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Cell Transmission Model for Mixed Traffic Flow
with Connected and Autonomous Vehicles
Yanyan Qin1and Hao Wang2
Abstract: The cell transmission model (CTM) is important for solving macroscopic traffic problems. Although some studies have been
conducted for multiclass CTM, a simplified CTM for traffic flow mixed with connected and autonomous vehicles (CAVs) is still needed. This
paper extends the homogenous CTM to form a mixed CTM under different CAV market penetration rates. To deal with this, the mixed
triangular fundamental diagram is derived with different proportions of various vehicle types. The maximum capacity and backward
wave speed of the mixed traffic flow are described by vehicle class proportions. Based on the mixed triangular fundamental diagram,
the mixed CTM is proposed for different CAV penetration rates. An example application to validate the usefulness of the proposed mixed
CTM is carried out. The mixed traffic flow in the application consists of cooperative adaptive cruise control (CACC), adaptive cruise control
(ACC), and human vehicles with random vehicle orders. It is defined that CAVs travel under ACC when following a human vehicle without
vehicle-to-vehicle (V2V) communication. Otherwise, the CACC function applies. The expected proportions of the CACC, ACC, and human
vehicles in the random mixed traffic flow are described by the CAV penetration rate. Mixed CTM numerical experiments are conducted
to evaluate the impacts of traffic accidents under different CAV penetration rates. Moreover, car-following simulations are performed to
show consistency between macroscopic CTM numerical experiments and microscopic car-following simulations. The example application
reveals that the proposed mixed CTM is simple and practical as applied to solve the traffic problem under different CAV penetration rates.
DOI: 10.1061/JTEPBS.0000238.© 2019 American Society of Civil Engineers.
Author keywords: Connected and autonomous vehicle; Cell transmission model; Mixed traffic flow; Fundamental diagram.
Introduction
Connected and autonomous vehicles (CAVs) are promoting the
development of traffic systems. Generally speaking, CAVs have
two types of operation. Under adaptive cruise control (ACC),
onboard sensors detect distance gaps and speed differences be-
tween the CAV and a preceding vehicle, which are usually used
to optimize CAV acceleration/deceleration. In contrast, vehicle-
to-vehicle (V2V) communication is helpful for obtaining acceler-
ation information from a preceding vehicle. In this situation, CAVs
travel under cooperative adaptive cruise control (CACC).
Research in modeling CAV traffic flow has become a hot topic
in recent years (Mahmassani 2016). Many studies (Ardakani and
Yang 2017;Gong et al. 2016;Gong and Du 2018;Jia and
Ngoduy 2016a,b;Qin and Wang 2018;van Arem et al. 2006;
Wang et al. 2018) have been conducted on microscopic CAV
models—a car-following model, for example. However, the re-
search is not well balanced because the literature on macroscopic
CAV models is relatively sparse. Macroscopic models are essen-
tially applied in real-time traffic management because real-time
applications require computational efficiency (Tiaprasert et al.
2017). The Lighthill-Whitham-Richards (LWR) model (Lighthill
and Whitham 1955;Richards 1956), based on kinematic wave
theory, is a first-order macroscopic model. Some studies extended
this model to a high-order macroscopic model for capturing more
complex traffic phenomena (Hoogendoorn and Bovy 2000;Jin
2017;Ngoduy 2013). Jin (2016) studied the equivalence between
a macroscopic continuum model and a microscopic car-following
model. As one of the most well-known discrete approximation
of the LWR model, Daganzo (1994,1995) presented the cell trans-
mission model (CTM). Since it requires less computation, CTM
has been widely used to solve many traffic problems (Han et al.
2017;Lo et al. 2001). This paper focuses on the CTM for mixed
CAV traffic flow.
For traditional mixed traffic flow, such as car-truck vehicular
flow, the original CTM (Daganzo 1994,1995) has been extended
to a multiclass CTM (Tuerprasert and Aswakul 2010). Lane-
changing behavior has also been explained in some studies
(Carey et al. 2015;Laval and Daganzo 2006). We refer readers
to Qian et al. (2017) for a more detailed understanding of CTM
extensions. What mainly concerns us here is a CTM of mixed traf-
fic flow consisting of CAVs and human vehicles. Levin and Boyles
(2016a) presented a CTM to evaluate traffic flow behaviors in
dynamic lane reversal with CAVs. A multiclass CTM was also
proposed that takes into consideration the different reaction times
of CAVs and human vehicles (Levin and Boyles 2016b). Tiaprasert
et al. (2017) proposed a multiclass CTM enhanced with overtaking,
lane-changing, and first-in first-out properties. However, most
studies from the literature make the original CTM complicated
(Daganzo 1994) in an effort to capture more accurate vehicle
behaviors, such as lane-changing and overtaking. From a macro-
scopic perspective, what is of concern is vehicular flow evolution
as a whole, regardless of local vehicle behaviors, under different
CAV market penetration rates, in mixed traffic flow. Therefore, a
simple and practical CTM for the mixed CAVs flow is still needed.
This is the subject of this paper.
