Application of Reinforcement Learning with Continuous State Space to Ramp Metering in Real-world Conditions

Transcription

1 Application of Reinforcement Learning with Continuous State Space to Ramp Metering in Real-world Conditions Kasra Rezaee, Member, IEEE, Baher Abdulhai, Member, IEEE, and Hossam Abdelgawad Abstract In this paper we introduce a new approach to Freeway Ramp Metering (RM) based on Reinforcement Learning (RL) with focus on real-life experiments in a case study in the City of Toronto. Typical RL methods consider discrete state representation that lead to slow convergence in complex problems. Continuous representation of state space has the potential to significantly improve the learning speed and therefore enables tackling large-scale complex problems. A robust approach based on local regression, named k nearest neighbors temporal difference (knn-td), is employed to represent state space continuously in the RL environment. The performance of the new algorithm is compared against the ALINEA controller and typical RL methods using a microsimulation testbed in Paramics. The results show that RM using the knn-td method can reduce total network travel time by 44% compared to the do-nothing case (without RM) and by 17% compared to ALINEA. I. INTRODUCTION Recurring traffic congestion on freeways occurs when traffic demand exceeds infrastructure capacity resulting in system degradation with time. If no suitable control measures are timely employed, this degradation can lead to escalating instability and severe congestion. In recent years, it is realized that the infrastructure expansions cannot provide a complete solution to congestion problems due to obvious economic and environmental reasons and, in metropolitan areas, simply due to lack of space. Alternatively, Intelligent Transportation Systems (ITS) offers dynamic traffic control methods, such as ramp metering, variable speed limits, and route guidance. These freeway control strategies have the potential to substantially improve efficiency of the transportation network [1]. Among these methods ramp metering is considered the most effective traffic control measure and is widely used [2]. A group of Ramp Metering (RM) algorithms focus on regulating a traffic density at a specified value, usually close to critical density. Since traffic flow maximizes at critical density [2], by keeping mainline density close to its critical value one can expect the freeway throughput to be close to freeway capacity. The most well-known ramp metering algorithm ALINEA [3], a simple PI controller with zero proportional gain, and its variations [4] are based on this approach. While these controllers are simple to design and K. Rezaee is with the Civil Engineering Department, University of Toronto, ON, M5S 1A4, Canada (phone: ; fax: ; kasra.rezaee@mail.utoronto.ca) B. Abdulhai is Director of Toronto ITS Centre and Testbed, Civil Engineering Department, University of Toronto, ON, M5S 1A4, Canada. ( baher.abdulhai@utoronto.ca). H. Abdelgawad is a post-doctoral fellow at the Civil Engineering Department, University of Toronto, ON, M5S 1A4, Canada ( hossam.abdelgawad@alumni.utoronto.ca). easy to implement, they neither seek nor guarantee optimality of the system performance. Furthermore their simple design limits the possibility of coordinating them effectively. Another group of controllers determine the metering rate that directly maximizes the network performance. These optimal controllers use a model of the network to estimate the results of different ramp metering policies and choose the one that maximizes the system performance. Model Predictive Control (MPC) is a commonly used technique to achieve optimal ramp metering control [5][6]. Since MPC relies on an accurate model of the network to calculate optimal metering rate, any uncertainty or mismatch in network model will result in suboptimal performance of the MPC. Reinforcement Learning (RL) [7], which attracted significant attention in recent years, has the potential to alleviate some of the aforementioned limitations. The major advantage of RL is that it does not require a model of the environment. RL agents continuously learn from interaction with the environment. In [8] the authors used RL to provide closed loop optimal control for a freeway-arterial corridor through RM and dynamic route guidance. Zhao et al. [9] employed RL for coordinated control of a freeway corridor serving four on-ramps. In order to design RL for such a large problem (in RL context) Dual Heuristic Programming (DHP), a model-based RL method, is used. Another recent application of RL for ramp metering is presented in [1], where the ramp metering problem with queuing consideration is investigated. Although in [9] and [1] RL is used for RM problem, a macroscopic simulation model, which cannot reproduce reallife complexities, is used for evaluation. In this paper we investigated the potential of RL in real-life RM problems using a microscopic simulator. To answer this question a freeway section in city of Toronto is modeled and thoroughly calibrated using loop detector measurement to represent the real-life conditions. The RL agent is trained and evaluated using this calibrated model. The present paper is the first step toward a coordinated and integrated freeway control system based on RL that can be implemented in actual freeways. This paper is organized as follows: Section II describes RL and in particular RL in traffic control problems. In Section III the Paramics micro-simulation testbed will be introduced and the implementation and calibration of the freeway corridor study area will be described. The designed RL controller for RM and its challenges are discussed in Section IV. Section V presents the results of applying RL ramp metering controller to the study corridor and compares its performance to the do-nothing case (i.e., no ramp metering) in addition to ALINEA controller. Finally, conclusions and directions for future works are summarized in Section VI.

