A Microscopic Traffic Model Considering Time Headway and Distance Headway

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Abstract

A microscopic traffic model is presented which employs differences in velocity to characterize driver behavior. The Intelligent Driver (ID) model is based on an acceleration constant which cannot capture different traffic conditions. Further, it is not based on traffic physics and so can produce inaccurate results. The proposed model is an improved ID model and both are evaluated on a 2000 m circular road. The results obtained show that the proposed model can appropriately characterize traffic flow and density. Further, the variations in flow and velocity are smoother than with the ID model. This is because the proposed model is based on actual traffic parameters rather than an unrealistic traffic exponent.

1. Introduction

Traffic modeling plays a vital role in the planning, management, and development of road networks [1]. This requires accurate and realistic characterization of traffic behavior [2,3]. Traffic models have been extensively used to mitigate congestion [4] because it is a key concern in urban areas around the globe. Congestion causes excessive fuel consumption, air pollution, and safety issues, and has an adverse economic impact [4,5,6].

It has been shown that nearly 88 billion dollars were lost due to congestion in the United States in 2019 [7]. The increasing number of vehicles on the roads adds to congestion. For example, there were 284.5 million cars registered in the United States in 2019, and this increased by 0.84% in 2020 [8]. Large traffic queues are created during congestion which impede the smooth flow of traffic. Traffic flow is influenced by the time and space required for vehicles to adjust to the environment [9]. These factors affect driver response and result in velocity differences.

Three types of models are commonly used to characterize traffic: macroscopic, microscopic, and mesoscopic. Macroscopic models consider collective vehicle behavior and are typically used to determine velocity, flow, and density [10]. Microscopic models consider individual vehicle behavior using parameters such as time and distance headways, position, and velocity [11]. They are used to predict vehicle dynamics and are often based on driver response [12]. Mesoscopic models are a hybrid of microscopic and macroscopic models and so share the properties of both [13].

Pipes [14] and Reuschel [15] were the first to introduce microscopic traffic models. Velocity was determined using the distance between following and leading vehicles. However, their models are simplistic and cannot adequately characterize traffic behavior [16]. Newell [17] introduced a model which considers distance headway and velocity. With this model, large distance headways can produce high velocities and low-density traffic [5]. However, it can also create excessive acceleration, which is unrealistic. In [18], an Optimal Velocity Model (OVM) was proposed which depicts a constant driver response regardless of the traffic conditions. This is not a realistic characterization as traffic and velocity differences are ignored, leading to unstable dynamics [9]. Moreover, the acceleration can be very high when the velocity is far from the equilibrium distribution, and the density is not considered. It has been shown that this model leads to traffic accidents because of the small distances between vehicles.

Helbing and Tilch [19] introduced the Generalized Force Model (GFM) which employs negative velocity differences considering following vehicles and the OVM. Unfortunately, it can produce unrealistic results as there are rapid changes in acceleration. This is because only aggressive drivers are considered so slow and typical driver behavior is ignored [9]. This model was improved in [20] using both negative and positive velocity differences [5]. Gipps [21] developed a different traffic model but the acceleration is not realistic [16]. In addition, the behavior may not correspond to the model parameters [22].

Treiber et al. [22] developed the Intelligent Driver (ID) model which is an improvement of the Gipps model [21]. Driver behavior is considered along with distance headway and velocity to provide smooth acceleration [23,24]. The ID model has been used to avoid collisions during emergencies [25], and the results are similar to those obtained when observing real traffic [26]. However, it employs an acceleration constant $\delta$ which is the same for all traffic conditions, so traffic physics is ignored. The ID model was modified in [27] for traffic at signalized intersections. However, a value of $\delta =4$ was employed, so real traffic conditions were neglected [28]. This constant was chosen as the best fit for general traffic environments. A similar approach was employed in [29] for deceleration at intersections. In this case, the distances between vehicles are very small even with a high velocity, which is unrealistic [9]. The ID model has been incorporated in MovSim to evaluate longitudinal vehicle movement [25] and also in PTV VISSIM and Simulation of Urban Mobility (SUMO). However, driver behavior is not considered in this model [30].

