) q (t) C ) t + λ (t)} (a) (b)

9 Wang and Jin where λ (t) and λ (t + t) are the queue lengths at t and t + t for HOT lane, and λ (t) and λ (t + t) are the queue lengths at t and t + t for GP lane, t is the time interval. And the figure below can graphically show the queuing time for a vehicle entering the queue at time t. It is clear that the geometric relation in the small triangle shows C π(t) = λ(t). FIGURE : Queuing time 0 So, when a queue forms, the queuing time on HOT and GP lane is: π (t) = λ (t) C π (t) = λ (t) C Then, after obtaining the queue lengths on two type of lanes, the travel time difference, denoted as w(t), can be calculated: w(t) = π (t) π (t) = λ (t) λ (t) () C C Integral Controller In this study, we will only use one integral(i) controller to estimate the true VOT, because we think the rate in change of price is influenced by the error signal. And here is the formulation of this controller. Continuous Controller (a) (b) For the Integral controller, we have : d dt v(t) = K I (C q (t) q (t)) K I (0 λ (t)) () where KI and K I are both non-negative, and denote the coefficients for the integral term. And v(t) is the calibrated VOT at time step t.

10 Wang and Jin In formula, we have two error terms, one is the difference between the current flow rate and the capacity of HOT lane, another is the queue length on HOT lane. When q (t) + q (t) < C, it shows HOT lane is under-utilized, the pricing rate is set too high. So, the rate should decrease when we find fewer cars are using this lane than expected, a negative sign is needed before the KI. For the second term, if there is a queue on HOT lane, it is obvious that the pricing rate is too low, and we should increase rate to eliminate the queue. When these two terms become zero, we will reach the steady state (or equilibrium state). 0 Discrete Controller Then, we just simply replace dt d v(t+ t) v(t) v(t) by t, the discrete process of estimating VOT is given by: v(t + t) = KI (C q (t) q (t)) t + KI λ (t) t + v(t) (0) where v(t + t) is the estimated VOT at time t + t. Lane Group Choice For the route choice, if we apply a multinomial logit (MNL) model to capture the turning proportion of paid SOV to HOT lane, the flow-rate on HOT lane should be: q (t) + q (t) = q (t) + q (t) exp(v HOT (t)) exp(v HOT (t)) + exp(v GP (t)) () 0 where V HOT (t) and V GP (t) represent the deterministic disutility of drivers using HOT and GP lanes at time step t. For the SOV drivers, the disutility for them to change to HOT lane is ((t f + π (t)) v + p(t)), and for those who stay on GP lane is (t f + π (t)) v, where t f is the free flow travel time, v is the true VOT. So, we have: q (t) = q (t) where p(t) is the pricing rate at time step t. Pricing Strategy + exp(p(t) w(t) v ) While in our analysis, we do not know the true VOT in the beginning. During the estimation process, at time t, the disutility for SOVs turning to HOT lane is ((t f + π (t)) v(t) + p(t)), and for those who stay on GP lane is (t f + π (t)) v(t), where v(t) is the estimated VOT. Based on MNL model, the lane choice should be formulated as follows: () q (t) q (t) q (t) = exp( ((t f + π (t)) v(t) + p(t))) exp( (t f + π (t)) v(t)) () If we take the log on both sides of formula, and simplify it, we can get: q (t) ln( ) = p(t) + w(t) v(t) () q (t) q (t)

11 Wang and Jin 0 When HOT lane is operated at optimal, we have q (t) + q (t) = C. At time t, the pricing rate should be set as: p(t) = w(t) v(t) + ln( q (t) (C q (t)) ) () (C q (t)) SIMULATION RESULTS For the simulation, the site is a freeway segment with lane drop downstream of the GP lane (see Figure ), and the capacity for one HOT and one GP lane is the same, with the value of 00 vehicle per hour per lane. A one-hour period is simulated, and the time interval is one second in order to match the requirement for Jin s point queue model. We assume the arriving flow-rate of GP lane is 00 veh/h, and 00 veh/h for HOV lane. We assume the true VOT is $0 per hour, and the initial VOT is $0 per hour. The simulation with KI = and K I = 0 is shown below. (a) Calibration of VOT under uniform arrival (b) Queue length and pricing rate under uniform arrival FIGURE : Homogeneous drivers under uniform arrival 0 We can see v(t) will converge at $0 per hour. Then we test different combinations of KI and KI. The result shows that when we increase K I, the HOT lane will have a smaller maximum queue length and higher throughput. And the larger KI /K I value will lead to a shorter queue, and faster queue elimination.

