A New Driver Model Based on Driver Response

Abstract

In this paper, a new microscopic traffic model based on forward and rearward driver response is proposed. Driver response is characterized using the distance and time headways. Existing models such as the Intelligent Driver (ID) model characterize traffic flow based on a constant acceleration exponent. This exponent reflects uniform driver behaviour during different conditions which is unrealistic. Driver response is slow with a large distance headway and quick with a short time headway. Conversely, it is quick with a small distance headway and slow with a long time headway. Thus, a new microscopic traffic model is proposed which incorporates driver response. Results are given that show the proposed model provides better traffic stability than the ID model as this stability is based on traffic physics. Further, for effective utilization of road infrastructure, shorter time and longer distance headways are preferred. The performance of the ID and proposed models was evaluated over an 800 m circular road with a string of $15$ vehicles for $120$ s. These models are numerically discretized using the Euler scheme. The results obtained show that traffic queue dissemination with the proposed model is more realistic than with the ID model and the changes in density with the proposed model are smaller. This is because traffic dissemination with the proposed model is based on traffic parameters rather than a constant exponent.

1. Introduction

Interest in urban traffic modelling has increased in recent years due to the prevalence of congestion, which is a pressing social and economic issue, particularly in developing countries [1]. High density traffic results in slow-moving vehicles and small distances between vehicles. This increases interactions between vehicles and thus the probability of accidents [2]. As urban populations increase, so does the number of vehicles. The number of vehicles has grown from $580$ million in $1990$ to $816$ million in $2010$ [3]. The number of vehicles is estimated to reach nearly 3 billion by $2040$ [4]. Large numbers of vehicles on the roads result in long traffic queues which dissipate slowly, i.e., traffic takes longer to attain a smooth flow. Traffic queues occur after the flow reaches a maximum. The velocity when the flow is maximum is called the critical velocity and corresponds to the maximum density. Beyond the maximum flow, traffic velocity reduces, and congestion develops. Effective traffic forecasting and management strategies are required to mitigate traffic problems such as congestion and improve road infrastructure utilization [5]. This requires realistic traffic prediction using traffic flow models [6,7,8]. The traffic flow is affected by the distance between vehicles and the time required to align to forward conditions. These factors affect driver response, resulting in velocity variations. Thus, they should be incorporated into traffic models to characterize traffic behaviour accurately and realistically.

Three types of models have been used to characterize traffic flow, microscopic, macroscopic, and mesoscopic. Microscopic models consider individual vehicle behaviour and are often based on driver physical and psychological responses [9]. They employ vehicle parameters such as position, velocity, and time and distance headways [10]. Microscopic models are used to predict vehicle dynamics. Macroscopic models deal with aggregate traffic behaviour and are used to determine average density, velocity, and flow [11]. Mesoscopic models take into account both individual and combined vehicle behaviour [12], so vehicles are characterized both individually and as a group.

The first microscopic traffic model was proposed by Gazis et al. [13]. This model considers driver response to forward vehicles based on the velocity difference. However, it ignores changes in driver behaviour due to traffic conditions; it assumes that in a string of vehicles, a driver adjusts to the speed of forward vehicles with a constant delay of $1.3$ s, which is unrealistic and does not follow traffic physics. Newell [14] proposed a microscopic traffic flow model that characterized vehicle behaviour in dense traffic. The distance headway is covered by vehicles while aligning to forward conditions. This is the distance between two consecutive vehicles. In this model, velocity depends on the distance headway. A larger distance headway means a lower traffic density and hence a greater velocity. However, this relationship between velocity and density can result in excessive acceleration, which is unsafe and therefore unrealistic [15]. However, this model was the first to include the time required to align to forward conditions [16]. Newell further suggested that the time to align to forward conditions is a constant [17], but this ignores the differences in driver behaviour.

Bando et al. [18] proposed an improvement to the Newell model but neglected the effects of velocity differences resulting in unstable behaviour. Moreover, deviations from the equilibrium velocity can result in high acceleration and deceleration, which is unrealistic. This is because density dependent traffic conditions are not considered. In reality, traffic adjustment is slow when the density is high and quick when the density is low. Further, a constant driver response is employed for all conditions so differences in driver behaviour are ignored. This model is based on data dependent constants, which is not a realistic approach for traffic characterization. The use of traffic physics is more appropriate for microscopic traffic characterization. In addition, the distances between vehicles are too small leading to accident conditions. Bosch [19] improved the Bando et al. model using regression techniques. However, this resulted in a very complex traffic characterization with $11$variables in the model.

Helbing and Tilch [20] developed a model that incorporated reaction to differences in velocity. Thus, it can accurately model velocity and time headway in congestion, but acceleration and deceleration occur over a very short time. This behaviour is typical of an aggressive driver, so slow and average driver behaviour is ignored. Gipps [21] proposed a model with realistic acceleration and deceleration and driver behaviour. However, it is not accurate for a wide range of parameters [15]. Several models are described in Table 1.

The Intelligent Driver (ID) model developed by Treiber, Henneck, and Helbing [15] is based on driver response. This model considers the velocity and distance headway of preceding vehicles and incorporates practical traffic parameters [22,23,24]. However, the acceleration exponent in this model is a constant and so is not based on traffic physics, which leads to unrealistic behaviour. Traffic flow with the ID model is typically estimated at a maximum speed of $30$ m/s and a jamming distance headway at standstill of $7$ m. The maximum acceleration is $1$m/s² while the maximum deceleration is $1.5$ m/s² [16]. The acceleration exponent is commonly set to 4 [15] and is the same for all traffic conditions. The ID model has been improved by incorporating driver response based on deceleration at intersections [25]. However, the distances between vehicles are small even with large velocities. Traffic alignment is the adjustment in velocity when a forward change in traffic occurs. Adaptive cruise control is designed to maintain the required distance headway for safe alignment to forward conditions [26], i.e., changes in velocity are small and within a maximum and minimum range. It has been shown that using the ID model in an adaptive cruise control system can result in accidents [27].

Several traffic prediction software programs are available, such as Simulation of Urban Mobility (SUMO) and PTV VISSIM. The SUMO program is a free package that is commonly used for research [28]. It can be combined with the ID model for traffic prediction, but driver behaviour is not included [29]. The PTV VISSIM program is a commercial package that can be used to predict microscopic traffic behaviour [30]. A revised ID model is used which incorporates driver response based on the distance headway between vehicles. However, changes in flow only occur with a significant change in this response [29]. Traffic capacity is the maximum flow for the conditions at a given location [31]. Connected vehicles communicate bidirectionally with the outside world. The prime goal of connectivity is to improve public safety [32]. The PTV VISSIM program has been used to predict freeway capacity and increase the proportion of connected and autonomous vehicles [33]. Traffic signal algorithms have been developed to mitigate traffic congestion [34]. However, the interaction between autonomous and human operated vehicles is difficult to characterize [35]. The advanced interactive microscopic simulator for urban and non-urban networks (AIMSUN) is a modelling package used to simulate traffic models [36]. It encompasses a collection of dynamic modelling tools and includes microscopic and mesoscopic simulators and dynamic traffic assignment models. AIMSUN is employed for offline traffic engineering and real time traffic management decision support. It provides solutions to traffic planning and operational problems [37].

