1.INTRODUCTION
In recent years, a microscopic simulation has received increasing attention from researchers in the field of transportation owing to its precise analysis capabilities of traffic in different scenarios (Yuan et al., 2017; Errampalli et al., 2013). It enables accurate estimation of realistic travel time and traffic flows of a road network by considering vehicle movements, such as carfollowing (CF) and lane changing (LC). In order to enhance the performance of the microscopic simulation, these two models have been extensively studied (Balal et al., 2016; Hidas, 2002; Pan et al., 2016; Yang and Koutsopoulos, 1996; Yeo and Skabardonis, 2007).
Early studies on CF and LC modeling mainly concentrated on physical movements of vehicles. For example, CF models calculate a vehicle’s position and speed at the next time step based on its relationship with a leading vehicle. Wellknown and extensively used CF models include the Gipps model (Gipps, 1981), and Newell’s model (Newell, 1965, 2002) and its variations, such as the Daganzo’s CF linear model (Daganzo, 2002, 2006). Similar to CF models, LC models also focused on a physical movement condition of a subject vehicle to prevent vehicle crashes. A wellknown model is the gap acceptance LC model used to measure a safety gap between a leading vehicle and a following vehicle on a target lane (Ahmed, 1999). Once the gap acceptance condition is satisfied, a subject vehicle changes its lane to the target lane.
Although these CF and LC models can illustrate the movements of both major vehicles in great detail, the driver’s reasoning process needs to be also considered for accurate modeling of the driver’s behavior. Specifically, to enhance the driving speed by taking an LC action, a driver should select a target lane before one’s own action (Yeo and Skabardonis, 2007). This is known as discretionary lane changing (DLC). A major challenge of the lane selection modeling in DLC is to mimic the driver’s reasoning process accurately. In general, a binary logit model in conjunction with speed or traffic volume of a target lane (Yang and Koutsopoulos, 1996; Hidas and Behbahanizadeh, 1998) has been used to illustrate the lane selection behavior of a driver (Ahmed, 1999). It computes an LC desire probability of drivers on a target lane with the highest speed or lowest traffic volume. In other words, under a selected target lane, the model chooses on whether to take into account the LC behavior or not, and does not involve a reasoning uncertainty for the selection of a target lane. Thus, the lane selection against multiple target lanes (e.g., right and left lanes) cannot be captured by the binary logit model. In addition, the binary logit model does not consider perception aspects of drivers on road conditions. To consider a degree of uncertainty of a driver in LC, Balal et al. (2016) proposed a Fuzzy logicbased LC model involving 51 rules to notify the timing of LC to a driver.
The goal of this study is to propose a realistic lane selection model to address the DLC behavior regarding the individual driver’s uncertain perception and reasoning processes, via the extended decision field theory (EDFT). EDFT is a psychological decisionmaking algorithm used to represent complex realtime deliberation of humans (Lee et al., 2008). Unlike the Fuzzy logic used in Balal et al. (2016)’s study, EDFT is able to numerically consider deliberation process over time and correlations between multiple options. In addition, the Bayesian network (BN) is employed to mimic limited perception capabilities of drivers on dynamic road conditions. The proposed EDFTbased lane selection model is implemented in AnyLogic^{®} agentbased simulation (ABS) platform in conjunction with a nextgeneration simulation (NGSIM) CF model (Yeo and Skabardonis, 2007) for experiments. The proposed model is compared to the NGSIM DLC model in terms of the modeling accuracy and execution time.
The organization of the paper is as follows: Section 2 reviews the existing LC models in ABS and provides a brief introduction to the EDFT and its key properties. In Section 3, the EDFTbased lane selection model for DLC is proposed. Section 4 suggests a calibration method regarding the inference accuracy of the proposed model based on an NGSIM traffic dataset on Peachtree Street in Atlanta, Georgia. Section 5 illustrates the proposed lane selection model under an ABS environment and demonstrates its performance under traffic congestion. Finally, Section 6 provides conclusion and future work comments on this topic.