12 Wang and Jin K I K I K I /K I Maximum Queue Length(veh) Converge Time(h) TABLE : Performance of different controller Here is the simulation result when the arrival rate is Poisson with an average of 00 veh/h for GP lane, and 00 veh/h for HOV lane. (a) Calibration of VOT under poisson arrival (b) Queue length and pricing rate under poisson arrival FIGURE : textbfhomogeneous drivers under poisson arrival We also make simulation when the arrival rate is random with an average of 00 veh/h for GP lane, and 00 veh/h for HOV lane. As shown in Figure, our proposed method still applies.

13 Wang and Jin (a) Calibration of VOT under random arrival (b) Queue length and pricing rate under random arrival FIGURE : Homogeneous drivers under random arrival Finally, we set true VOT to be around a constant value to reflect the characteristics of heterogeneous drivers. And the average arrival rate is the same as the simulations above. Here are the results.

14 Wang and Jin (a) Heterogeneous VOT under random arrival (b) Queue length and pricing rate FIGURE : Heterogeneous drivers under random arrival At the same time, we set up the simulation in Simulink. Figure is the setup under constant arrival rates. Function 0 represents the integral controller; function shows the process of calculating tolling rate; function is the lane group choice model; function is the application of the point queue model; function calculates the travel time difference between two types of lanes. And the VOT will finally converge to $0 per hour as well.

15 Wang and Jin FIGURE : Simulink setup for constant arrival pattern CONCLUSION Summary In this study, we provide one new approach to estimate drivers VOT, it is simple and easy to implement. A unified approach of point queue model proposed by Jin () is applied to capture traffic dynamics. For the whole system, the framework can be regarded as an integration of control and estimation problem, which has received little attention in literature. The goal of operating HOT lane is to maximize flow rate while providing free flow speed to drivers. We can design a control system to estimate the VOT with these two goals serving as the inputs. The outputs of the system are queue lengths on two types of lanes, and the actual turning in-flux for SOVs. And at 0 each time step, the system state variables are estimated VOT, tolling rate and travel time difference. The feedback process is controlled by an integral controller, and two error terms are the difference between the actual in-flux turning from GP lane and optimal turning in-flux and the queue length on the HOT lane. Stability analysis for the system is provided, and the true VOT will be obtained when these error terms become zero. The dynamic tolling rate can be presented by the product of travel time difference between two types of lanes and the estimated VOT at time step t. The simulation results of a simple freeway segment with capacity drop are provided. The results show that we can eventually get the true value of drivers VOT and corresponding tolling rate if the HOT lane operates at optimum. And for the simulation results, when we have KI =,K I = 0 for a 0-minute simulation, 0 the average queue length on HOT lane is. veh, and the average throughput is veh/h. While in the self-learning approach of Yin and Lou (), the average queue length is. veh, and the average throughput is veh/h. At the same time, we can converge to the real VOT in a

16 Wang and Jin 0 shorter time. So, our method should be more efficient in a sense. Future Work The current study is set on a simple freeway segment with one lane drop on GP lane, and a MNL model is used to capture drivers lane group choice behavior. In the future, we will first modify our formulation, and try to show how user equilibrium (UE) and system optimal (SO) principles will influence the pricing strategy in this simple scenario. Then, we will consider some more complex traffic conditions. First, we need to consider the effect of on-ramp and off-ramp that exists downstream. If SOV drivers are aware of the traffic condition changes ahead, will they have different lane group choices, and how will the real-time pricing strategies be modified? Second, we need to think about lane changing effect, because it can create voids in traffic, and reduce the throughput of freeways. Real-time ramp metering (RM) has been applied for decades and it is considered an efficient method for freeway operation. In the next step, we will see if there is any opportunity to coordinate HOT lane with RM strategy to provide a superior traffic condition inside a merge zone of local freeway segment.