Table 1. Comparison of traffic models.

Model	Mathematical Description	Interpretation
Newell [14]	$v\left(t+T\right)=v\left(s\left(t\right)\right)$ $\mathrm{where}$       $v\left(s\right)=v_{max}\left[1-\exp \left[-\frac{\left(s-s_0\right)}{v_{max}T}\right]\right]$ $v\left(t+T\right)$ = velocity attained by a vehicle at time $(t+T)$, $v_{max}$ = maximum velocity, $T$ = time headway, $s$ = distance headway, and $s_0$ = jam spacing.	A car-following model that characterizes vehicle behaviour in dense traffic. However, the acceleration can be very high which is unrealistic.
Gipps [21]	$v\left(t+T\right)=\min \left[v_{F,a}\left(t+T\right), v_{F,b}\left(t+T\right)\right]$ where $v_{F,a}\left(t+T\right)=v_F\left(t\right)+2.5a_{F, max}T\left[1-\frac{v_F\left(t\right)}{v_{F,max}}\right]\sqrt{0.025+\frac{v_F\left(t\right)}{v_{F,max}}}$ $v_{F,b}\left(t+T\right)=B_{F}T+\sqrt{B_F^2{T}^2-B_F\left[\begin{array}{c}2\left(s_L\left(t\right)-s_F\left(t\right)-\left(l_L+s_0\right)\right)\\ -v_F\left(t\right)T-\frac{v_L^2\left(t\right)}{\widehat{B}}\end{array}\right]}$ $v\left(t+T\right)$= velocity attained by a vehicle at time $(t+T)$, $a$ = acceleration, $b$ = deceleration, $v_F$ = following vehicle velocity, $a_{F,max}$ = maximum acceleration of the following vehicle, $T$ = time headway, $v_{F,max}$ = maximum velocity of the following vehicle, $B_F$ = braking response of the following vehicle, $s_L$ = leading vehicle distance headway, $l_L$ = leading vehicle length, $s_F$ = following vehicle distance headway, $v_L$ = leading vehicle velocity, $s_0$ = jam spacing, and $\widehat{B}$ = braking response of the leading vehicle.	A model that incorporates braking to provide realistic vehicle behaviour and avoid accidents. However, this is a probabilistic model.
Intelligent Driver (ID) [15]	$\frac{dv}{dt}=a\left(1-{\left(\frac{v}{v_{max}}\right)}^{\delta }-{\left(\frac{D}{s}\right)}^2\right)$ where        $D=s_j+Tv+\frac{v\Delta v}{2\sqrt{ab}}$	A model that employs acceleration to characterize driver response. Driver behaviour is related to the velocity and distance headway of leading vehicles. It characterizes changes in traffic in an accident-free environment. However, a constant acceleration exponent is used for all traffic conditions.
Revised ID [33]	$a=a_{max}\left(1-{\left(\frac{v}{v_{max}}\right)}^{\delta }-{\left(\frac{D}{s}\right)}^2\right)$ where    $D=s_j+s_1\sqrt{\frac{v}{v_{max}}}+c\frac{v^2}{b}+\max \left[0, vT+\frac{v\Delta v}{2\sqrt{a_{max}b}}\right]$ $c$ = safety impact factor, $s_{1 }$= non-linear jam spacing, and $a_{max}$ = maximum acceleration	A model that maintains large distances between vehicles to characterize freeway traffic. It has been used for connected and autonomous vehicles.
Proposed	$\delta ={\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f$   $\frac{dv}{dt}=a\left(1-{\left(\frac{v}{v_{max}}\right)}^{{\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f}-{\left(\frac{D}{s}\right)}^2\right)$  $D=(s_j+Tv){\left(1-{\left(\frac{v}{v_{max}}\right)}^{{\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f}\right)}^{-1/2 }$	A model that improves on the ID model by incorporating both forward and rearward vehicle conditions. The acceleration exponent is a variable based on traffic conditions.

Table 1. Comparison of traffic models.

Model	Mathematical Description	Interpretation
Newell [14]	$v\left(t+T\right)=v\left(s\left(t\right)\right)$ $\mathrm{where}$       $v\left(s\right)=v_{max}\left[1-\exp \left[-\frac{\left(s-s_0\right)}{v_{max}T}\right]\right]$ $v\left(t+T\right)$ = velocity attained by a vehicle at time $(t+T)$, $v_{max}$ = maximum velocity, $T$ = time headway, $s$ = distance headway, and $s_0$ = jam spacing.	A car-following model that characterizes vehicle behaviour in dense traffic. However, the acceleration can be very high which is unrealistic.
Gipps [21]	$v\left(t+T\right)=\min \left[v_{F,a}\left(t+T\right), v_{F,b}\left(t+T\right)\right]$ where $v_{F,a}\left(t+T\right)=v_F\left(t\right)+2.5a_{F, max}T\left[1-\frac{v_F\left(t\right)}{v_{F,max}}\right]\sqrt{0.025+\frac{v_F\left(t\right)}{v_{F,max}}}$ $v_{F,b}\left(t+T\right)=B_{F}T+\sqrt{B_F^2{T}^2-B_F\left[\begin{array}{c}2\left(s_L\left(t\right)-s_F\left(t\right)-\left(l_L+s_0\right)\right)\\ -v_F\left(t\right)T-\frac{v_L^2\left(t\right)}{\widehat{B}}\end{array}\right]}$ $v\left(t+T\right)$= velocity attained by a vehicle at time $(t+T)$, $a$ = acceleration, $b$ = deceleration, $v_F$ = following vehicle velocity, $a_{F,max}$ = maximum acceleration of the following vehicle, $T$ = time headway, $v_{F,max}$ = maximum velocity of the following vehicle, $B_F$ = braking response of the following vehicle, $s_L$ = leading vehicle distance headway, $l_L$ = leading vehicle length, $s_F$ = following vehicle distance headway, $v_L$ = leading vehicle velocity, $s_0$ = jam spacing, and $\widehat{B}$ = braking response of the leading vehicle.	A model that incorporates braking to provide realistic vehicle behaviour and avoid accidents. However, this is a probabilistic model.
Intelligent Driver (ID) [15]	$\frac{dv}{dt}=a\left(1-{\left(\frac{v}{v_{max}}\right)}^{\delta }-{\left(\frac{D}{s}\right)}^2\right)$ where        $D=s_j+Tv+\frac{v\Delta v}{2\sqrt{ab}}$	A model that employs acceleration to characterize driver response. Driver behaviour is related to the velocity and distance headway of leading vehicles. It characterizes changes in traffic in an accident-free environment. However, a constant acceleration exponent is used for all traffic conditions.
Revised ID [33]	$a=a_{max}\left(1-{\left(\frac{v}{v_{max}}\right)}^{\delta }-{\left(\frac{D}{s}\right)}^2\right)$ where    $D=s_j+s_1\sqrt{\frac{v}{v_{max}}}+c\frac{v^2}{b}+\max \left[0, vT+\frac{v\Delta v}{2\sqrt{a_{max}b}}\right]$ $c$ = safety impact factor, $s_{1 }$= non-linear jam spacing, and $a_{max}$ = maximum acceleration	A model that maintains large distances between vehicles to characterize freeway traffic. It has been used for connected and autonomous vehicles.
Proposed	$\delta ={\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f$   $\frac{dv}{dt}=a\left(1-{\left(\frac{v}{v_{max}}\right)}^{{\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f}-{\left(\frac{D}{s}\right)}^2\right)$  $D=(s_j+Tv){\left(1-{\left(\frac{v}{v_{max}}\right)}^{{\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f}\right)}^{-1/2 }$	A model that improves on the ID model by incorporating both forward and rearward vehicle conditions. The acceleration exponent is a variable based on traffic conditions.