2 II. REINFORCEMENT LEARNING RL addresses the question of how an autonomous agent senses and acts in its environment to learn the optimal actions to achieve its long-term goals. An RL agent learns by trialand-error interaction with its dynamic environment [7, 11]. At each time step, the agent perceives the complete state of the environment and takes an action, which causes the environment to transit into a new state. The agent receives a scalar reward signal that evaluates the quality of this transition. The mapping from the environment state to agent s action is the agent s policy which defines the agent s behavior. The RL agent s goal is to find the policy that maximizes the reward received over time. RL is being adopted in many research areas and has shown promising results in applications related to transportation systems control where optimal real-time adaptive control is a key element in improving the effectiveness and efficiency of the system [12]. Numerous algorithms with plausible convergence speed and easilycustomized parameters are used to solve the single-agent RL task; the most notable of which is the Q-learning approach of Watkins [13] that has shown promising result in the field of transportation [14, 15]. The Q-learning algorithm is designed such that a function assigns a scalar, also called Q-value, to each state-action combination which represents the quality of that combination. In the function, set is the set of possible states and set is the set of agent s possible actions. At the beginning of learning process the function is considered immature to respond to varying states in the system as it is usually set to output a fixed value depending on the designer decision. This function will be updated with every new training sample according to: ( ) ( ) ( ) ( ( ) ( )) where is the reward received after performing action at state and moving to the new state, is the discount factor,, that defines the importance of expected future rewards that is calculated by ( ), and ( ) is the learning factor that typically decreases over time to ensure convergence of the function ( ). If designed properly, the Q-learning algorithm is expected to converge to the optimal Q-values regardless of the policy used for action selection [16]. After the function ( ) is converged, at each state the agent evaluates the quality of different actions and chooses the action corresponding to the best quality to achieve optimal policy. Q-learning in its conventional form uses a table to represent the function. This approach limits the practicality of applying Q-learning to more complex problems with large number of states. This issue becomes more complicated in problems with continuous state space. In these problems, there is always a trade-off between the system performance and the learning time because a finer discretization will result in a better performance but at the cost of intractable state space and longer training time. One approach to overcome this limitation is to use a general function approximator instead of the discretized table. Although Neural Networks (NN) has been used in many RL implementations of realworld problems, there have been also reports of divergence and sub-optimality of these methods [17]. Another class of algorithms that can incorporate the continuous state space representation in Q-learning are based on local regression. Although local regression algorithms cannot reduce the unknown parameters to the same extent as NN does, they are more robust compared to NN in RL applications due to their local nature. In this paper we employ a promising algorithm of this class, called NN-TD [18, 19] that is based on the k- nearest neighbor (k-nn) technique. In NN-TD, a set of points is generated in the state space. For each of these points an explicit Q-value is assigned. To estimate the Q-value of a new point in the state space, the first step is to find the set which contains k points from that are the k-nearest neighbors of. These k points are chosen such that the calculated distance between and all points in is the smallest k distances. A weight value is then assigned to each of the points in : ( ) The probabilities weights : are calculated by normalizing ( ( ) ) The Q-value of a point and an action is the weighted average of the Q-values of the points in set with weights : ( ) ( ) Similarly, during the learning, the k-nearest neighbors are updated. Given that the agent performed action in state and the new state is, let be the set of k-nearest neighbors of with probabilities of, and be the set of k-nearest neighbors of with probabilities of. According to Q-learning algorithm the error is: ( ) ( ) Thus each point in the set its probability: can be updated according to ( ) ( ) III. STUDY AREA AND SIMULATION ENVIRONMENT A testbed microscopic simulation model is developed using Paramics [2] to replicate the real environment. Paramics models individual vehicle based on car following, gap acceptance and lane changing models and therefore can provide a replica of real-life transportation networks. The selection of the study area forms an important element in the experimental design we conduct in this paper. Highway 41 is considered the busiest freeway in North America carrying around 42, a day at its busiest segments. Along the City of Toronto, HWY 41 has collector/express sections ranging from 8 to 16 lanes which results in a unique combination of on and off-ramps with many close divergence and