One application of the ID model is with Connected and Autonomous Vehicles (CAVs) to improve traffic safety and passenger comfort [31,32]. Li et al. [33] employed this model to characterize the car-following behavior of CAVs, while Schakel et al. [34] proposed an improved ID model to explore CAV traffic stability. In this case, there is an abrupt decrease in velocity when the density reaches half its maximum (critical density), which is unrealistic as the velocity should be smooth. A modified ID model was proposed in [35] to better characterize real traffic conditions. However, CAV behavior with this model is not realistic because under actual traffic conditions, the drivers take longer to achieve a smooth car-following behavior and the variations in headway are greater [36]. The ID model has also been incorporated into cooperative and Adaptive Cruise Control (ACC) systems [37,38]. Unfortunately, the safe distance between vehicles with this model is too small when the velocity is high, which can result in accidents when employed in ACC and related systems.

A model is introduced here to realistically characterize traffic flow based on velocity differences and driver sensitivity. This improves the ID model which can produce unrealistic traffic behavior because of the constant acceleration exponent. Driver response to changes in velocity is known as sensitivity. The interval for a vehicle to adapt to variations in traffic conditions is referred to as the distance headway, while the time required to traverse this interval is known as the time headway. Both the proposed and ID models are evaluated for a platoon of 52 vehicles on a circular road of length 2000 m with periodic boundary conditions. The results obtained demonstrate that the proposed model provides better traffic characterization than the ID model.

2. Traffic Models

The ID model is a car-following model based on the behavior of leading vehicles. The acceleration is determined by the velocity, distance headway, and driver response which is $\frac{v}{v_d}$, where $v$ and $v_d$ are the average and the desired velocities, respectively. The acceleration is [22]

$$\dot{v}=a_m\left(1-{\left(\frac{v}{v_d}\right)}^{\delta }-{\left(\frac{s^{*}}{h}\right)}^2\right),$$

(1)

where $a_m$ is the maximum acceleration, $\delta$ is the acceleration constant, and $h$ is the distance headway as illustrated in Figure 1. The desired distance headway is [22]

$$s^{*}=J+\tau v+\frac{v∆v}{2\sqrt{a_{m}b}}$$

(2)

where $J$ is the jam spacing during congestion, $\tau$ is the time headway, $∆v$ is the change in velocity, and $b$ is the minimum acceleration.

With the ID model, traffic behavior is determined by the exponent $\delta$. However, this fixed value is inadequate for diverse traffic scenarios. Further, it is not based on real vehicle dynamics so poor results can occur. Hence, a variable exponent is proposed which is based on the difference in velocity between forward and rearward vehicles. This difference is given by

$$V=v_f-v_r$$

(3)

where $v_f$ is the forward vehicle velocity and $v_r$ is the rearward vehicle velocity as shown in Figure 2. The product of velocity and density is the traffic flow [23]

$$F=\rho v$$

(4)

and substituting Equation (4) in Equation (3) gives

$$V={\left(\frac{F}{\rho }\right)}_f-{\left(\frac{F}{\rho }\right)}_r$$

(5)

A high density results in slow vehicles and a small distance headway $h$. Conversely, a low density results in a large distance headway between vehicles and large vehicle speeds [9]. Further, the time headway $\tau$ is inversely proportional to the flow [39], so Equation (5) can be expressed as

$$V=\frac{h_f}{{\tau }_f}-\frac{h_r}{{\tau }_r}$$

(6)

Driver sensitivity can be expressed as

$$\zeta =\frac{\tau }{{\tau }_s}$$

(7)

where ${\tau }_s$ is the safe time headway required to avoid accidents. When $\tau$ is less than ${\tau }_s$, vehicles are slow and there is a small distance between them so drivers respond quickly. Conversely, when the time headway is greater than the safe time headway, the flow is smooth so the distance headway is large and driver response is slow. Substituting Equation (7) in Equation (6) gives the proposed acceleration exponent

$$\delta =\frac{\tau }{{\tau }_s}\left(\frac{h_f}{{\tau }_f}-\frac{h_r}{{\tau }_r}\right)$$