In this paper, a new model is proposed which characterizes acceleration based on the forward and rearward driver response and the time and distance headways. Driver response is the reaction to traffic conditions. The response to forward conditions is called the forward response and the response to rearward conditions is called the rearward response. Time headway is the time required for a vehicle to align after a change in traffic conditions. The time for the safe adjustment of velocity is called the typical time headway and the distance travelled during this time is the typical distance headway. Typical headway has been shown to affect driver response [38]. The behaviour of the proposed and ID models is evaluated over an $800$ m circular road to illustrate the advantages of our approach. Traffic behaviour is observed for a string of $15$ vehicles for $120$ s. The proposed and ID models are discretized for simulation using the Euler technique.

The rest of the paper is organized as follows. Section 2 presents the ID and proposed model and a stability analysis is given in Section 3. Section 4 presents the discretization using the Euler technique. The performance of these models is evaluated in Section 5. Finally, Section 6 provides some concluding remarks.

2. Traffic Flow Models

The ID model provides a microscopic characterization of traffic based on forward vehicles [39]. In this model, acceleration is a function of driver response, the distance between vehicles, and the time required for vehicles to align to forward conditions. Driver response is based on the ratio of average velocity to maximum velocity. This ratio is nominally 1 for a smooth flow with low acceleration and deceleration. In this case, traffic velocity is near the maximum.

Acceleration in the ID model is given by [15]

$$\frac{dv}{dt}=a\left(1-{\left(\frac{v}{v_{max}}\right)}^{\delta }-{\left(\frac{D}{s}\right)}^2\right),$$

(1)

where $a$ is the maximum acceleration, $v_{max}$is the maximum velocity, $v$ is the average velocity, and δ is the acceleration exponent. The term $D$ is the distance headway during traffic alignment. During alignment, vehicles adjust their velocity according to forward conditions in order to achieve the equilibrium velocity. The distance headway can be expressed as [15]

$$D=s_j+Tv+\frac{v\Delta v}{2\sqrt{ab}},$$

(2)

where $s_j$ is the bumper-to-bumper distance between vehicles during congestion (zero velocity), $T$ is the time headway, $b$ is the minimum acceleration, $\Delta v$ is the change in velocity required to align to a forward vehicle, and $s$ is the bumper-to-bumper distance between vehicles as shown in Figure 1. The ID model employs (1) and (2) to characterize traffic flow based on the driver response and distance headway during transitions.

In the ID model, driver response to a change in traffic conditions is characterized with the acceleration exponent δ. This exponent cannot characterize traffic behaviour in different conditions because it is a constant. It indicates that driver behaviour is the same for all conditions, which is unrealistic. Further, it is not based on traffic physics. Therefore, a variable exponent is proposed here to model changes in traffic flow based on the forward and rearward driver responses. A driver responds slowly to leading vehicles when the distance headway is large. Conversely, the response is quick when this headway is small [38] and also when the time headway is short [5].

The acceleration exponent δ is impacted by both the distance headway and time headway. Thus, the exponent in the proposed model is a function of the ratio of distance headway $h$ to typical distance headway $h_N$, and time headway$T$

$$\delta ={\left(\left(1-\frac{h}{h_N}\right)T\right)}_r+{\left(\left(\frac{h}{h_N}\right)T\right)}_f,$$

(3)

where ${\left(\left(1-\frac{h}{h_N}\right)T\right)}_r$ is the rearward driver response and ${\left(\left(\frac{h}{h_N}\right)T\right)}_f$ is the forward driver response. These responses are illustrated in Figure 2. The subscript $r$ denotes rearward traffic and $f$ denotes forward traffic. The minimum distance headway $h_s$ is the distance required between vehicles to avoid accidents. The transition distance headway is the distance required to align to forward conditions. The distance headway $h$ includes the minimum distance headway and transition distance headway $h_t$ [40] and so is expressed as

$$h=h_s+h_t.$$

(4)

The transition distance is covered during the time headway so that

$$h_t=Tv.$$

(5)

Substituting (5) in (4) gives

$$h=h_s+Tv,$$

(6)

and the proposed acceleration exponent is obtained by Substituting (6) in (3)

$$\delta ={\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f.$$

(7)

Substituting this expression for δ in (1) gives the proposed model

$$\frac{dv}{dt}=a\left(1-{\left(\frac{v}{v_{max}}\right)}^{{\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f}-{\left(\frac{D}{s}\right)}^2\right).$$

(8)

In free flow traffic, the typical distance headway is required to avoid accidents and smoothly align to forward vehicles so that $h=h_N$. In congested traffic, $h<h_N$ and is reduced to zero in a traffic jam. In free flow, the traffic density is low, and vehicles can maintain a larger distance headway. In this case, there is minimal acceleration and deceleration. In the proposed model (8), alignment due to changes in traffic is based on the rearward and forward conditions and so is more realistic than with a constant$\delta$.

The traffic density is $\mathsf{\rho }=\frac{1}{D}$ [16] where $D$ is given by (2). At equilibrium, $\Delta v=0$ so the distance headway at equilibrium for the ID model is

$$D=(s_j+Tv){\left(1-{\left(\frac{v}{v_{max}}\right)}^{\delta }\right)}^{-1/2},$$

(9)

and for the proposed model is

$$D=(s_j+Tv){\left(1-{\left(\frac{v}{v_{max}}\right)}^{{\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f}\right)}^{-1/2 }.$$

(10)

Equation (9) indicates that the distance between vehicles at equilibrium with the ID model is based on a constant acceleration exponent and so is the same for all conditions. Conversely, (10) shows that the corresponding distance with the proposed model is based on the forward and rearward traffic conditions and so is not a constant.

Traffic flow is the product of density and velocity [22]

$$Q=\frac{v}{D}.$$

(11)

Substituting (9) in (11) gives

$$Q=\frac{v}{(s_j+Tv){\left(1-{\left(\frac{v}{v_{max}}\right)}^{\delta }\right)}^{-1/2}},$$

(12)

therefore, traffic flow with the ID model is based on a constant acceleration exponent. Now, substituting (10) in (11) results in

$$Q=\frac{v}{(s_j+Tv){\left(1-{\left(\frac{v}{v_{max}}\right)}^{{\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f}\right)}^{-1/2}}$$

(13)

which indicates that traffic flow with the proposed model is based on forward and rearward driver responses. A slower response produces a smaller flow whereas a quicker response gives a larger flow.