1Ph.D. Candidate, School of Transportation, Southeast Univ., Si Pai
Lou 2, Nanjing 210096, China. Email: qinyanyan@seu.edu.cn
2Professor, School of Transportation, Southeast Univ., Si Pai Lou 2,
Nanjing 210096, China (corresponding author). Email: haowang@seu
.edu.cn
Note. This manuscript was submitted on April 18, 2018; approved on
October 22, 2018; published online on February 27, 2019. Discussion
period open until July 27, 2019; separate discussions must be submitted
for individual papers. This paper is part of the Journal of Transportation
Engineering, Part A: Systems, © ASCE, ISSN 2473-2907.
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A trapezoidal/triangular fundamental diagram (Newell 1993)
is essential in a CTM. Moreover, a mixed triangular fundamental
diagram can describe the maximum capacity and backward wave
speed of mixed traffic flow under various CAV penetration rates.
Levin and Boyles (2016a,b) studied capacity and backward wave
speed in designing a multiclass CTM without lane-changing
and overtaking behaviors. However, they presented it from the
perspective of a car-following model incorporating driver reaction
time. To the best of our knowledge, little research has been
conducted on a CTM of mixed CAV-human traffic flow from
the perspective of a general triangular fundamental diagram with
different CAV penetration rates. Therefore, this paper deals with
this research gap by providing another modeling idea for a CTM
of mixed CAV flow.
The organization of this paper is as follows. Mixed triangular
fundamental diagrams are first derived for different proportions
of vehicle types. Then the mixed CTM is proposed and its useful-
ness is validated in an example application. Finally, some conclu-
sions are offered.
Mixed Triangular Fundamental Diagram
A CTM assumes a trapezoidal/triangular fundamental diagram
(Newell 1993) for traffic flow. This section presents a triangular
fundamental diagram of mixed flow with CAVs. The diagram
requires a linear relationship between spacing and speed at equi-
librium for both homogenous and mixed traffic flow. Equilibrium
spacing is usually derived as a function of equilibrium speed as
follows:
sm¼vTmþdmð1Þ
where sm= spacing of vehicle class m;v= equilibrium speed;
Tm= reaction time of human vehicles or the time gap of CAVs
belonging to class m; and dm= jam spacing, usually including
vehicle length.
Eq. (1) describes the desired spacing of an individual vehicle
depending on its vehicle class and traffic speed. It can be seen that
each individual vehicle’s spacing is determined by its class and by
its traffic speed, regardless of vehicle order.
In mixed traffic flow, vehicle speeds are the same at equilibrium,
while individual vehicle spacing may vary. The subscript mof
Eq. (1) stands for the vehicle class in terms of Tmand dm, which
includes different vehicle types and heterogeneous characteristics
of individual vehicles. Assume that there are Mclasses in the mixed
traffic flow. Then the density of the mixed flow is calculated as
follows:
k¼1
PM
m¼1pmðvTmþdmÞð2Þ
where pm= proportion of class min the mixed traffic flow.
From Eq. (2), it can be seen that the density of mixed traffic flow
is calculated based on equilibrium speed. Equilibrium speed is con-
sidered to be constant for each vehicle class in mixed traffic flow
(Ge and Orosz 2014;Jia et al. 2018), which means that all vehicles
have the same equilibrium speed under a certain traffic condition.
In order to calculate the q-krelation, we make an adjustment for
Eq. (2), which means that speed vcan be described as a function of
density k
v¼1
kPM
m¼1ðpmTmÞ
−PM
m¼1ðpmdmÞ
PM
m¼1ðpmTmÞð3Þ
Therefore, the q-krelation of the mixed fundamental diagram
can be obtained as
q¼1
PM
m¼1ðpmTmÞ
−PM
m¼1ðpmdmÞ
PM
m¼1ðpmTmÞkð4Þ
The q-krelation in Eq. (4) describes the congested state of the
fundamental diagram, while the free flow state can be added to
obtain a triangular fundamental diagram, as illustrated in Fig. 1.
In Fig. 1,vfis the free-flow speed, qmax is the maximum flow,
kcis the critical density at qmax ,kjis the jam density, and wis the
backward wave speed. For the triangular fundamental diagram, the
critical density kcis related to the free-flow speed vf. Substituting
v¼vfinto Eq. (2) to calculate kc, we obtain
kc¼1
PM
m¼1pmðvfTmþdmÞð5Þ
Then the maximum flow qmax is
qmax ¼vf
PM
m¼1pmðvfTmþdmÞð6Þ
The jam density kjcan be calculated when vis equal to zero in
Eq. (2), which yields
kj¼1
PM
m¼1ðpmdmÞð7Þ
The backward wave speed is the slope of the triangular funda-
mental diagram. Based on Eq. (4), we have
w¼−PM
m¼1ðpmdmÞ
PM
m¼1ðpmTmÞð8Þ
Eq. (8) shows that the properties of the mixed triangular funda-
mental diagram, such as maximum capacity qmax and backward
wave speed w, are related to the proportion pmof specific class m
in the mixed traffic flow.