(8)

and replacing $\delta$ in Equation (1) with Equation (8), the proposed model is obtained as

$$\dot{v}=a_m\left(1-{\left(\frac{v}{v_d}\right)}^{\frac{\tau }{{\tau }_s}\left(\frac{h_f}{{\tau }_f}-\frac{h_r}{{\tau }_r}\right)}-{\left(\frac{s^{*}}{h}\right)}^2\right),$$

(9)

This model employs the distance and time headways to account for vehicle alignment due to changes in traffic. This is more accurate than the ID model that relies on an arbitrary constant regardless of the traffic conditions. Further, this constant is not based on traffic physics.

Traffic density is the inverse of the distance headway and at steady state $\rho =1/h_e$ where $h_e$ is the equilibrium distance headway [40]. At steady state, $∆v=0$ so substituting Equation (2) in Equation (1) and solving for $h$ gives

$$h_e=\left(J+\tau v\right){\left(1-{\left(\frac{v}{v_d}\right)}^{\delta }\right)}^{-0.5}$$

(10)

for the ID model and

$$h_e=\left(J+\tau v\right){\left(1-{\left(\frac{v}{v_d}\right)}^{\frac{T}{T_s}\left(\frac{h_f}{T_f}-\frac{h_r}{T_r}\right)}\right)}^{-0.5}$$

(11)

for the proposed model. Equation (10) indicates that the ID model headway is based on an arbitrary fixed value δ and so is unrealistic. On the other hand, Equation (11) is based on actual traffic parameters such as the distance and time headways and so δ is not a constant.

The steady-state traffic flow is given by $F=\frac{v}{h_e}$ which for the ID model is

$$F=\frac{v}{\left(J+\tau v\right){\left(1-{\left(\frac{v}{v_d}\right)}^{\delta }\right)}^{-0.5}}$$

(12)

and for the proposed model is

$$F=\frac{v}{\left(J+\tau v\right){\left(1-{\left(\frac{v}{v_d}\right)}^{\frac{T}{T_s}\left(\frac{h_f}{T_f}-\frac{h_r}{T_r}\right)}\right)}^{-0.5}}$$

(13)

This indicates that the steady-state traffic flow for the proposed model incorporates the time and distance headways. With a small headway, driver response is quick as there are significant interactions between vehicles and drivers have little time to react. In this case, the traffic flow is low [9,41]. Conversely, with a large headway there are few vehicle interactions and driver response is slow. In addition, drivers have more time to react so the flow is high [9,41].

3. Performance Results

The proposed and ID models are evaluated using the Euler scheme [40] implemented in MATLAB. A circular road of length 2000 m is considered with periodic boundary conditions. The simulation time is 200 s and the time step is 0.5 s. The desired velocity is 30 m/s and the maximum and the minimum deceleration (negative deceleration) are 0.5 m/s² and 3 m/s², respectively [29]. The jam spacing is 2 m [37], and the forward and rearward distance headways are both 25 m [2]. The time headway varies depending on the traffic conditions and is usually between 0.5 s and 2.6 s [42]. Here, the forward time headway is 1.5 s, the rearward time headway is 1.6 s, and ${\tau }_s$ is $1.4$ s. The ID model is evaluated for $\tau =2$ s [6], while the proposed model is evaluated for $\tau =0.6,1,1.5,2,$ and $2.2$ s. The acceleration exponent varies from 1 to $\infty$ and is commonly set to 4 [22], so $\delta =1,4$ and $100$ is considered. There are 52 vehicles on the road, each with a length of 4.5 m [43]. The simulation parameters are given in

Table 1. Simulation Parameters.