3. Stability Analysis

In this section, the stability of the ID and proposed model is examined. A homogeneous infinite road with identical drivers and vehicles is considered so the alignment to forward conditions is the same, i.e., the acceleration during alignment is uniform, and vehicles maintain the same distance headway $D$ and corresponding equilibrium velocity $v_e\left(D\right)$. Small changes in distance headway are denoted by $y$ and the corresponding small changes in velocity by $u\left(t\right).$ The distance headway during a change in velocity is then

$$s=D+y,$$

(14)

and the corresponding velocity is

$$v=v_e\left(D\right)+u.$$

(15)

Temporal changes in the distance headway produce changes in velocity during alignment [35] which can be characterized as

$$y\left(t\right)=\frac{dy}{dt}=u_L-u_F,$$

(16)

where the subscripts $L$ and $F$ denote the leading and following vehicles, respectively. $u\left(t\right)$ is a function of the change in distance headway [41] and for a smooth flow is given by

$$u\left(t\right)=\frac{du}{dt}=f_s{y}_F+(f_v+f_{\Delta v})u_F-f_{\Delta v}{u}_L,$$

(17)

with $f_s=\frac{\partial f}{\partial s}$, $f_v=\frac{\partial f}{\partial v}$, and $f_{\Delta v}=\frac{\partial f}{{\partial }_{\Delta v}}$.

For small changes in distance headway and velocity

$$y\left(t\right)=\widehat{y}{e}^{\lambda t+ik},$$

(18)

$$u\left(t\right)=\hat{u}{e}^{\alpha t+ik},$$

(19)

Equations (18) and (19) can be expressed as

$$\left(\begin{array}{c}y\left(t\right)\\ u\left(t\right)\end{array}\right)=\left(\begin{array}{c}\widehat{y}\\ \hat{u}\end{array}\right)e^{\alpha t+ik},$$

(20)

where $\alpha =\sigma +i\omega$ is the growth rate of change during alignment with $i=\sqrt{-1}$. The real part $\sigma$ is the change in amplitude, the imaginary part $\omega =2\pi /T$ is the corresponding frequency, and $k$ is the phase shift, i.e., the delay between traffic waves that a driver experiences [41]. $\widehat{y}$ and $\hat{u}$ are the amplitudes of the changes in distance headway and velocity, respectively. Substituting (20) in (16) and (17) gives

$$y\left(t\right)=u-u{e}^{ik},$$

(21)

$$u\left(t\right)=f_{s}y{e}^{ik}+\left(f_v+f_{\Delta v}\right)u{e}^{ik}-f_{\Delta v}u.$$

(22)

A model is stable if the eigenvalues of the Jacobian matrix $J$ have negative real parts. The eigenvalues can be obtained from

$$\left|J-\lambda I\right|=0,$$

(23)

where $I$ is the identity matrix and $J$ is the Jacobian matrix given by

$$J=\left(\begin{array}{cc}{j}_{11}& j_{12}\\ j_{21}& j_{22}\end{array}\right),$$

where $j_{11}$ and $j_{21}$ are the differentials of (21) and (22) with respect to $y$ and $j_{12}$ and $j_{22}$ are the differentials of (21) and (22) with respect to $u$ which gives

$$J=e^{ik}\left(\begin{array}{cc}0& e^{-ik}-1\\ f_s& \left(f_v+f_{\Delta v}\right)-f_{\Delta v}{e}^{-ik}\end{array}\right)$$

(24)

Substituting this in (23) results in

$$\left|e^{ik}\left(\begin{array}{cc}0& e^{-ik}-1\\ f_s& \left(f_v+f_{\Delta v}\right)-f_{\Delta v}{e}^{-ik}\end{array}\right)-\left(\begin{array}{cc}\lambda & 0\\ 0& \lambda \end{array}\right)\right|=0$$

$$\left(\begin{array}{cc}\lambda & 1-e^{-ik}\\ -f_s& \lambda -f_v-f_{\Delta v}+f_{\Delta v}{e}^{-ik}\end{array}\right)=0,$$

(25)

therefore, the eigenvalues are the solutions of

$${\lambda }^2+\left(-f_v-f_{\Delta v}+f_{\Delta v}{e}^{-ik}\right)\lambda +f_s\left(1-e^{-ik}\right)=0.$$

(26)

Let $A\left(k\right)=-f_v-f_{\Delta v}+f_{\Delta v}{e}^{-ik}$ and $B\left(k\right)=f_s\left(1-e^{-ik}\right)$ which gives

$${\lambda }^2+A\left(k\right)\lambda +B\left(k\right)=0.$$

(27)

The eigenvalues are then

$${\lambda }_{1,2}=-\frac{A\left(k\right)}{2}\left( 1\pm \sqrt{1-\frac{4B\left(k\right)}{A^2\left(k\right)}}\right),$$

(28)

A group of two or more vehicles maintaining a distance headway is called a string [42]. A traffic model is string stable if the real parts of the eigenvalues (28) are negative [41]. Conversely, a model is unstable if changes in the string increase such as stop and go traffic [16]. With a stable string, changes in flow do not grow. A stable string is necessary for efficient and effective traffic control, and indicates that congestion can be controlled in a flow that maintains the desired distance headway [42]. Moreover, traffic models become unstable as $k\to 0$, i.e., when there is no delay between traffic waves which occur in congestion. In this case, the wavelengths and periods increase, and thus so do the distance headway and velocity [41].

The coefficients of (27) can be expressed as Taylor series. For small $k$, $A\left(k\right)$ can be approximated as [41]

$$A\left(k\right)\approx -f_v-f_{\Delta v}+f_{\Delta v}{e}^{-i\left(0\right)}-\frac{i{f}_{\Delta v}{e}^{-i\left(0\right)}}{1}k$$

$$=-f_v-i{f}_{\Delta v}k,$$

(29)

and $B\left(k\right)$ can be approximated as [41]

$$B\left(k\right)\approx f_s\left(1-e^{-i\left(0\right)}\right)+\frac{f_{s}i{e}^{-i\left(0\right)}}{1}k+\frac{f_s{e}^{-i\left(0\right)}}{2}{k}^2$$

$$=i{f}_{s}k+\frac{f_s}{2}{k}^2.$$

According to the steady-state (equilibrium) condition, the interdependence of $f_s$ and $f_v$ [41] is

$$f_s=-v_e^{\prime }\left(D\right)f_v.$$

(30)

where $v_e^{\prime }\left(D\right)$ is the change in velocity for a given distance headway at equilibrium. Substituting $f_s$ from (30) in $B\left(k\right)$ gives

$$B\left(k\right)=-i{v}_e^{\prime }\left(D\right)f_v-\frac{v_e^{\prime }\left(D\right)}{2}{f}_v$$

(31)

Let

$$\begin{gathered} A\left(k\right)=a_0+a_{1}k, \\ B\left(k\right)=b_{1}k+b_2{k}^2 \end{gathered}$$

(32)

With

$$\begin{gathered} a_0=-f_v, \\ {a}_1=-i{f}_{\Delta v}, \\ {b}_1=-i{v}_e^{\prime }\left(D\right)f_v=i{v}_e^{\prime }\left(D\right)a_0, \\ {b}_2=-\frac{v_e^{\prime }\left(D\right)}{2}{f}_v=\frac{v_e^{\prime }\left(D\right)}{2}{a}_0. \end{gathered}$$

(33)

Expressing the square root in (28) as a Taylor series gives the approximation

$$\sqrt{1-\frac{4B\left(k\right)}{A^2\left(k\right)}}\approx 1-\frac{2B\left(k\right)}{A^2\left(k\right)}-\frac{2B^2\left(k\right)}{A^4\left(k\right)}.$$