2.BACKGROUND
2.1.Lane Changing Model
There are two major types LC models: (1) the mandatory lane changing (MLC), and (2) the discretionary lane changing (DLC). First, MLC occurs when a vehicle needs to follow another roadway associated with its route plan. In this case, the driver cannot have an opportunity to select a target lane since the target lane is determined by a driver’s route plan. For example, when a vehicle tries to follow an exit ramp on a highway, it has to change its current lane to the rightmost lane. Thus, MLC only includes gap acceptance models (i.e., lead gap and lag gap acceptance models) to prevent crashes between a subject vehicle and vehicles on a target lane. The lead gap model considers a safety distance between a lead vehicle on a target lane and a subject vehicle, and the lag gap model regards a safety distance between a following vehicle on a target lane and a subject vehicle. Eq. (1) and Eq. (2) show both gap acceptance models (Yeo and Skabardonis, 2007). Let the (n+1)'^{th} vehicle and the (n+1)'^{th} vehicle be the leading and following vehicles to the n^{th} vehicle on a target lane, respectively. If any of the following two conditions are not satisfied, the n^{th} vehicle is not allowed to change its lane.
where
x_{n}, v_{n}, d_{n}, are position, speed, and the stopping distance of the n^{th} vehicle, respectively. Δt is the simulation time step, τ_{n} is the wave travel time (or perception reaction time) for the n^{th} vehicle, ${g}_{n}^{jam}$ is the jam spacing for the n^{th} vehicle, and l_{n} is the length of the n^{th} vehicle. Eq. (1) guarantees a minimum gap in the case where vehicle n responds to the deceleration of vehicle (n1)'. Similarly, if the following vehicle (n+1)' decelerates in response to vehicle n’s deceleration in LC, Eq. (2) formulates the safety condition to avoid a crash. Herein, it is assumed that the LC vehicle emits a signal beforehand so the upstream vehicle in a target lane can start to decelerate immediately.
DLC is optional and allows a vehicle to increase speed or improve position (Laval and Leclercq, 2008; Yeo and Skabardonis, 2007). In addition to the gap acceptance models for MLC, DLC involves the driver’s decisionmaking process. For instance, a NGSIM DLC model proposed by Yeo and Skabardonis (2007) computes a probability π of LC from a current lane to an adjacent lane j to increase the speed or improve position [see Eq. (3)].
where φ_{n} ≤ Δt is a timeunit sensitivity factor that represents how sensitive a driver is to the relative speed. In addition, v^{j}(t) is the average speed of the vehicles within a certain range ahead of a subject vehicle, and ${v}_{n}^{f}$ is the freeflow speed of the n^{th} vehicle at time t. Eq. (3) is also known as the rulebased DLC model used to compute LC desirability. The rulebased DLC model only considers the selected target lane with the highest speed. Similarly, Das and Bowles (1999) selected the target lane based on the congestion level of the lane.
There is another approach, called a binary logit model suggested by Ahmed et al. (1996) for MLC, applied in order to consider multiple attributes to compute desirability of MLC [see Eq. (4)].
where X_{n}(t) is a vector of explanatory variables, β is a vector of parameters, and α is a parameter of v_{n}(t). Although Eq. (4) computes desirability of MLC, the model can be applied for the DLC by selecting X_{n}(t) associated with DLC. The binary logit model can be extended with artificial intelligence models, such as fuzzy logic (Das and Bowles, 1999) and artificial neural networks (Dumbuya et al., 2009) in order to capture the uncertain perception of the drivers on attributes related to the desirability computation. However, it still cannot be used to compute selection probabilities of multiple target lanes.
2.2.Extended Decision Field Theory
Although there are many desirability computational models of DLC (mentioned in Section 2.1), a new model is needed for the modeling of the lane selection behavior of drivers. In particular, correlations between options should be considered to mimic the lane selection behaviors of drivers among multiple lanes. Instead of assuming that a selected target lane always has the highest speed or utility against other lanes, a theoretical mechanism is needed to represent an uncertain driver’s reasoning process (i.e., a probabilistic model for the selection).
Decision field theory (DFT) has developed as a mathematical framework to analyze the human cognitive deliberating process, and is based more on psychological principles rather than economic principles (Busemeyer and Townsend, 1993). DFT obtains the human preferences among multiple options by evaluating them via different aspects (i.e., attributes). Lee et al. (2008) extended DFT to deal with a dynamically changing environment. Specifically, EDFT updates preferences of options based on observed values of attributes over time, and then selects an option with the highest preference value after it has spent a certain amount of deliberation time (i.e., the final decision time). EDFT describes the evolution of a human preference in accordance to Eq. (5).