17 Wang and Jin REFERENCES [] Nash, C. et al., UNITE, UNIfication of accounts and marginal costs for Transport Efficiency Final Report for Publication. European Commission, th Framework Transport RTD, 00. [] Chang, G.-L. and H. Xiang, The Relationship between Congestion Levels and Accidents, 00. [] Schrank, D., B. Eisele, and T. Lomax, 0 URBAN MOBILITY SCORECARD. Texas Transportation Institute, 0. [] Barth, M. and K. Boriboonsomsin, Real-world carbon dioxide impacts of traffic congestion. Transportation Research Record: Journal of the Transportation Research Board,, No. 0, 00, pp.. [] Kwon, J. and P. Varaiya, Effectiveness of California s high occupancy vehicle (HOV) system. Transportation Research Part C: Emerging Technologies, Vol., No., 00, pp.. [] Pigou, A. C., The economics of welfare. Macmillan and Co., London, 0. [] Knight, F. H., Some fallacies in the interpretation of social cost. The Quarterly Journal of Economics,, pp. 0. [] Sorensen, P., M. Wachs, E. M. Daehner, A. Kofner, L. Ecola, M. Hanson, A. Yoh, T. Light, and J. Griffin, Reducing Traffic Congestion in Los Angeles, 00. [] Dahlgren, J., High-occupancy/toll lanes: where should they be implemented? Transportation Research Part A: Policy and Practice, Vol., No., 00, pp.. [0] Office, U. G. A., Road Pricing Can Help Reduce Congestion, but Equity Concerns May Grow, 0. [] Li, J. and S. Govind, An optimization model for assessing pricing strategies of managed lanes. In Proc., nd Annual Meeting of the Transportation Research Board, 00. [] Zhang, G., Y. Wang, H. Wei, and P. Yi, A feedback-based dynamic tolling algorithm for high-occupancy toll lane operations. Transportation Research Record: Journal of the Transportation Research Board, Vol. 0, 00, pp.. [] Yin, Y. and Y. Lou, Dynamic tolling strategies for managed lanes. Journal of Transportation Engineering, Vol., No., 00, pp.. [] Lou, Y., Y. Yin, and J. A. Laval, Optimal dynamic pricing strategies for high-occupancy/toll lanes. Transportation Research Part C: Emerging Technologies, Vol., No., 0, pp.. [] Laval, J. A. and C. F. Daganzo, Lane-changing in traffic streams. Transportation Research Part B: Methodological, Vol. 0, No., 00, pp.. [] Laval, J. A., H. W. Cho, J. C. Muñoz, and Y. Yin, Real-time congestion pricing strategies for toll facilities. Transportation Research Part B: Methodological, Vol., 0, pp.. [] Vickrey, W. S., Congestion theory and transport investment. The American Economic Review, Vol., No.,, pp. 0. [] Lam, T. C. and K. A. Small, The value of time and reliability: measurement from a value pricing experiment. Transportation Research Part E: Logistics and Transportation Review, Vol., No., 00, pp.. [] Brownstone, D., A. Ghosh, T. F. Golob, C. Kazimi, and D. Van Amelsfort, Drivers willingness-to-pay to reduce travel time: evidence from the San Diego I- congestion pric-

18 Wang and Jin ing project. Transportation Research Part A: Policy and Practice, Vol., No., 00, pp.. [0] Papageorgiou, M., H. Hadj-Salem, and J.-M. Blosseville, ALINEA: A local feedback control law for on-ramp metering. Transportation Research Record, Vol. 0,. [] Kuwahara, M. and T. Akamatsu, Decomposition of the reactive dynamic assignments with queues for a many-to-many origin-destination pattern. Transportation Research Part B: Methodological, Vol., No.,, pp. 0. [] Jin, W.-L., Point queue models: A unified approach. Transportation Research Part B: Methodological, Vol., 0, pp..