(34)

Substituting (34) in (28) and considering the negative sign for the square root results in

$${\lambda }_1=\frac{-B\left(k\right)A^2\left(k\right)-B^2\left(k\right)}{A^3\left(k\right)}.$$

Substituting $A\left(k\right)$ and $B\left(k\right)$ from (32) gives

$${\lambda }_1=-\frac{b_1}{a_0}k+\left(\frac{b_1{a}_1}{a_0^2}-\frac{b_2}{a_0}-\frac{b_1^2}{a_0^3}\right)k^2,$$

(35)

and then substituting (33) in (35) results in

$$\begin{gathered} {\lambda }_1=-\frac{i{v}_e^{\prime }\left(D\right)a_0}{a_0}k+\left[\frac{i{v}_e^{\prime }\left(D\right)a_0\left(-i{f}_{\Delta v}\right)}{a_0^2}-\frac{v_e^{\prime }\left(D\right)a_0}{2a_0}-\frac{{\left(i{v}_e^{\prime }\left(D\right)a_0\right)}^2}{a_0^3}\right]k^2 \\ =-i{v}_e^{\prime }\left(D\right)k+\frac{v_e^{\prime }\left(D\right)}{f_v}\left[\frac{-2f_{\Delta v}-f_v}{2}-v_e^{\prime }\left(D\right)\right]k^2. \end{gathered}$$

(36)

The real part of (36) describes the wave growth and the traffic flow is string stable if this term is negative. As

$$v_e^{\prime }\left(D\right) \ge 0\mathrm{and}{f}_v<0,$$

(37)

the term $\left[\frac{-2f_{\Delta v}-f_v}{2}-v_e^{\prime }\left(D\right)\right]$ gives the following string stability criteria [27]

$$v_e^{\prime }\left(D\right)\le -\frac{f_v}{2}-f_{\Delta v}$$

(38)

From (37) and (38), the product of $\left[-\frac{2f_{\Delta v}-f_v}{2}-v_e^{\prime }\left(D\right)\right]$ and $\frac{v_e^{\prime }\left(D\right)}{f_v}$ results in a negative real part of ${\lambda }_1$ in (38).

The criterion in (38) indicates that the model is very stable for a driver following large changes in traffic. That is, the traffic flow is smooth and changes in velocity are small. Further, traffic is stable when a driver tracks changes in velocity. Drivers not doing this may produce stop and go traffic. In this case, the model is not string stable. The partial derivatives $f_v$ and $f_{\Delta v}$ of (1) at steady state with traffic moving at the equilibrium velocity $v_e\left(D\right)$can be expressed as

$$f_v=a\left(-\frac{\delta v_e{\left(D\right)}^{\delta -1}}{v_{max}^{\delta }}-\frac{2T\left(s_j+v_e\left(D\right)T\right)}{D^2}\right),$$

(39)

$$f_{\Delta v}=-\frac{v_e\left(D\right)}{D}\sqrt{\frac{a}{b}}\left(\frac{s_j+v_e\left(D\right)T}{D}\right)$$

(40)

Substituting (39) and (40) in (38) gives the string stability criteria

$$v_e^{\prime }\left(D\right)\le \frac{a\left(\delta D^2{v}_e{\left(D\right)}^{\delta -1}+2T{s}_j{v}_{max}^{\delta }+2v_e\left(D\right)T^2{v}_{max}^{\delta }\right)}{2D^2{v}_{max}^{\delta }}+\frac{v_e\left(D\right)\sqrt{ab}\left(D+T{v}_e\left(D\right)\right)}{D^{2}b}.$$

(41)

Changes in velocity for the ID model are based on a constant exponent $\delta$, and thus so are changes in the distance headway. To ensure string stability, a large value of $\delta$. is employed in practice which results in large changes in velocity. Then, vehicles will leave congestion quickly while maintaining the desired distance headway. Using a large value of $\delta$ is not based on traffic physics but rather is a means of ensuring stability. Thus, it results in an unrealistic and inadequate characterization. In reality, changes in velocity occur based on the rearward and forward driver responses. Therefore, these responses must be considered, and this should ensure traffic string stability. Thus, for the proposed model, replacing $\delta$ in (41) with (7) gives

$$\begin{gathered} v_e^{\prime }\left(D\right)\le \frac{a\left(\begin{array}{c}{\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f{D}^2{v}_e{(D)}^{[{\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f]-1}+2T{s}_j{v_{\max }}^{{\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f}\\ +2v_e(D)T^2{v_{\max }}^{{\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f}\end{array}\right)}{2D^2{v_{\max }}^{{\left(\left(1-\frac{h_s+Tv}{h_N}\right)T\right)}_r+{\left(\left(\frac{h_s+Tv}{h_N}\right)T\right)}_f}} \\ +\frac{v_e(D)\sqrt{ab}\left(D+T{v}_e(D)\right)}{D^{2}b} \end{gathered}$$

(42)

With the proposed model, changes in velocity are based on the forward and rearward traffic conditions, i.e., traffic stability is affected by the forward and rearward time and distance headways. With a long time headway, vehicles move slowly and stay longer in stop and go traffic. In this case, the flow is not string stable. However, a driver maintaining a short time headway will leave stop and go traffic quickly so the flow is string stable. Further, if the distance headway is large, the time headway will be small as the conditions are predictable. Thus, a large distance headway will also ensure quick movement from stop and go traffic, as the driver is more responsive. Conversely, a driver is less responsive with a small distance headway and large time headway and vehicles stay longer in stop and go traffic. Then the flow with the proposed model is not stable which is realistic. In a transportation network, increased utilization of road infrastructure is a limiting factor. Road infrastructure is effectively utilized with a short time headway for traffic alignment to forward conditions. The string stability of the proposed model from (42) suggests that there will be minimal changes in traffic flow with a short time headway and large distance headway. This is as expected with a microscopic traffic model for realistic and effective traffic evolution [43].

4. Numerical Models

The Euler technique is employed to evaluate the performance of the ID and proposed models. This is a simple method to solve ordinary differential systems and is widely employed in the literature. This overcomes the problem of obtaining analytic solutions, which is not tractable [16]. With the Euler technique, time is divided into intervals and the models are used to approximate vehicle positions, velocity, and acceleration each interval.

Temporal changes in distance cause changes in velocity which gives

$$\frac{dD}{dt}=v,$$

(43)

and the temporal change in velocity $\frac{dv}{dt}$ results in changes in acceleration. For simplicity, denote the right-hand side of (2) and (8) by Ω so then

$$\frac{dv}{dt}=\mathsf{\Omega }.$$

(44)

According to the Euler technique, position, and velocity for the ID model (9) and (1), and proposed model (10) and (8), can be obtained as

$$D_n^{k+1}=D_n^k+\Delta t\times v_n^k$$

(45)

$$v_n^{k+1}=v_n^k+\Delta t\times {\Omega }_n^k,$$

(46)

where $k$ is the previous time step and $k+1$ is the next time step. $D_n^k$, $v_n^k$ and ${\Omega }_n^k$ represent the position, velocity, and acceleration, respectively, of the $n$th vehicle at the $k$th time interval

$$t=k\Delta t,$$

(47)

and ∆t is the duration of a time step. ${\Omega }_n^k$ is obtained for the ID and proposed model as a function of $D_n^k$ and $v_n^k$ from (1) and (8), respectively.