In Eq. (5), P(t) is a m×1 vector representing preference on each of the m options at time t (t < T_{D}), where T_{D} is the final decision time and h is the time step. M(t) (m×n matrix, where m is the number of options and n is the number of attributes for each option) is the value of the matrix representing a decisionmaker’s subjective perception on each attribute for each option at time t. W(t) is a n×1 vector that allocates weight to each attribute at time t. The weight vector reflects the attention of the decisionmakers on each attribute. S (m×m matrix) is the stability matrix in which the diagonal elements represent the memory effect from the previous preference state, and the offdiagonal elements denote the inhibitory interactions among the competing options. For instance, if s_{ii} = 0.9 and s_{ij} = −0.01, this means that the memory effect decays slowly and there is no significant interaction between options. To maintain the stability of this linear system, the eigenvalues λ_{i} of S are assumed to be less than one in magnitude (λ_{i}  < 1). C (m×m matrix) is the contrast matrix comparing the weighted evaluations of each option, M(t)W(t). In the contrast matrix, c_{ii} = 1 and c_{ij} = −1/(m− 1) for i ≠ j. Thus, increasing a preference for one option will decrease the preference for the alternative options.
There are three major advantages of EDFT in the modeling of the decision making for the behavior of humans. First, it provides evolution trajectories of preferences within the final decision time (T_{D}) so that researchers are able to understand the reasoning process of a human in greater detail. Second, multiple attributes with different weights can be considered via M(t) W(t). Third, a correlation between options is regarded via the contrast matrix (C). Thus, it selects an option that has a relatively higher preference value than those of other options. Owing to these modeling capabilities of EDFT, it is able to accurately represent the reasoning process of drivers in DLC.
3.EXTENDED DECISION FIELD THEORYBASED LANE SELECTION MODEL
In this study, the lane selection behavior in DLC is modeled via EDFT (see Section 2.2). The proposed EDFT based lanes selection model is going to choose one lane among multiple lanes (e.g., LC to the left or right lanes) so that it is quite different from the NGSIM DLC model that computes a desire choice probability of LC from a current lane. In addition, three attributes are considered for the evaluations of each lane i: 1) the predicted speed of a lane i (v_{n, i}) at time t+t_{LC}, 2) the predicted lead gap between a subject vehicle on a current lane and a leading vehicle on a target lane i (${g}_{n,\text{\hspace{0.17em}}i}^{lead}$) at time t+t_{LC}, and 3) the predicted lag gap between a subject vehicle on a current lane and a following vehicle on a target lane i${g}_{n,\text{\hspace{0.17em}}i}^{lag}$ at time t+t_{LC}. Herein, t_{LC} is the travel time for LC. However, the proposed EDFTbased lane selection model can consider other attributes in addition to the three attributes. Eq. (6) shows an example of M(t, n) when a driver is on a roadway that has three lanes (see Figure 1). The entry values of M(t, n) represent attribute conditions at time t.
To represent the drivers’ uncertain inference based on the three attributes under a dynamic traffic environment, the BN is applied to EDFT. BN is a directed acyclic graph (DAG) in conjunction with conditional probabilities between variables. Because it is able to infer states of each variable from observed (or unserved) variables in DAG via the conditional probabilities, it is extensively used to represent an uncertain inference behavior of a human (Neapolitan, 2004). Figure 2 shows a BN topology used in the proposed EDFTbased lane selection model.
Three attributes are considered as the child nodes in BN for the five parent nodes. For a target lane i, states of three attributes at t + t_{LC} are inferred by the observed states of the five parent nodes at t with the following conditional probabilities:
According to Eq. (7), the future speed of lane i after the subject vehicle has changed a lane [i.e., v_{n,k}(t+t_{LC})] is inferred based on the current perception on the lead gap ${v}_{n,\text{\hspace{0.17em}}k}\left(t+{t}_{LC}\right)$, the lag gap $\left[{g}_{n,\text{\hspace{0.17em}}k}^{lead}\left(t\right)\right]$, speed of the leading $\left[{v}_{n1,\text{\hspace{0.17em}}i}\left(t\right)\right]$, the following $\left[{v}_{n+1,\text{\hspace{0.17em}}i}\left(t\right)\right]$, and subject [v_{n}(t)] vehicle speeds. Similarly, a lead gap and a lag gap are inferred from Eq. (8) and Eq. (9), respectively.
Once all BN states for all attributes in each lane are inferred, M(t, n) [see Eq. (6)] is constructed and used for the EDFT evaluation with a weight vector, W(t), configured based on a driver type, such as aggressive or conservative drivers (see Section 5.2 for more details). EDFT selects a lane with the highest preference value at the final decision time (T_{D}). Owing to the randomness of the BN inference, this selection process of EDFT needs to be executed at multiple times (i.e., multiple sampling) in order to obtain statistically convincing results from BN. These selections are normalized as the selection probability of each lane i based on the following equation:(10)
where N_{i} is a selected number of lanes i, and N is the total sampling number. If the current lane has the highest choice probability, a vehicle is likely to stay in the current lane. Otherwise, a vehicle moves to a selected target lane. In this way, the proposed model represents the real driver’s uncertain perception and reasoning processes.