3 convergence points. This unique nature of HWY 41 makes it a representative testbed for the proposed algorithm. The morning peak period of the eastbound 41 freeway at the merging point of the on-ramps from Keele Street was selected as the study area for this paper. As shown in Fig. 1, the study area includes two on-ramps. On-ramp #1 has significant demand that causes a bottleneck along the mainline traffic stream while no bottlenecks are present immediately downstream of this on-ramp. The place marks in the aerial photo are the location of loop detectors that are available on the freeway. The Paramics model was extracted from a comprehensive model that included the entire freeways series in the Greater Toronto Area (GTA) [21]. Although the original model was plausibly calibrated using loop detector counts as discussed in [21], the calibration process was conducted at the network level and was not intended for dynamic control application. Therefore, the simulation model required another round of calibration to capture the dynamic behavior of the vehicles at a refined time scale with location-specific measures. The refined calibration process conducted in this paper ensured that the simulation model replicates the fundamental diagram dynamics when compared to real loop detector at the freeway. In particular, capacity and critical density of the freeway were the two measures governed the calibration process. It was also desired to compare the modeled speed and density pattern over time against the loop detector freeway measures to capture the congestion dynamics. reported by the loop detectors was used as an alternative to density; however, it was observed that occupancy-flow relationship considerably differs from the expected densityflow relationship in the congested region. In order to investigate this behavior one can write the occupancy based on the speed of individual vehicles as: where is the occupancy, and are vehicle length and speed respectively, is number of vehicles passed, and is the sampling interval. In sub-critical conditions, where density is below critical density, different cars pass the loop detector with similar speed and the relationship can be written as: If the right term, which is the average length of vehicles passed, and left constants are neglected the remaining is reminiscent of relationship between density, flow, and speed. On the other hand, in congested conditions traffic is unstable and cars pass the loop detector at different speeds; therefore, occupancy cannot be approximated as (9) and it is not one-toone with density. In the second approach the density was estimated by dividing the flow by speed based on the relationship between flow, density, and speed as shown in Fig. 2. Although the loop detector speed is averaged over time at a single point whereas the speed should be calculated by spatially averaging over a section of road, the resulting density-flow relation was much closer to ideal fundamental diagram than the occupancy-flow relation. The estimated density,, is calculated based on the following equation. k Veh Km Km u Hour q Veh Hour On-ramp # 1 On-ramp # 2 Figure 1. Aerial Photo of the Study Area and the Testbed Simulation Model The loop detector data required for the calibration were retrieved from ONE-ITS (one-its.net) on the morning peak period of July 6 th 211. The loop detector data were available in 2 sec intervals for each lane. Each sample contains the number of vehicles detected, average speed, and occupancy (percentage that the loop was occupied in the last 2 sec). Traffic density is one of the most commonly used variable in traffic control; yet not readily available from loop detector data. Two approaches were tested to estimate the density from loop detectors: 1) using occupancy instead of density, 2) estimating density from speed and flow based on fundamental diagram. In the first approach the occupancy q k u Figure 2. A freeway section showing the relationship between speed, density and flow Using the measurement from loop detectors, a two stage calibration process, as described in [22], was performed to ensure that the Paramics model can reproduce the condition of the actual freeway. First the global parameters of Paramics including reaction time and headway were calibrated. Second, the local parameters of the sections with on-ramps and sections with lane drop (due to off-ramp) were modified. Fig. 3 shows the Paramics network results after calibration in comparison to real freeway measurements. It is clear from the figure that Paramics was able to reproduce a fundamental diagram in the region around the critical density similar to the one from real freeway. However, the density during congestion was found slightly higher than the actual measured freeway speed. The parameters of the fundamental diagram obtained from Paramics network and the actual freeway are illustrated in Table I.