5. Performance Results

The performance of the ID and proposed models was evaluated on a circular road of length $800$ m for $120$ s using the Euler technique with time step $0.5$ s [16]. This technique was employed as it has been implemented in SUMO and AIMSUN for traffic prediction [16]. On a circular road, vehicle congestion can occur temporally, and large traffic changes can be expected. This is one of the worst conditions to evaluate traffic models. The simulation parameters are given in

Table 2. Simulation Parameters.

Parameter	Value
Maximum velocity, $v_{max}$	$33.3$ m/s
Maximum normalized density, ${\rho }_m$ $\mathrm{at}v=0$ m/s	$\frac{1}{s_j}=0.143$
Average velocity, $v$	$15$ m/s
Typical distance headway, $h_N$	$25$ m
Time headway for the ID model,$T$	$1.6$ s
Time headway for the proposed model, $T$	$0.1, 0.5, 1, 1.6,$$\mathrm{and}2.7$ s
Jam spacing, $s_j$	$7$ m
Maximum acceleration	$0.73$ m/s2
Maximum deceleration	$1.67$ m/s2
Vehicle length, $l$	$5$ m
Minimum distance headway, $h_s$	$21$ m
Time step, $\Delta t$	$0.5$ s
Rearward velocity	$33.3$ m/s
Rearward time headway	$4$ s
Forward velocity	$24.5$ m/s
Forward time headway	$5$ s

. The maximum velocity was $33.3$ m/s, the average velocity was $15$ m/s, and the jam spacing was $7$ m [16]. The maximum normalized density was $\frac{1}{s_j}=0.143$. The ratio of current density to the road capacity is called the normalized density. The ID model was evaluated with a time headway of $1.6$ s [15], while the proposed model was evaluated with time headway values of $0.1, 0.5, 1, 1.6,$ and $2.7$ s for uniform driver behaviour. Non-uniform driver behaviour was also examined with rearward velocity $33.3$ m/s, rearward time headway $4$ s, forward velocity $24.5$ m/s, and forward time headway $5$ s. The maximum acceleration and deceleration were $0.73$ m/s² and $1.67$ m/s², respectively [15]. The vehicle length was set to $5$ m [16] as this typically varies between $2.7$ and $5.6$ m [44]. The values for $h_s$ and $h_N$ were $21$ m and $25$ m, respectively [45]. The acceleration exponent for the ID model ranges from $1$ to $\infty$ and is typically $4$ [15], so $\delta =1$, $4$, and $20$ were considered here. A string of $15$ vehicles was investigated. As with most studies of vehicle behaviour, roadside data to compare with the numerical results is unavailable.

The flow and velocity behaviour for the ID and proposed models are given in Figure 3 and Figure 4, respectively, and the results are tabulated in

Table 3. Maximum flow, maximum density, and critical velocity for the ID model.

Acceleration Exponent $\mathit{\delta }$	Maximum Flow (veh/s)	Maximum Density	Critical Velocity (m/s)
$1$	$0.36$	$0.030$	$12.4$
$4$	$0.48$	$0.027$	$17.7$
$20$	$0.53$	$0.020$	$27.2$

and

Table 4. Maximum flow, maximum density, and critical velocity for the proposed model.

Time Headway $\mathit{T}$ (s)	Maximum Flow (veh/s)	Maximum Density	Critical Velocity (m/s)
$0.1$	$0.49$	$0.028$	$17.6$
$0.3$	$0.60$	$0.038$	$15.7$
$0.5$	$0.59$	$0.041$	$14.4$
$1$	$0.51$	$0.038$	$13.4$
$1.6$	$0.41$	$0.031$	$13.4$
$1.7$	$0.40$	$0.030$	$13.6$
$2$	$0.36$	$0.025$	$14.0$
$2.7$	$0.30$	$0.021$	$14.1$

. For the ID model with $\delta =1$, the maximum flow was $0.36$ veh/s at density $0.03$, and the corresponding velocity was $12.4$ m/s. With$\delta =4$, the maximum traffic flow was $0.48$ veh/s at density $0.027$, and the corresponding velocity was $17.7$ m/s. With $\delta =20$, the maximum traffic flow was $0.53$ veh/s at density $0.02$, and the corresponding velocity was $27.2$ m/s. Table 3 indicates that for the ID model the maximum velocity and flow were smaller with a lower $\delta$ while the density was larger. Further, Figure 3 shows that as $\delta$ increased the maximum flow increased while the density decreased, and the corresponding velocity increased as shown in Figure 4.

For the proposed model, the maximum flow with $T=0.1$ s and $T=0.3$ s was $0.49$ and $0.60$ veh/s, respectively, at density $0.028$ and$0.038$. The corresponding velocities were $17.6$ and $15.7$ m/s, respectively. With $T=0.5$ s and $T=1$ s, the maximum flow was $0.59$ and $0.51$ veh/s, respectively, at density $0.041$ and $0.038$. The corresponding velocities were $14.4$ and $13.4$ m/s, respectively. With $T=1.6$ s and $T=1.7$ s, the maximum flow was $0.41$ and $0.40$ veh/s, respectively, at density $0.031$ and $0.030$. The corresponding velocities were $13.4$ and $13.6$ m/s, respectively. With $T=2$ s and $T=2.7$ s, the maximum flow was $0.36$ and $0.30$ veh/s, respectively, at density $0.025$ and $0.021$. The corresponding velocities were $14$ and $14.1$ m/s, respectively. Table 4 indicates that for the proposed model, the flow was a maximum at $T=0.3$ s. Between $T=0.1$ and $0.5$ s, the density increased with $T$, and above $T=0.5$ s it decreased. For $T$ between $0.1$ and $1.7$ s, the velocity decreased with $T$ and above this value it increased.

The spatial and temporal evolution of the queue caused by traffic congestion for the ID and proposed models is shown in Figure 5 and the results are tabulated in

Table 5. Velocity and time for the ID and proposed models during and after the queue.

Parameter	Velocity during Queue (m/s)	Time (s)	Velocity after Queue (m/s)	Time (s)
$\delta =1$	$0$	$0–45.0$	$1.7$	$45.0$
$\delta =4$	$0$	$0–43.5$	$1.6$	$43.5$
$\delta =20$	$0$	$0–43.5$$\mathrm{and}86.0–120$	$1.4–6.4$	$43.5–86.0$
$T=0.1$ s	$0$	$0–120$	Queue does not dissipate	-
$T=0.5$ s	$0$	$0–45.0$	$1$	$45.0$
$T=1$ s	$0$	$0–39.5$	$0.9$	$39.5$
$T=1.6$ s	$0$	$0–41.0$	$0.8$	$41.0$
$T=2.7$ s	$0$	$0–45.5$	$0.6$	$45.5$
Non-uniform traffic	$0$	$0–40.5$	$0.7$	$40.5$

. The traffic velocity during the queue was zero, which is shown in yellow, and blue denotes the maximum velocity attained. For the ID model with $\delta =1$, the queue dissipated at $45$ s as shown in Figure 5a, while with $\delta =4$ it dissipated at $43.5$ s, as shown in Figure 5b. Table 5 indicates that with $\delta =1$, the velocity after the queue was $1.7$ m/s, while with $\delta =4,$ it was $1.6$ m/s. With $\delta$ = 20, the queue exists from $0$to $43.5$ s and the corresponding velocity was $0$ m/s. The velocity after the queue (after $43.5$ s), varied between $1.4$ m/s and $6.4$ m/s as indicated in Table 5. A traffic queue again developed from $86$ s to $120$ s between $0$ and $10$m, as shown in Figure 5c. Moreover, with $\delta$ = 1, the maximum velocity was $20$ m/s, while with $\delta =4$ it was $25$ m/s and with $\delta =20$ it was $30$ m/s, as shown in Figure 5a–c. These results indicate that the maximum velocity increased with $\delta$.