3.1.Computational Complexity
As mentioned in Section 2.2, EDFT updates the preferences of the options based on the observed values of the attributes over time until a decisionmaker selects an option, after it has spent a certain amount of deliberation time (T_{D}) (Lee et al., 2008). If n iterations are needed to compute the preferences within T_{D}, the computational complexity of EDFT is O(n). When the value of the matrix M changes during the deliberation time T_{D} (e.g., n^{(1)}, n^{(2)}, …, n^{(q)}), the computational complexity of EDFT significantly increases. However, this computational complexity can be reduced if we are only interested in stabilized preference values when T_{D}→∞ (i.e., the expected preference values). Lee et al. (2008) has derived two theorems from the original DFT about the expected preference value calculation for the two options of the decision making problem, using EDFT that y_{i}elds the minimum amount of time steps required for the stabilization of the preference values. Let V(t) = CMW(t), v_{i}(t) be ith row element of the V(t) matrix, and s_{ij} be an element of the stability matrix S. A preference value of option l at deliberation time h is then listed below (Lee et al., 2008)(11)(13)(16)
where(14)
K is the number of attributes and L is the number of options. If the deliberation time is 2h, the expected preference can be expressed in accordance to the equation listed below:(17)
where D = s_{1}s_{2}. Eq. (18) can be generalized with multiple deliberation time steps (nh, n ≥ 1).
If n → ∞,
Equation (20) can also be applied in the case where the value matrix M changes over time (n^{(1)}, n^{(2)}, …, n^{(q)}) (Lee et al., 2008). Correspondingly, q is considered as the number of inferences via BN. From Eq. (20),(21)(22)
Thus,(23)
As a result, the expected preference of EDFT is(24)
Let us name the original approach of waiting for convergence as the timestep approach [i.e., p_{l}(h)] and the approach that uses the expected preference value as the expectedvalue approach {i.e., E[p_{l}(h)]}. Since the expected value approach directly computes its preference without considering multiple iterations [n^{(i)}], the computation time becomes O(q). Table 1 shows the execution time between the timestep approach and the expectedvalue approach. All execution times of the expectedvalue approach are less than the timestep approach. Moreover, the gap of the execution time increases as the number of options is increased. Thus, the expectedvalue approach is more efficient than the timestep approach. The experiments are performed on a dualcore desktop processor, namely, an Intel Core 2 Duo P8400 running at 2.26 GHz.
4.MODEL CALIBRATION
4.1.Description of a LC Dataset
The proposed EDFTbased lane selection model was calibrated with a vehicle trajectory dataset collected from 12:45 pm to 1:00 pm, and from 4:00 pm to 4:15 pm, on Peachtree Street in Atlanta, Georgia on November 8, 2006 [see Systematics (2007) for more details]. Vehicle trajectory data were processed from the NGSIM video images. The data provide the X and Y coordinates of each vehicle, every 1/10^{th} of a second in relative space and in NAD83 (the units are US Survey feet). However, the SI units are used in this study.
Table 2 and Table 3 list the lane selection behaviors of drivers on Peachtree Street, Atlanta, Georgia. The total number of lane changes was 2,541 (1,219 from 12:45 pm to 1:00 pm and 1,322 from 4:00 pm to 4:15 pm). In both tables, all vehicles on lane 0 changed to a right lane since the only the right lane was available for LC. Owing to the same reason, all vehicles on lane 3 changed to a left lane. Vehicles on lane 1 (81.95% of vehicles from 12:35 pm to 1:00 pm and 82.42% of vehicles from 4:00 pm to 4:15 pm) tended to occupy a left lane even though the left lane had a similar average speed. This is because the left lane had higher average lead and lag gaps than those of lane 1. However, vehicles on lane 2 showed a different LC pattern. Most vehicles (96.97%) from 12:35 pm to 1:00 pm occupied their left lane which had a high average speed than that of the right lane. In total, 92.91% of vehicles from 4:00 pm to 4:15 pm also occupied their left lane since it had a higher average speed than their right lane. From these results, the speed of a lane is a significant factor for LC as Yeo and Skabardonis (2007) claimed. However, the lead and lag gaps are also significant factors for LC, especially when each lane has a similar speed.