4 Flow (veh/hour/lane) TABLE I. Network FUNDAMENTAL DIAGRAM PARAMETERS OF THE PARAMICS NETWORK AND THE REAL FREEWAY Fundamental Diagram Parameters Capacity Critical density (Veh/Hour/Lane) (Veh/Km/Lane) Real Freeway Paramics IV. REINFORCEMENT LEARNING RAMP METERING AGENT DESIGN There are two on-ramps in the study area. After examining the system and the demand on the two on-ramps, it was realized that the upstream on-ramp (from southbound Keele to eastbound 41) exhibits considerably lower demand and does not have a significant effect on the freeway traffic, specifically if the downstream on-ramp is efficiently metered. Therefore, the upstream on-ramp was not metered and only the downstream on-ramp was controlled. The design process of RL agents consists of defining the appropriate states, actions, and reward in addition to tuning the Q-learning parameters to achieve convergence. Two Q-learning agents were designed and compared in this paper: 1) conventional Q-learning with discrete state pace, 2) NN-TD with continuous representation of state space Density (veh/km/lane) Paramics (fitted curve) Paramics Real Freeway (fitted curve) Real Freeway Figure 3. Paramics vs Real Freeway Loop Detector Measurements (upstream of the on-ramp merging point) A. States The number of states in RL directly affects the learning speed of the agent. Therefore, it is desired to choose the least number of variables that can describe the system. After examining different variables, mainline density downstream of the on-ramp and on-ramp traffic flow were chosen as system state variables. In the conventional Q-learning, the downstream density was normalized and discretized into 12 intervals with the following density values:. The on-ramp flow was normalized and discretized into 1 intervals, such that the center of bins were the flows associated with discrete metering rates, which are described in the next section. For the NN-TD agent, the fixed points with explicit Q- values were chosen to be the same as the center of the bins in the discrete case. Therefore, the combination of downstream densities and ramp flows resulted in 12 points. B. Actions In freeways, RM is implemented by placing a traffic light on the on-ramp and controlling the flow entering the freeway by changing the green and red phases of the traffic light. In this paper one-car-per-green policy [23] was employed to meter the on-ramp traffic. Therefore, the green phase was set at 2 seconds, and the red phase was selected from the set {, 2, 3, 4, 5, 6, 7, 8, 1, 13}. The resulting on-ramp flow can vary from 24 to 18 h. C. Reward Generally the goal of traffic control systems is to minimize the total travel time (TTT) of the whole network. However, in real-life drivers perceive the travel time on the ramp differently from the freeway mainline travel time. To account for this behavior, the weight of the on-ramp travel time was considered slightly higher than 1. Assigning higher weight to on-ramp travel time will also ensure that on-ramp will be cleared as soon as possible when there is capacity on the freeway mainline. Using a similar approach as the one presented in [1], we can remove the constant terms in the total travel time and find the equivalent equation in terms of traffic flow. The weighted total travel time,, can be defined as: ( ) ( ) ( ) where ( ) and ( ) are the number of cars at time step in the mainline and on-ramp, respectively. Based on the conservation of vehicles we can rewrite ( ) and ( ) as: ( ) ( ) ( ( ) ( ) ( )) ( ) ( ) ( ( ) ( )) where ( ) and ( ) are demand and exit flow at time step, respectively. The terms ( ), ( ), ( ), and ( ) are independent of the control action; therefore, they can be removed from the weighted total travel time with no effect on system performance. By substituting (12) and (13) into (11) and after removing independent terms, we obtain: ( ( ) ( )) Therefore minimizing is equivalent to maximizing mainline early exit flows in addition to on-ramp early exit flows with a weight. Consequently, the reward in each time step was defined as the number of cars passed the loop detector downstream of the on-ramp in addition to number of cars passed on-ramp loop detector with weight of. In this study, was considered to be.1. D. Q-learning Parameters The following parameters should be chosen carefully to ensure proper performance of the RL agent: control cycle, learning factor, discount factor, and action selection policy. In this study a control cycle of 2 minutes was selected. The control cycle was chosen to be long enough so that the freeway conditions can be estimated with enough