Parameter	Value
Desired velocity, $v_d$	$30$ m/s
Time headway for ID model, $\tau$	$2$ s
Forward and rearward distance headways, $h_f$ and $h_r$	$25$ m
Forward time headway, ${\tau }_f$	$1.5$ s
Rearward time headway, ${\tau }_r$	$1.6$ s
Safe time headway, ${\tau }_s$	$1.4$ s
Jam spacing, $J$	$2$ m
Maximum deceleration, $a_m$	$0.5$ m/s2
Minimum deceleration, $b$	$3$ m/s2
Time headway for the proposed model, $\tau$	$0.6$, $1$, $1.5,$ 2, and $2.2$ s
Vehicle length, $L$	$4.5$ m
Acceleration exponent, $\delta$	$1,4,$ and $100$
Time step, $∆t$	$0.5$ s
Maximum normalized density, ${\rho }_m=1/J$	$0.5$

.

Figure 3 presents the ID model velocity on a $2000$ m circular road for $\delta =1,4$, and $100$. When $\delta =1$, the initial velocity is $6.7$ m/s and increases to $22.5$ m/s at $97$ s. It then decreases to $0.86$ m/s at $127$ s and then increases to $8.5$ m/s at $200$ s. When $\delta =4$, the velocity is $3.1$ m/s at $138.5$ s, increases to $15.9$ m/s at $200$ s, and then decreases to $0.07$ m/s at $107.5$ s. The highest velocity is $27.1$ m/s at $78$ s. When $\delta =100$, the velocity increases from $0.74$ m/s at $1.5$ s to $29.8$ m/s from $62$ s to $73.5$ s. It decreases to $0.12$ m/s at $101.5$ s, increases to $1.5$ m/s at $127$ s, and is $16.7$ m/s at $200$ s.

The velocity on a $2000$ m circular road with the proposed model for $\tau =0.6$, $1$, $1.5,$ 2, and $2.2$ s is shown in Figure 4. When $\tau =0.6$ s, the initial velocity is $8.1$ m/s and increases to $19.5$ m/s at $127.5$ s. It decreases to $11.0$ m/s at $163$ s and then increases to $15.0$ m/s at $200$ s. When $\mathsf{\tau }=1$ s, the initial velocity is $9.74$ m/s and increases to $21.5$ m/s at $108$ s. It is $9.5$ m/s at $142$ s and $15.0$ m/s at $200$ s. When $\tau =1.5$ s, the velocity increases from $10.7$ m/s at $95$ s to $23.0$ m/s at $25.5$ s. It decreases to $3.8$ m/s at $125.5$ s and then increases to $13.2$ m/s at $200$ s. When $\tau =2$ s, the initial velocity is $10.1$ m/s and increases to $24.1$ m/s at $87.5$ s. It decreases to $0.44$ m/s at $119$ s and then increases to $11.1$ m/s at $200$ s. When $\tau =2.2$ s, the initial velocity is $10.4$ m/s and increases to $24.3$ m/s at $86$ s. It is $0.12$ m/s at $86$ s and increases to $9.9$ m/s at $200$ s.

Figure 5 presents the flow with the ID model on a $2000$ m circular road for $\delta =1,4,$ and $100$. When $\mathsf{\delta }=1$, the initial flow is $0.04$ veh/s and increases to $0.54$ veh/s at $122.5$ s. It is $0.21$ veh/s at $128$ s and increases to $0.34$ veh/s at $199.5$ s. When $\delta =4$, the initial flow is $0.05$ veh/s and increases to $0.56$ veh/s at $104.5$ s. It decreases to $0.01$ veh/s at $108.5$ s and then increases to $0.31$ veh/s at $140$ s and $0.37$ veh/s at $200$ s. When $\delta =100$, the initial flow is $0.04$ veh/s, increases to $0.57$ veh/s at $99$ s, and then decreases to $0.002$ veh/s at 104 s.