For better understanding on the impact of the speed on LC, the individual vehicle lane selection behavior is analyzed. In a dataset collected from 12:45 pm to 1:00 pm, 75.47% of 1,219 vehicles selected a target lane to attain a higher vehicle speed. The choice probabilities of lane selections associated with the vehicle speed in each lane were 85.47% in lane 0, 66.48% in lane 1, 73.76% in lane 2, and 75.00% in lane 3. However, in a dataset collected from 04:00 pm to 04:15 pm, only 57.79% of 1,322 vehicles selected a target lane for increasing the speed of the vehicle. The choice probabilities of lane selections associated with vehicle speed in each lane were 61.12% in lane 0, 50.09% in lane 1, 72.73% in lane 2, and 79.17% in lane 3. These results imply that only the consideration of the increase of the vehicle’s speed may not be enough to accurately explain the lane selection behavior of the drivers. Thus, multiple attributes (e.g., lead and lag gaps) in addition to speed increases from the acquired speed in a current lane need to be considered for accurate lane selection modeling. The results also show that road conditions change over time so that the speed of a lane after LC can be different from the perceived speed of a lane before LC. Figures 3 and 4 reveal distributions of LC vehicles under different speed categories. The blue color and the red color represent the “before” and “after” cases.
In Figure 3 and Figure 4, the ‘Before LC’ results refer to average lane speeds of LC vehicles when the drivers selected a target lane, and the ‘After LC’ results show average lane speeds after the drivers changed their lane. If ‘Before LC’ and ‘After LC’ have the same distribution, we can conclude that driver’s experienced speed of a target lane after LC is exactly same as their expectation. Pvalues of the paired ttest for similarity comparison between the average lane speed before and after LC are 2.81×1031 in Figure 3, and 1.42×10^{240} in Figure 4. Effectively, there were differences in the average lane speed for both the before and after the LC cases at the 0.05 confidence interval. This implies that drivers need to expect future road conditions based on their past observations and utilize the predicted information for their target lane selections. This perceptionbased inference behavior of drivers will be captured by BN in the proposed EDFTbased lane selection model.
4.2.Parameter Estimation of Bayesian Network
The advantage of BN is that it infers states of attributes in a probabilistic manner. To this end, BN has a complex conditional probability involving all possible combinations of variables associated with the attributes. For example, if there is a binary attribute influenced by five other binary variables, a conditional probability between the six variables should y_{i}eld (2)^{6} states. Thus, when BN has many variables with multiple states, a large memory space is required to infer the states of an attribute. This problem is a significant issue in ABS, especially when each agent has its own BN structure. It can be resolved by discretizing continuous variables with the minimum number of states (Dougherty et al., 1995). This study employs a concept of an errorbased discretization approach (Maass, 1994) that aims to identify the number of states (θ_{i}) of a discretized variable (y_{i}) that minimizes the inference error of BN. Eq. (25) shows an errorbased discretization approach for BN.
where θ = [θ_{1}, …, θ_{n}] and θ_{i} ≥ 2. N is the number of attributes (i.e., n = 3 in this study), R is the number of observations in a dataset, π_{ik} is a set of parent variables of a discretized attribute y_{ik}, and G is the topology of BN. During the optimization, variables are discretized with respect to θ_{i}. The conditional probability of a discretized attribute y_{i} is computed with the following formula:
where N_{ijk} is the number of cases in a dataset with a configuration j for π _{i} when the variable y_{i} has value k. In addition, r_{i} is a possible value assignment of y_{i}. Utilization of kfold crossvalidation (CV) — that refers to an assessment method of the statistical model to avoid an overfitting problem (Kohavi, 1995) — allows the appropriate number of states of each variable to be easily determined. Table 4 shows the discretization results (k = 10).
In Table 4, parentheses represent the range of observed values (left value is the observed minimum value and the right value is the observed maximum value). As shown in Eq. (25) and Eq. (26), the number of states is determined by the value range of each variable. The lead gap has the lowest inference error because it has the highest number of states among the three variables and smaller value range than lag gap. The average inference errors of speed, lead gap, and lag gap are 0.14, 0.07, and 0.09, respectively.
4.3.Parameter Estimation of EDFT
Once the BN model is developed to represent the driver’s inference behavior on attributes, the EDFTbased lane selection model is calibrated with an observed choice probability for each lane. According to Eq. (19), a preference for a lane l can be illustrated as follows:
where ${m}_{lk}=\text{Pr}\left({y}_{lk}=k{\pi}_{lk}=j\right)\text{Pr}\left({\pi}_{lk}=j\right)$ determined by BN. Eq. (28) shows the EDFTbased lane selection model that chooses a lane with the highest preference among alternatives.