For the proposed model with $T=0.1$ s, the traffic queue did not dissipate. At $0$ s, the velocity between $-96.5$ m and $-3.3$ m was $0$ m/s. At $60$ s, the velocity was $0$ m/s between $-49.0$ m and $-96.5$ m, while at $120$ s the velocity was $0$ m/s from $-89.0$ m to $-96.5$ m, as shown in Figure 5d. With $T=0.5$ s and $1$ s, the queue dissipated at $45.0$ s and $39.5$ s, respectively, as shown in Figure 5e,f. With $T=0.5$ s, the velocity after the queue was $1$m/s, while with $T=1$ s, this velocity was $0.9$ m/s, as indicated in Table 5. With $T=1.6$ s and $2.7$ s, the queue dissipated at $41.0$ s and $45.5$ s, respectively, as shown in Figure 5g,h. At $T=1.6$ s, the velocity increased from $0$ m/s to $0.8$ m/s after $41.0$ s as given in Table 5. At $T=2.7$ s, the velocity increased from $0$ m/s to 0.6 m/s after $45.5$ s. The maximum velocity attained with the proposed model at $T=0.1$ s was $10$ m/s, while with $T=0.5, 1, 1.6,$ and $2.7$ s it was $20$ m/s, as shown in Figure 5d–h. Note that with $T=0.1$ s, the traffic queue did not dissipate. The largest velocity with the proposed model after the queue was $1$ m/s, which was observed with $T=0.5$ s. The results for non-uniform traffic conditions are given in Figure 5i. This shows that the queue dissipated at $40.5$ s. The velocity after $40.5$ s was $0.7$ m/s, increased to $8.7$ m/s at $74.0$ s, and then decreased to $2.4$ m/s at $119.5$ s.

The time-space vehicle trajectories for the ID and proposed models are given in Figure 6 and the results are summarized in

Table 6. Position of the $1$st, $4$th, and $10$th vehicles at $30$ s with the ID and proposed models.

Parameter	1st Vehicle Position (m)	4th Vehicle Position (m)	10th Vehicle Position (m)
$\delta =1$	$261.9$	$80.8$	$-58.9$
$\delta =4$	$314.5$	$101.1$	$-58.3$
$\delta =20$	$317.7$	$101.3$	$-58.3$
$T=0.1$ s	$88.7$	$-17.5$	$-63.0$
$T=0.5$ s	$207.4$	$64.7$	$-60.9$
$T=1$ s	$262.2$	$93.2$	$-42.2$
$T=1.6$ s	$289.7$	$94.1$	$-58.3$
$T=2.7$ s	$307.0$	$75.8$	$-61.2$

. This shows the velocity changes for a platoon of $15$ vehicles with an initial average velocity of $15$ m/s over an $800$ m circular road. The red trajectory corresponds to the first vehicle that starts moving at $t=0$ s, while the black trajectories are the following $14$ vehicles. With $\delta =1$ and $4$, the queue existed for $45.0$ s and $43.5$ s, respectively, as shown in Figure 6a,b. Further, the $15$th vehicle was in the queue for $45.0$ s at $-91.8$ m and $43.5$ s at $-93.1$m, respectively, as shown in Figure 6a,b. With $\delta =20$, the queue existed for $43.5$ s. The first vehicle was in the queue for $2.0$ s at $1.2$ m, while the 15th vehicle was in the queue for $43.5$ s at $-93.1$m. A queue again appeared at $86.0$ s at $50.0$m and increased up to $120$ s with the queue between $59.4$ m and $94.7$ m, as shown in Figure 6c.

For the proposed model with $T=0.1$ s, the queue existed for $120$ s and gradually dissipated. The first vehicle was in the queue for $3.0$ s at $1.6$ m, while the 15th vehicle was in the queue for $120$ s at $-97.3$ m, as shown in Figure 6d. With $T=0.5$ s and $1$ s, the queue existed for $45$.0 s and $39.5$ s, respectively, as shown in Figure 6e,f. Thus, the $15$th vehicle stayed for $45.0$ s at $-95.9$m and $39.5$ s at $-96.0$ m, respectively. With $T=1.6$ s and $2.7$ s, the queue existed for $41.0$ s and $45.5$ s, respectively, as shown in Figure 6g,h. The 15th vehicle was in the queue for $41.0$ s at $-96.05$m and $45.5$ s at $-96.06$m, respectively.

The positions of the $1$st, $4$th and $10$th vehicles at $30$ s with the ID and proposed models are now examined. With the ID model and $\delta =1$, the position of the $1$st and $4$th vehicles was $261.9$m and $80.8$ m, respectively, whereas the position of the $10$th vehicle was $-58.9$ m as shown in Figure 6a. With $\delta =4$, the position of the $1$st and $4$th vehicles was $314.5$ m and $101.1$ m, respectively, as shown in Figure 6b. With $\delta =20$, the position of the $1$st and $4$th vehicles was $317.7$ m and $101.3$ m, respectively, as shown in Figure 6c. The position of the $10$th vehicle with $\delta =4$ and $\delta =20$ was $-58.3$ m, as shown in Figure 6b,c.

With the proposed model and $T=0.1$ s, the position of the $1$st and $4$th vehicles was $88.7$ m and $-17.5$ m at $30$ s, respectively, and the position of the $10$th vehicle was $-63.0$ m, as shown in Figure 6d. With $T=0.5$ s, the position of the $1$st and $4$th vehicles was $207.4$ m and $64.7$ m, respectively, and the position of the $10$th vehicle was $-60.9$ m at $30$ s, as shown in Figure 6e. With $T=1$ s, the position of the $1$st and $4$th vehicles was $262.2$ m and $93.2$ m at $30$ s, respectively, and the position of the $10$th vehicle was $-42.2$ m as shown in Figure 6f. With $T=1.6$ s, the position of the $1$st and $4$th vehicles was $289.7$ m and $94.1$ m, respectively, as shown in Figure 6g. With $T=2.7$ s, the position of the $1$st and $4$th vehicles was $307$ m and $75.8$ m at $30$ s, respectively, as shown in Figure 6h. The position of the $10$th vehicle with $T=1.6$ and $T=2.7$ was $-58.3$ m and $-61.2$ m, respectively. Table 6 indicates that as the acceleration exponent increased, the distance travelled by the $1$st and $4$th vehicles increased. However, the distance travelled by the $10$th vehicle decreased. Further, a decrease in the time headway decreased the distance travelled by the vehicles.