The flow on a $2000$ m circular road with the proposed model for $\tau =0.6,1,1.5,2,$ and $2.2$ s is given in Figure 6. When $\mathsf{\tau }=0.6$ s, the flow is $0.07$ veh/s at $130.5$ s, increases to $0.83$ veh/s at $161.5$ s, and then decreases to $0.44$ veh/s at $199.5$ s. When $\tau =1$ s, the flow is $0.05$ veh/s at $106$ s, increases to $0.72$ veh/s at $140$ s, and then decreases to $0.49$ veh/s at $200$ s. When $\tau =1.5$ s, the flow is $0.06$ veh/s at $99$ s and increases to $0.66$ veh/s at $122.5$ s. It decreases to $0.39$ veh/s at $134.5$ s and then increases to $0.42$ veh/s at $200$ s. When $\tau =2$ s, the flow is $0.07$ veh/s at $98$ s and then increases to $0.54$ veh/s at $115.5$ s. It decreases to $0.15$ veh/s at $120$ s and then increases to $0.36$ veh/s at $200$ s. When $\tau =2.2$ s, the flow is $0.04$ veh/s at $90.5$ s and increases to $0.51$ veh/s at $114$ s. It is $0.05$ veh/s at $119$ s and increases to $0.33$ veh/s at $200$ s.

Figure 7 presents the trajectories of a platoon of $52$ vehicles with the ID model on a $2000$ m circular road, and the results at $90$ s are given in

Table 2. Position of the $1$st, $15$th, $30$th, and $50$th vehicles at $90$ with the ID model on a $2000$ m circular road.

Acceleration Exponent $\mathit{\delta }$	1st Vehicle Position (m)	15th Vehicle Position (m)	30th Vehicle Position (m)	50th Vehicle Position (m)
$1$	$1286$	$266.9$	$-23.1$	$-96.0$
$4$	$1639$	$383.1$	$-14.0$	$-96.0$
$100$	$1762$	$396.0$	$-2.3$	$-96.0$

. The thick black line is the trajectory of the $1$st vehicle while the pink lines show the trajectories of the following $51$ vehicles. When $\delta =1$, Figure 7a shows that the position of the 1st and $15$th vehicles is $1286$ m and $266.9$ m, respectively, while the position of the $30$th and $50$th vehicles is $-23.1$ m and $-96.0$ m, respectively. When $\delta =4$, Figure 7b shows that the position of the $1$st and $15$th vehicles is $1640$ m and $383.1$ m, respectively, while the position of the $30$th and $50$th vehicles is $-14.0$ m and $-96.0$ m, respectively. When $\delta =100$, Figure 7c shows that the position of the $1$st and $15$th vehicles is $1762$ m and $396.0$ m, respectively, while the position of the $30$th and $50$th vehicles is $-2.3$ m and $-96.0$ m, respectively.

The trajectories of a platoon of $52$ vehicles with the proposed model on a $2000$ m circular road are presented in Figure 8, and the results at $90$ s are given in

Table 3. Position of the $1$st, $15$th, $30$th, and $50$th vehicles at $90$ s with the proposed model on a $2000$ m circular road.

Time Headway $\mathit{\tau }$ (s)	1st Vehicle Position (m)	15th Vehicle Position (m)	30th Vehicle Position (m)	50th Vehicle Position (m)
$0.6$	$956.5$	$395.0$	$69.8$	$-96.0$
$1$	$1171$	$422.8$	$78.1$	$-95.9$
$1.5$	$1325$	$426.4$	$47.0$	$-96.0$
$2$	$1432$	$327.8$	$-5.7$	$-96.0$
$2.2$	$1443$	$338.3$	$-20.9$	$-96.0$