In Eq. (27), since (1D^{n}/1D) is a common term between formulae for different lanes, Eq. (28) is simplified by cancelling out the common term. Thus, p_{l}(h) can be used instead of p_{l}(nh).(29)
Thus, an appropriate M matrix can be found using the following equation:(30)
where t_{i} is an observed lane selection and g_{i}[W(t)] is the inferred lane selection from EDFT. Similar to the parameter learning of BN mentioned in Section 4.2, kfold CV can be also utilized to find an appropriate W(t) that elicits the minimum estimation errors in accordance to the following equation:(31)
Table 5 shows the results of the CV test. Similar to the NGSIM DLC model, “EDFT with speed” only regards the speed of lanes as its attribute (i.e., W(t) = [1 0 0]). The result shows that the EDFTbased lane selection model with three attributes has a higher inference accuracy (i.e., the average inference accuracy is 0.89) compared to those of the “EDFT with speed” case (i.e., the average inference accuracy is 0.68). This means that the speed is not the only attribute to explain the LC behaviors of the drivers. Thus, the calibrated EDFTbased lane selection model with three attributes can be used for the illustration of LC behaviors of drivers
5.SIMULATION STUDY
5.1.Scenario
The calibrated EDFTbased lane changing model in Section 4.3 is illustrated with a single oneway link involving five lanes (see Figure 5). In Figure 5, all vehicles drive from a right side (origin) to a left side (destination). A control signal is installed at 762 m where queues start to build up. The control zone is cleared at t = 600 s. In this scenario, behaviors of three different types of drivers (see Section 5.1.1) are investigated under various levels of traffic congestions. Therefore, the computational complexity of the EDFTbased lane changing model is discussed in terms of the execution time. The ABS model employs the NGSIM CF model to realize the speed and position adjustment of the vehicles under different traffic congestions. Parameter values used in the CF model are listed in Table 6. All the experiments are performed on a dualcore desktop processor, namely, an Intel Core 2 Duo P8400 running at 2.26 GHz.
5.1.1.Agent Categorization
In the model, we have considered three types of agents with different lane selection characteristics as de scribed below:

Nominal drivers follow the EDFTbased lane selection algorithm (see Section 3) that considers three attributes in the decisionmaking process. In addition to the average lane speed within a certain range ahead (i.e., lane speed), drivers take into account the lead and lag gaps (see Section 4.3). The calibrated weight vector W(t) (see Table 7) from the NGSIM dataset is used in the EDFTbased lane selection model..

Aggressive drivers follow EDFTbased lane selection and attribute the highest weight to the lane speed among the aforementioned three attributes.

Conservative drivers follow the EDFTbased lane selection and attribute the highest weight to the lead and lag gaps of a target lane.

NGSIM DLC drivers follow the NGSIM DLC model [see Eq. (3)] where the lane speed is the only attribute considered in the decisionmaking process. In this study, φ_{n} = Δ t
5.2.Validation of NGSIM CarFollowing Model
To represent the physical movements of vehicles along a roadway in ABS, an NGSIM CF model (Yeo and Skabardonis, 2007) has been employed. At every simulation time step Δt, the model updates the position of the n^{th} vehicle based on: 1) its own capabilities (i.e., maximum acceleration/deceleration rate, freeflow speed, and current speed), and 2) the action of the (n1)^{th} vehicle (i.e., a vehicle leading the n^{th} vehicle). The n^{th} vehicle’s position at time t+Δt is given by Eq. (32).
Here, ${x}_{n}^{U}$ and ${x}_{n}^{L}$ are the upper and lower bounds of the position at t+Δt, respectively. Correspondingly, ${x}_{n}^{U}$ is restricted by the four safety constraints: 1) Newell’s CF rule involving the length of the (n1)^{th} vehicle (l_{n1}) and jam spacing of the n^{th} vehicle (${g}_{n}^{jam}$), 2) the maximum acceleration capability (${a}_{n}^{U}$), 3) the freeflow speed (${v}_{n}^{f}$), and 4) the maximum safety driving distance ($\Delta {x}_{n}^{s}$).(33)
where $\Delta {x}_{n}^{s}$ is the safety driving distance derived from Gipps (1981) and expressed in accordance to Eq. (34).
where τ_{n} is the wave travel time (or perception reaction time) for the n^{th} vehicle, and n 1 d − is the stopping distance of the (n1)^{th} vehicle with a maximum deceleration rate as given by Eq. (35).
The lower bound of position ${x}_{n}^{L}$ is constrained by the vehicle’s current position and the maximum deceleration capability is shown in Eq. (36).
The implemented NGSIM CF model in ABS is validated with several literature results via the densityflow diagram (see Figure 6), and the densityspeed diagram (see Figure 7).