Figure 7 presents the spatial and temporal traffic density behaviour with the ID and proposed models. This shows that with $\delta =1$ for the ID model, the density between $0$ m and $163.9$ m at $119.5$ s was $0.014$. It increased to $0.02$ at $369.6$ m and then decreased to $0.016$ at $602.7$ m, as shown in Figure 7a. The density between $0$ s and $34$ s was $0.14$ at $-96.57$m, which indicates a traffic queue. After the queue dissipated, the density at $117$ s gradually decreased to $0.03$ at $296.3$ m from $0.1$ at $-93.2$ m at $41.5$ s. With $\delta =4$, the density between $0$ m and $66.6$ m at $120$ s was $0.01$. It increased to $0.03$ at $306.3$ m and then decreased to $0.015$ at $602.7$ m, as shown in Figure 7b. The density between $0$ s and $33.5$ s at $-96.5$ m was $0.14$, which indicates a queue. After the queue dissipated, the density at $115$ s gradually decreased to $0.03$ at $259.7$ m from $0.09$ at $-93.24$ m at $41.5$ s. With $\delta =20$, the density between $0$ m and $49.9$ m at $119.5$ s was $0.01$. It increased to $0.14$ between $63.27$ m and $89.9$ m and then decreased to $0.008$ at $592.7$ m as shown in Figure 7c. The density between $0$ s and $33.5$ s at $-96.6$m was $0.14$, which indicates a queue. After the queue dissipated, the density at $68.5$ s gradually decreased to $0.07$ at $-23.3$ m from $0.1$ at $-93.2$ at $40.5$ s. It then increased to $0.14$ at $77.5$ s at $20.0$ m and was constant between $63.3$ m and $86.5$ m to $120$ s. These results show that as $\delta$increased, the change in density also increased with time as shown in Figure 7a–c, so the ID model was not stable.

For the proposed model with $T=0.1$ s, the density at $0$ m was $0.03$ at $119.5$ s. It decreased to $0.01$ at $466.2$ m and then to $0.009$ at $559.4$ m as shown in Figure 7d. The density at $0$ s between $-96.5$ m and $-3.3$ m was $0.14$ and was constant up to $120$ s at $-109$ m, which indicates a queue. With $T=0.5$ s, the density between $0$ m and $452.8$ m at $119.5$ s was $0.01$ and increased to $0.03$ at $606$ m, as shown in Figure 7e. The density between $0$ s and $37.5$ s at $-96.5$ m was $0.14$, which indicates a queue. After the queue dissipated, the density at $108$ s gradually decreased to $0.015$ at $-149.8$m from $0.09$ at $-93.2$ m at $44$ s. It then increased to $0.02$ at $116$ s at $-153.1$ m.

With $T=1$ s, the density at $0$ m was $0.02$ at $120$ s. It decreased to $0.015$ between $173.1$ m and $446.2$m and then increased to $0.027$ at $602.7$ m, as shown in Figure 7f. The density between $0$ s and $32.5$ s at $-96.5$ m was $0.14$, which indicates a queue. After the queue dissipated, the density at $93.5$ s gradually decreased to $0.015$ at $-173.1$ m from $0.09$ at $-93.3$ m at $40$ s. It then increased to $0.02$ at $118$ s.

With $T=1.6$ s the density between $0$ m and $156.5$ m at $119$ s was $0.01$. It increased to $0.29$ at $356.3$ m and then decreased to $0.01$ at $606.0$ m, as shown in Figure 7g. The density between $0$ s and $33.5$ s at $-96.5$ m was $0.14$, which indicates a queue. After the queue dissipated, the density at $114.5$ s gradually decreased to $0.01$ at $-179.8$ m from $0.08$ at $-93.2$ m at $42.5$ s. With $T=2.7$ s, the density at $0$ m was $0.04$ at $119.5$ s and then decreased to $0.01$ at $362.9$ m, as shown in Figure 7h. The density between $0$ s and $36.5$ s at $-96.5$ m was $0.14$, which indicates a queue. After the queue dissipated, the density at $56.5$ s gradually decreased to $0.06$ at $-89.9$ m from $0.1$ at $-93.2$ m at $45.5$ s. It then decreased to $0.03$ at $117.5$ s at $-46.6$ m.

These results indicate that with the proposed model, changes in density were small and the velocity and density evolve realistically over time based on the time headway. Conversely, the velocity and density with the ID model are based on a constant $\delta ,$which produced unrealistic traffic behaviour, as shown in the results with $\delta =20$. In this case, the changes in density increased over time whereas they should have decreased. Further, position changes with the ID model are based on a constant and vehicles are slower with a larger acceleration exponent $\delta$. With the proposed model, position changes are based on the time headway and vehicles are faster with a smaller headway. Thus, following vehicles achieve a smooth flow based on time headway, which makes the platoon more realistic than with the ID model. The results for the ID and proposed models are summarized in

Table 7. Summary for the ID and proposed models.

Traffic Parameter(s)	ID Model	Proposed Model
Flow and Velocity	As $\delta$ increases, the maximum flow and velocity increase based on this constant.	The flow and velocity vary based on the forward and rearward driver responses.
Queue dissipation	The queue dissipation is based on the constant $\delta$. The queue appears again with a larger $\delta$ and the congestion increases over time which is unrealistic.	The queue dissipation is based on the time and distance headways and evolves realistically over time.
Position changes	The position changes are based on the constant $\delta$. They are slower with a larger $\delta$ and hence vehicles move slower.	The position changes are based on the time and distance headways. Vehicles are faster with a smaller time headway and hence achieve a smooth flow which makes the platoon more realistic than with the ID model.
Density	Density is based on the constant $\delta$ which results in unrealistic traffic behaviour. A larger $\delta$ produces significant changes in density which increase over time whereas they should decrease.	Changes in density are small and evolve realistically over time.

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6. Conclusions

A new microscopic traffic flow model based on forward and rearward driver responses is proposed. Driver response is a function of distance and time headways. The proposed model was compared with the Intelligent Driver (ID) model for different time headways over a circular road. The results obtained show that with a large exponent $\delta$, traffic queues with the ID model do not dissipate which is unrealistic. This is due to an inadequate characterization of traffic behaviour. With the proposed model, queue dissemination is quick with a large time headway, while queues exist longer with a short time headway. Further, velocity and density evolve realistically over time based on the headway. Conversely, the ID model results in unrealistic behaviour due to the use of a constant exponent. Moreover, position changes with the proposed model are faster with a smaller time headway. Hence, following vehicles achieve a smooth flow which is realistic. With the ID model, position changes are based on a constant exponent, which results in an unrealistic flow.

With the proposed model, a large distance headway ensures quick movement from stop and go traffic, whereas with a smaller distance headway, vehicles stay longer in stop and go traffic. Hence, traffic stability with the proposed model is more realistic than with the ID model, as expected. This suggests that the proposed model can be employed in adaptive cruise control systems, and with connected and automated vehicles for effective traffic control. Further, it can be used to obtain realistic flows in traffic software packages. The proposed model can characterize traffic flows in different traffic conditions as compared to the ID model. Existing research has primarily considered homogeneous traffic flows. Future research can investigate heterogeneous flows based on real-time data obtained from road measurements.