. The first vehicle is denoted by a thick pink line, while the following $51$ vehicles are shown by black lines. When $\tau =0.6$ s, Figure 8a shows that the $1$st vehicle position is $956.5$ m and the $15$th vehicle position is $395.0$ m, whereas the $30$th and $50$th vehicle positions are $69.8$ m and $-96.0$ m, respectively. When $\tau =1$ s, Figure 8b shows that the $1$st vehicle position is $1171$ m and the $15$th vehicle position is $422.8$ m, whereas the $30$th and $50$th vehicle positions are $78.1$ m and $-95.9$ m, respectively. When $\tau =1.5$ s, Figure 8c shows that the $1$st vehicle position is $1325$ m and the $15$th vehicle position is $426.4$ m, whereas the $30$th and $50$th vehicle positions are $47.0$ m and $-96.0$ m, respectively. When $\tau =2$ s, Figure 8d shows that the $1$st vehicle position is $1432$ m and the $15$th vehicle position is $327.8$ m, whereas the $30$th and $50$th vehicle positions are $-5.7$ m and $-96.0$ m, respectively. When $\tau =2.2$ s, Figure 8e shows that the $1$st vehicle position is $1443$ m and the $15$th vehicle position is $338.3$ m, whereas the $30$th and $50$th vehicle positions are $-20.9$ m and $-96.0$ m, respectively.

Figure 9 presents the density with the ID model over time and space on a $2000$ m circular road and

Table 4. ID model density and time during and after congestion.

Acceleration Exponent $\mathit{\delta }$	Time during Which Congestion Occurs (s)	Density during Congestion	Density after Congestion Dissipates	Time after Congestion Dissipates (s)
$1$	0–101	$0.50$	$0.29$	$125.5$
$4$	0–103	$0.50$	0.26–0.50	110.5–200
$100$	0–200	$0.50$	Congestion does not dissipate	-

gives the congestion results. When $\delta =1$, Figure 9a shows that there is congestion from $0$ s to $101$ s as the density is $0.50$. At $-93.0$ m, it decreases to $0.29$ at $125.5$ s and $0.15$ at $198.5$ s. At $199$ s, the density is $0.15$ at $-9.0$ m and decreases to $0.012$ at $1791$ m. When $\delta =4$, Figure 9b shows that there is congestion from $0$ s to $103$ s as the density is $0.50$. It is between $0.26$ and 0.50 from $110.5$ s to $200$ s. At $198$ s, the density is $0.5$0 at $-132.0$ m and decreases to $0.10$ at $-54.0$ m and $0.02$ at $1785$ m. When $\delta =100$, Figure 9c shows that there is congestion between $-99$.0 m and $-3.0$ m as the density is $0.50,$ and at $-144.0$ m this continues until $200$ s. At $198$ s, the density is $0.5$ at $-144.0$ and decreases to $0.07$ at $-21.0$ m and $0.01$ at $1791$ m.

The density with the proposed model over time and space on a $2000$ m circular road is given in Figure 10 and

Table 5. Proposed model density and time during and after congestion.

Time Headway $\mathit{\tau }$ (s)	Time during Which Congestion Occurs (s)	Density during Congestion	Density after Congestion Dissipates	Time after Congestion Dissipates (s)
$0.6$	0–102	$0.50$	$0.01$	$146$
$1$	0–89.5	$0.5$0	$0.02$	$129$
$1.5$	0–93	$0.5$0	$0.15$	$124$
$2$	0–96	$0.5$0	$0.36$	$118.5$
$2.2$	0–103	$0.5$0	$0.45$	$117$