Figure 6 depicts the densityflow diagram of the developed ABS model. In the figure, the flow becomes zero when the density is zero (i.e., when there is no vehicle on a roadway) or when the maximum or jam densities occur (i.e., vehicles cannot move). Generally, the maximum flow (i.e., 752 veh/h) occurs between the zero density and the maximum density zones. Owing to these three major characteristics, the densityflow diagram should show the triangular relationship between the density and the flow (Banks, 1989; Hall et al., 1986).
In Figure 7, the densityspeed relationship also matches the result elicited from the macroscopic trafficflow analysis (Greenberg, 1959). When the density is zero, the average speed of a lane is the freeflow speed [i.e., normal (60, 5) km/h]. On the other hand, the average speed becomes zero when a lane has the maximum density (i.e., 87 veh/km/lane). Because the elicited results in both figures match those of the fundamental diagrams of traffic flow, the developed ABS model can be used to evaluate the performance of the proposed lane selection model.
5.3.Comparing Execution of Lane Selection Models
In this section, the computational complexity of the proposed EDFTbased lane selection model is addressed in terms of execution time of simulation. There are three lane selection models for this experiment: (1) the NGSIM DLC model [see Eq. (3)], (2) the EDFTbased lane selection model using a timestep approach, and (3) the EDFTbased lane selection model using the expectedvalue approach. The real execution time of the simulation is divided by the number of vehicles in the simulation (i.e., s/veh). For the two cases that use the EDFTbased lane selection model, all drivers are set to be of the normal driver type (i.e., calibrated EDFTbased lane selection model). The comparison results under different arrival rates of vehicles are depicted in Table 8.
First, the average execution time of the drivers in all three cases increases with higher arrival rates. At each arrival rate, the EDFTbased lane selection with the timestep approach requires a much greater execution time than the time required by the other two implementations. One reason for this is that BN inference on the states of the three attributes is implemented in the EDFTbased lane selection model so that it has a higher execution time than that of the NGSIM DLC model that directly calculates an LC desire probability of drivers on a target lane. In addition, the EDFTbased lane selection model takes time for the preference values to converge if a timestep approach is used (see Section 3.1). However, the EDFTbased lane selection model with an expected value approach has a similar execution time to that of the NGSIM DLC model when the arrival rate is lower than 30 veh/min. The Pvalue of the paired ttest between both models at a 30 veh/min arrival rate is 0.0247 (≤ 0.05). If the arrival rate of the vehicles is larger, the EDFTbased lane selection model based on an expectedvalue approach requires slightly more execution time owing to its complexity as mentioned above. The Pvalue of the paired ttest between both models at an arrival rate of 36 veh/min is 0.1065 (≥ 0.05). However, given the inference accuracy of the EDFTbased lane selection model shown in Table 5, the gap in the execution time will not be the problem. In the following sections, all the EDFTrelated modeling will use the expected value approach.
5.4.Comparison of Lane Selection Behaviors
In this section, the impact of three attributes on DLC behaviors is analyzed with three types of drivers as addressed in Section 5.1.1. In addition, the behaviors of EDFTbased lane selection are compared with the NGSIM DLC model. Figure 8 and Figure 9 show the cumulative LC attempts and LC actions, respectively, for all aggressive, normal, conservative, and NGSIM DLC driver types. The arrival rate of vehicles in ABS is 60 veh/min (see Table 6).
In Figure 8 and Figure 9, despite the differences in the individual response types, the overall patterns of the LC attempts and LC actions are similar. Both attempts and actions increase at the beginning of the simulation. Subsequently, they start to decrease at t = 300 s since the red control signal causes congestion on the road way and reduces the acceptable gaps for LC (i.e., lead and lag gaps) between vehicles. As soon as the congestion is released at t = 600 s, all the drivers try to change lane to increase speed or improve position. At t = 1,140 s, the numbers of LC attempts and LC actions are drastically reduced because all lanes tend to have similar speed and enough space for the LC (see Table 9). In other words, the traffic congestion in ABS is released.
In both figures, aggressive drivers elicit the largest cumulative number of LC attempts and LC actions. This is because they attribute the highest weight value to lane speed. Thus, even though a target lane has small lag and lead gaps for LC, drivers try to improve their speed via LC as frequently as possible. On the other hand, conservative drivers have the lowest number of LC attempts owing to the small lead and leg gaps under hightraffic congestion. In the normal driver case, the number of LC attempts is in the middle of the range between both driver types associated with W(t), as shown in Table 7.