5 Flow (Veh/Hour) Total Travel Time (Veh.Hour) accuracy, but at the same time not very long control cycle to enable capturing the effect of the control cycle on the measurements. The discount factor was chosen to be.8. The learning factor,, at each step through the learning process was determined based on the number of visits to the state-action pair by the following equation: ( ( )) where ( ) is the times that state-action pair ( ) has been visited by the agent. For action selection, the soft-max action selection policy was used. ( ) ( ( ) ) ( ( ) ) where is a positive scalar that affects the level of randomness in action selection. At the extreme, a zero value for will result in totally random action selection and as increases to infinity the action selection will become greedy. In this study the parameter was defined separately for each state as the number of visits to that state divided by the possible actions. This dependence will ensure that the softmax policy will become greedy policy after all state-action pairs in a state have been visited enough. V. CASE STUDY RESULT Paramics was used to evaluate the performance of different scenarios in a microscopic simulation environment. There are numerous sources of randomness in Paramics; therefore, various simulation runs with different seeds but with exactly same inputs will result in different outcomes. Even though it is a natural phenomenon and it actually happens in real life, for the purpose of evaluation we decided to eliminate any source of uncertainty and randomness by conducting 3 different simulation runs for each controller the results were averaged across these runs. The two RL agents were trained for 9 epochs. The learning performance of the two agents is shown in Fig. 4. As can be seen in this figure, the NN-TD agent is converged to optimal Q-values after about 3 epochs while the conventional Q-learning agent achieved same result after 9 epochs. To compare the proposed NN-TD ramp metering agent with other traffic-responsive ramp metering algorithms, ALINEA controller [3] was considered as the benchmark. ALINEA controller is a PI controller with the proportional gain set to zero as shown in the following equation: ( ) ( ) ( ( )) In (17) ( ) and ( ) are metering rates at current control cycle and next control cycle, respectively. Integral gain is the controller parameter, is the desired density and ( ) is the current measured density. In practice the desired density is usually set slightly lower than critical density to avoid falling into the congested side of fundamental traffic flow diagram. ALINEA controller is very simple to design and does not require knowledge of the network for tuning the controller gain. In this paper, the desired density and the controller gain was found by trial and error. Different values for desired density and controller gain were examined and the one that resulted in the lowest total travel time for the whole network was selected Figure 4. Learning convergence for NN-TD and conventional Q- learning in RL ramp metering problem In addition to ALINEA, the network without ramp metering is also simulated to show the improvement that can be achieved through deployment of RM in the case study. The demands from the 3-lane freeway mainline and on-ramp, calculated based on actual freeway loop detector counts obtained from ONE-ITS, are shown in Fig Figure 5. The demand profile from on-ramp and freeway mainline Table II summarizes the freeway network performance under the three control cases. NN-TD ramp metering has significantly reduced total travel time by 44% compared to the case with no ramp metering. Although ALINEA controller was successful overall in reducing the total travel time by around 27%; it aggressively kept the mainline travel time low at the expense of sever congestion and long queue at the on-ramp. TABLE II. Performance Measures FREEWAY NETWORK PERFORMANCE USING DIFFERENT CONTROLLERS No ramp metering Control Method ALINEA controller TTT (Veh.Hour) TTT savings - 27% 44% Mainline TTT (Veh.Hour) Average on-ramp queue (Veh) epcohs Conventional Q-learning knn-td Time of Day (Hour) Mainline Demand On-ramp demand NN-TD ramp metering

6 On-ramp queue length (Veh) Density (Veh/Km/Lane) For illustration purposes, the freeway mainline density downstream of the on-ramp and on-ramp queue for one representative run of the 3 simulation runs for the three control cases are shown in Fig. 6. It can be seen from the figure that NN-TD ramp metering accepts higher densities compared to ALINEA while achieving the compromise between mainline travel time and the on-ramp queue length No ramp metering ALINEA knn-td ramp metering Time of Day (Hour) Time of Day (Hour) Figure 6. Top graph: the meainline density downstream of the on-ramp. Down graph: on-ramp queue length VI. CONCLUSION AND FUTURE WORK There are numerous challenges associated with implementation of an RL-based RM control system in realworld conditions. In this paper we have