gives the congestion results. When $\tau =0.6$ s, Figure 10a shows that there is congestion from $0$ s to $102$ s as the density is $0.50$. It decreases to $0.01$ at $78.0$ m and $146$ s and then increases to $0.09$ at $564.0$ m and $200$ s. At $199$ s, the density is $0.02$ between $0$ m and $477.0$ m and decreases to $0.02$ at $1722$ m. When $\tau =1$ s, Figure 10b shows that the density is $0.50$ from $0$ s to $89.5$ s which indicates congestion. After the congestion dissipates, the density is $0.02$ at $129$ s and $33.0$ m, and increases to $0.09$ at $199.5$ s and $525.0$ m. Between $0$ m and $462$ m, the density is $0.02$ at $198$ s. It increases to $0.09$ at $525.0$ m and then decreases to $0.0$2 at $1779$ m. When $\tau =1.5$ s, Figure 10c shows there is congestion from $0$ s to $93$ s as the density is $0.50$. The density decreases to $0.15$ at $-54.0$ m and $124$ s and then increases to $0.18$ at $129.0$ m and $199.5$ s. At $200$ s, the density between $0$ m and $66.0$ m is $0.15$ and decreases to $0.02$ at $1773$ m. When $\tau =2$ s, Figure 10d shows there is congestion between $0$ s and $96$ s as the density is $0.50$. At $-96.0$ m, it decreases to $0.36$ at $118.5$ s and $0.20$ at $198$ s. At $198.5$ s, the density is $0.20$ at $-42.0$ m and decreases to $0.06$ at $99.0$ m and $0.01$ at $1746$ m. When $\tau =2.2$ s, Figure 10e shows there is congestion between $0$ s and $103$ s as the density of $0.5$. It decreases to $0.45$ at $-99.0$ m and $117$ s and $0.19$ at $-75.0$ m and $199$ s. At $199$ s, the density is $0.07$ at $18$.0 m and $0.01$ at $1767$ m.

Figure 11 presents the ID model flow for $\delta =1,4$, and $100$. When $\delta =1$, Figure 11a shows that at $-204$.0 m the flow is zero from $0$ s to $100$ s. It is $0.54$ veh/s at $123$ s and $0.55$ veh/s at $196.5$ s. At $-12.0$ m, the flow is $0.52$ veh/s at $199.5$ s, and decreases to $0.37$ veh/s at $93.0$ m and $0.25$ veh/s at $1743$ m. When $\delta =4$, Figure 11b shows that at $-180.0$ m the flow is zero from $0$ s to $90$ s. It increases to $2.13$ veh/s at $112.5$ s and varies between $0.67$ veh/s and $6.05$ veh/s from $131.5$ s to $198.5$ s. At $196$ s, the flow is $0.24$ veh/s at $-96$.0 m and increases to $0.38$ veh/s at $144.0$ m where it remains. When $\delta =100$, Figure 11c shows that at $-141.0$ m, the flow is zero from $0$ s to $84$ s. It increases to $4.11$ veh/s at $106.5$ s and varies between $9.17$ veh/s and $3.20$ veh/s from $124.5$ s to $200$ s. At $200$ s, the flow is $0.24$ veh/s at $-114.0$ m and increases to $0.37$ veh/s at $174.0$ m where it remains.

Figure 12 presents the flow with the proposed model on a $2000$ m circular road for $\tau =0.6$, 1, $1.5$, $2,$ and $2.2$ s. When $\tau =0.6$ s, Figure 12a shows that at $-174.0$ m, the flow is zero from $0$ s to $118$ s. It increases to $1.04$ veh/s at $225$ s and then decreases to $1.00$ veh/s at $199$ s. At $198$ s, the flow is $0.40$ veh/s between $0$ m and $276.0$ m and increases to $1.01$ veh/s at $540$.0 m. It then decreases to $0.43$ veh/s at $753.0$ m and remains constant. When $\tau =1$ s, Figure 12b shows that at $-201.0$ m, the flow is zero from $0$ s to $111$ s. It increases to $0.82$ veh/s at $142.5$ s and is approximately constant until $200$ s. At $198$ s, the flow is $0.36$ veh/s between $0$ m and $102$.0 m and increases to $0.84$ veh/s at $510.0$ m. It decreases to $0.49$ veh/s at $864.0$ m and $0.40$ veh/s at $1734$ m. When $\tau =1.5$ s, Figure 12c shows that at $-159$.0 m the flow is zero from $0$ s to $91$ s. It increases to $0.69$ veh/s at $125.5$ s and is approximately constant until $200$ s. At $199$ s, the flow is $0.32$ veh/s between $0$ m and $63.0$ m and increases to $0.69$ veh/s at $129.0$ m. It decreases to $0.48$ veh/s at $450.0$ m and $0.33$ veh/s at $1767$ m. When $\tau =2$, Figure 12d shows that at $-201.0$ m, the flow is zero from $0$ s to $84$ s. It increases to $0.54$ veh/s at $115.5$ s and is approximately constant until $200$ s. At $199$ s, the flow is $0.56$ veh/s at $-48.0$ m and decreases to $0.41$ veh/s at $96$.0 m and $0.28$ veh/s at $1785$ m. When $\tau =2.2$ s, Figure 12e shows that at $-171.0$ m, the flow is zero from $0$ s to $83$ s. It increases to $0.50$ veh/s at $113.5$ s and $0.52$ veh/s at $197$ s. At $199.5$ s, the flow is $0.52$ veh/s at $-81.0$ m and decreases to $0.35$ veh/s at $24.0$ m and $0.27$ veh/s at $1773$ m.