The behavior of NGSIM DLC drivers is similar to that of aggressive drivers. Especially, at t = 5,000 s, the average number of attempts of NGSIM DLC drivers and aggressive drivers respectively elicit a mean of 4.0664 with a variance of 0.2508, and a mean of 3.8337 with a variance of 0.3532. The Pvalue of the paired ttest for attempt number comparison between both types is 0.2233 (≥ 0.05). Therefore, it is difficult to assess on whether both driver types y_{i}eld different number of attempts. Similarly, the average cumulative number of actions of NGSIM DLC drivers and aggressive drivers at t = 5,000 s are 0.8102 with a variance of 0.0055, and 0.6566 with a variance of 0.0061, respectively. Although the Pvalue of the paired ttest for action number comparison between both types is lower than the significance level of 0.05 (i.e., 0.0001), Figure 9 shows similar patterns between both driver types. This is because both driver types mainly consider the lane speed for their DLC. However, before t = 600 s, the NGSIM DLC drivers tend to y_{i}eld a lower number of attempts and actions than aggressive drivers. In fact, the patterns are similar to those of conservative drivers. This is because the NGSIM DLC model considers the difference between a current lane and a target lane based on the freeflow speed of a roadway in order to compute the desire probability of DLC [see Eq. (3)]. Thus, if the amount of speed improvement is relatively small, the NGSIM DLC model has a similar DLC pattern as that of the conservative drivers in the proposed lane selection model. However, considering the fact that normal drivers represent the DLC behaviors of the drivers in the NGSIM dataset, the NGSIM DLC model provides quite different behavioral patterns from the NGSIM dataset.
Figure 10 demonstrates that the cumulative ratio of the LC actions to the LC attempts changes over time. Similar to Figure 8 and Figure 9, the ratios of the four driver types show similar evolution patterns. The converged ratios for all driver types at t = 5,000 s are less than 0.5. This means that the LC actions are restricted by the physical condition of LC (i.e., the NGSIM gap acceptance condition). Among the three driver types, conservative drivers y_{i}eld the largest converged ratio (i.e., 0.3919). This is because they attribute high weight values to the lead and lag gaps. Invariably, this translates to a high probability of meeting the NGSIM gapacceptance condition. On the other hand, aggressive drivers y_{i}eld the lowest converged ratio value of 0.1724 because they do not significantly consider the lead and lag gaps for their DLC actions. The converged ratios of the normal drivers and NGSIM DCL drivers are 0.3192 and 0.1999, respectively. As mentioned earlier, the NGSIM DCL drivers tend to elicit similar behavioral patterns to those of aggressive drivers. However, quite different behavioral patterns are elicited from these drivers compared to those elicited by normal drivers. This means that both the lead and lag gaps are needed to be considered, in addition to the speed of a target lane, for accurate DLC behavioral modeling. Furthermore, the proposed EDFTbased lane selection model is able to demonstrate various behavioral patterns of drivers regarding multiple attributes. In other words, it can be used to represent LC patterns of young and old drivers (or drivers in a city and in a rural area). Thus, in ABS, the effects of various drivers on a road network can be addressed by the proposed lane selection model.
6.CONCLUSION
In this paper, the EDFTbased lane selection model has been proposed to mimic the discretionary LC behavior of the drivers. The proposed lane selection model takes into account attributes associated with driving safety (e.g., lead and lag gaps of a lane) in addition to a driving performance attribute (e.g., speed of a lane). In order to represent the drivers’ uncertain perception and inference on the three attributes, BN has been utilized. In other words, drivers select a target lane based on their anticipation on three attributes (i.e., speed, lead gap, and lag gap of a target lane) of lanes. The proposed model has been calibrated with NGSIM traffic dataset on Peachtree Street in Atlanta, Georgia. The calibration results have shown that the proposed model with the three attributes accurately represents LC behaviors of drivers in the dataset. In addition, the calibrated model has been implemented using the AnyLogic^{®} ABS software with microscopic CF and LC models. Experiments have been conducted with a straight roadway model involving five lanes in order to compare the execution time of the proposed model with the NGSIM DLC model and understand the impact of the three attributes on LC behaviors under congested traffic conditions. As a result, the proposed EDFTbased lane selection model is able to efficiently select a target lane and represent various behavioral patterns of drivers regarding multiple attributes. In addition, this implies that the model can be also applied to autonomous driving systems such as a lane departure warning system and adaptive cruise control and lane detection systems. Since the model is able to predict various LC patterns of adjacent vehicles, it provides warning signs to the autonomous systems, which will find a way to avoid potential collisions with adjacent vehicles.
For future work, humanintheloop experiments via a driving simulator are needed to find additional significant factors to improve the prediction accuracy of the proposed lane selection model. Moreover, the proposed model needs to be implemented in a largescale ABS to accurately evaluate performance (e.g., travel time and the level of congestions) of realroad networks.