demonstrated how the measurements from loop detectors can be used for the training and deployment phase of the RL agent using real-life data from Highway 41 in Toronto. A microscopic traffic simulation model is used as a replica of the real-world and a testbed for evaluating three scenarios: do-nothing (no metering), ALINEA controller, and knn-td controller. The experimental result have shown the superior performance of the proposed RL-based RM control strategy compared to ALINEA algorithm. Furthermore, the continuous and discrete representations of state space were compared and it has been shown that the RL agent with continuous state space learns much faster than the one with discrete states. This work is in fact the first step of a work in progress toward an integrated closed loop optimal freeway traffic control. The algorithm presented in this paper is a local ramp metering algorithm and has the limitations of the other local algorithms. Our next step is to coordinate local agents in order to improve the overall performance of the freeway network in addition to providing equity for drivers using different on-ramps. Another research direction is to expand the application of RL to other traffic control measures such as variable speed limit and route guidance. REFERENCES [1] M. Papageorgiou, et al., Review of road traffic control strategies. Proceedings of the IEEE, (12): p [2] M. Papageorgiou and A. Kotsialos, Freeway ramp metering: An overview. IEEE Transactions on Intelligent Transportation Systems, 22. 3(4): p [3] M. Papageorgiou, H. Hadj-Salem, and J. M. Blosseville, ALINEA: A local feedback control law for on-ramp metering. Transportation Research Record, : p [4] E. Smaragdis and M. Papageorgiou, Series of new local ramp metering strategies. Freeways, High-Occupancy Vehicle Systems, and Traffic Signal Systems 23, 23(1856): p [5] I. Papamichail, et al., Coordinated ramp metering for freeway networks - A model-predictive hierarchical control approach. Transportation Research Part C-Emerging Technologies, (3): p [6] T. Bellemans, B. De Schutter, and B. De Moor. Model predictive control with repeated model fitting for ramp metering. in 5th International IEEE Conference on Intelligent Transportation Systems, 22. [7] R. S. Sutton and A. G. Barto, Reinforcement Learning: An Introduction. 1998, Cambridge, MA: MIT Press. [8] C. Jacob and B. Abdulhai, Machine learning for multi jurisdictional optimal traffic corridor control. Transportation Research Part a-policy and Practice, (2): p [9] Z. Dongbin, et al., DHP Method for Ramp Metering of Freeway Traffic. IEEE Transactions on Intelligent Transportation Systems, (4): p [1] M. Davarynejad, et al. Motorway ramp-metering control with queuing consideration using Q-learning. in 14th International IEEE Conference on Intelligent Transportation Systems (ITSC), 211. [11] L. P. Kaelbling, M. L. Littman, and A. W. Moore, Reinforcement learning: A survey. Journal of Artificial Intelligence Research, : p [12] B. Abdulhai and L. Kattan, Reinforcement learning: Introduction to theory and potential for transport applications. Canadian Journal of Civil Engineering, 23. 3(6): p [13] C. J. C. H. Watkins, Learning from Delayed Rewards. 1989, Cambridge University. [14] B. Abdulhai, R. Pringle, and G. J. Karakoulas, Reinforcement learning for true adaptive traffic signal control. Journal of Transportation Engineering, (3): p [15] S. El-Tantawy and B. Abdulhai. An agent-based learning towards decentralized and coordinated traffic signal control. in 13th International IEEE Conference on Intelligent Transportation Systems (ITSC). Madeira, Portugal, 21. [16] C. Watkins and P. Dayan, Q-learning. Machine Learning, (3-4): p [17] A. Gosavi, Reinforcement Learning: A Tutorial Survey and Recent Advances. Informs Journal on Computing, (2): p [18] J. A. Martin, J. de Lope, and D. Maravall, The knn-td Reinforcement Learning Algorithm, in Methods and Models in Artificial and Natural Computation, Pt I, J. Mira, et al., Editors. 29, Springer-Verlag Berlin: Berlin. p [19] J. A. Martin, J. de Lope, and D. Maravall, Robust high performance reinforcement learning through weighted k-nearest neighbors. Neurocomputing, (8): p [2] Quadstone Paramics. [cited 212 March 1]; Available from: [21] H. Abdelgawad, et al., Simulation of Exclusive Truck Facilities on Urban Freeways. Journal of Transportation Engineering-Asce, (8): p [22] M. Zhang, J. Ma, and H. Dong, Developing Calibration Tools for Microscopic Traffic Simulation Final Report Part II: Calibration Framework and Calibration of Local/Global Driving Behavior and Departure/Route Choice Model Parameters. 28, California PATH Research Report. [23] M. Papageorgiou and I. Papamichail, Overview of Traffic Signal Operation Policies for Ramp Metering. Transportation Research Record, 28(247): p