4. Discussion

The results presented in Figure 3 indicate that the variations in velocity with the ID model increase with the acceleration exponent $\delta$. Further, Figure 4 illustrates that the variations in velocity with the proposed model increase with the time headway. Figure 5 shows that the variations in flow with the ID model increase over time as $\delta$ increases. Similarly, the results for the proposed model presented in Figure 6 indicate that the variations in flow over time increase with a larger time headway.

Figure 7 and Figure 8 present the vehicle trajectories in time and space with the ID and proposed models, respectively, and the results are summarized in Table 2 and Table 3. Table 2 indicates that as $\delta$ increases, the distance traveled by the $1$st and $15$th vehicles increases, while that traveled by the $30$th vehicle decreases. However, the distance traveled by the $50$th vehicle is approximately the same. Table 3 shows that a decrease in the time headway decreases the distance traveled by the $1$st vehicle, while the distance traveled by the $15$th vehicle increases between $0.6$ s and $1.5$ s, decreases between $1.5$ s and $2.0$ s, and then increases. Between $0.6$ s and $1$.0 s, the distance traveled by the $30$th vehicle increases and then decreases between $1.0$ s and $2.2$ s. The distance traveled by the $50$th vehicle decreases between $0.6$ s and $1.0$ s and then increases. These results show that the vehicle positions with the ID model obtained using an arbitrary constant are not based on real traffic conditions. Conversely, the positions according to the proposed model are based on the time headway. This results in a smooth flow with the platoon of vehicles which is more realistic than the ID model.

Figure 9 shows that as $\delta$ increases, the ID model density increases over time so there is congestion with a large $\delta$. Moreover, Figure 11 indicates that the flow increases with $\delta$. Figure 10 shows that the density with the proposed model is small for a small time headway and vice versa. In addition, Figure 12 indicates that the changes in flow are proportional to the time headway.

Overall, the results given indicate that the traffic behavior with the proposed model is more realistic than with the ID model. In particular, the flow and velocity are smoother, and the variations in flow and density over time are smaller. The ID model can produce unrealistic traffic behavior such as the significant congestion shown in Figure 9c. This is because the ID model employs an arbitrary constant whereas in the proposed model this constant is replaced with a variable based on the time headway. The results presented highlight the importance of considering real traffic parameters in traffic flow models to accurately characterize traffic behavior.

5. Conclusions

A microscopic traffic model was proposed which improves the well know ID model. The proposed model considers the velocity differences between forward and rearward vehicles. It was evaluated on a $2000$ m circular road for different values of time headway. The results obtained demonstrate that the traffic density and flow with the proposed model are realistic. In contrast, the ID model provides poor results due to inappropriate traffic characterization. In particular, constant vehicle speeds are considered which do not reflect real driving behavior. The proposed model incorporates time headway to provide a more accurate characterization of vehicle movement. As a result, the variations in flow and velocity are smoother than with the ID model.

The proposed microscopic model can be employed in ACC and cooperative ACC systems, as well as in automated vehicles, to make traffic safe and efficient. In addition, it can be incorporated into traffic simulators to provide realistic traffic predictions. Future research can consider heterogeneous traffic based on real-time data acquired from roadside units.