Mixed Cell Transmission Model
Here the mixed triangular fundamental diagram just derived is used
to form the CTM of mixed traffic flow. Like Daganzo (1994), we
discretize the time into steps Δtand the road into cells Δx,so
Δx¼Δt×vfshould be satisfied. The cells are labeled i, which
k
q
kckj
qmax
w
vf
Fig. 1. Mixed triangular fundamental diagram with different pm.
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is larger when the cell is located downstream. The mixed triangular
fundamental diagram in Eq. (4) assumes that all vehicles travel at
the same speed. Although some studies have focused on different
vehicle speeds in the CTM (Tiaprasert et al. 2017;Tuerprasert and
Aswakul 2010), assuming the same speed is reasonable because a
vehicle might match the speed of surrounding vehicles, especially
in the crowded state (Levin and Boyles 2016a,b). Traffic speed vis
determined by the free-flow speed, capacity, and backward wave
speed. According to Fig. 1, we can calculate vas follows:
v¼minvf;qmax
k;
−wðkj−kÞ
kð9Þ
where k= mixed flow density.
Substituting Eqs. (6)–(8) into Eq. (9) yields
v¼minvf;vf
kPM
m¼1pmðvfTmþdmÞ;PM
m¼1ðpmdmÞ
PM
m¼1ðpmTmÞ
×1
kPM
m¼1ðpmdmÞ
−1 ð10Þ
Based on CTM theory (Daganzo 1994), cell transition flow can
be calculated as follows:
yiðtÞ¼minni−1ðtÞ;Qmax;
−w
vf
½N−niðtÞð11Þ
where yiðtÞ= vehicles entering from cell i−1to cell iat time t;
ni−1= vehicles in cell i−1at time t;ni= vehicles in cell iat time t;
Qmax = maximum capacity limitation; and N= maximum vehicles
that can fit in each cell.
Based on the mixed triangular fundamental diagram derived in
Eq. (4), we have
Qmax ¼qmaxΔt¼Δx
PM
m¼1pmðvfTmþdmÞð12Þ
and
N¼kjΔx¼Δx
PM
m¼1ðpmdmÞð13Þ
Then the cell occupancy update is described as
niðtþΔtÞ¼niðtÞþyiðtÞ−yiþ1ðtÞð14Þ
Based on Eq. (14), we can obtain vehicles in each cell over time
and then calculate density, speed, and flow over time for mixed
traffic flow at different proportions pmof vehicle class m. The
proposed CTM assumes that class-specific density is uniformly dis-
tributed throughout the cells, which extends the assumption of a
single-class CTM (Levin and Boyles 2016a). Levin and Boyles
(2016b) considered Newell’s car-following model (Newell 2002)
to start the multiclass CTM. Newell’s model is directly related to
the triangular fundamental diagram, which is the linkage between
this paper and Levin and Boyles (2016b).
Application
We consider a specific scenario of mixed traffic flow in order to
perform an example application using the proposed mixed CTM.
In addition, the corresponding car-following models are used for
microscopic simulations, which are shown to be consistent with
macroscopic CTM numerical experiments.
Mixed Flow Scenario
As mentioned previously, CAVs monitor a preceding vehicle’s ac-
celeration with V2V communication under CACC, but only detect
the distance gap and speed difference to the lead vehicle using on-
board sensors under ACC. Moreover, the time gap of CAVs under
CACC can be smaller than that under ACC, which has benefits in
capacity. CAVs are able to travel under CACC, but CACC cannot
work without V2V communication. Therefore, the mixed traffic
flow in the application is defined as follows: (1) when traffic flow
mixes CAVs and human vehicles in random vehicle order, CAVs
have V2V communication equipment but human vehicles do not;
(2) when CAVs follow human vehicles, they have ACC; otherwise,
have CACC.
To avoid confusion, CAVs under CACC and ACC are called
CACC vehicles and ACC vehicles, respectively. Let the symbol
pbe the market penetration rate of CAVs in the random mixed traf-
fic flow. From a mathematical expectation perspective, the proba-
bility that one CAV follows one human vehicle is the product of
their proportions—namely, pð1−pÞ. This means that the expected
ACC proportion is pð1−pÞ. Then the expected CACC proportion
equals the proportion of CAVs minus that of ACC vehicles—
namely, p−pð1−pÞ¼p2. Therefore, the expected proportions
of CACC, ACC, and human vehicles can be described by the
market penetration rate pof CAVs, as follows:
pc¼p2
pa¼pð1−pÞ
ph¼1−pð15Þ
where pc,pa, and ph= expected proportions of CACC, ACC, and
human vehicles, respectively.
Without loss of generality, Eq. (15) can also describe homog-
enous human flow if p¼0, while homogenous CACC flow is
described if p¼1.
Macroscopic CTM Numerical Experiments
CTM is useful for evaluating the macroscopic dynamics of traffic
flow with less computation. The mixed traffic flow defined in this
paper is focused on evaluating flow evolution using the proposed
mixed CTM. The mixed CTM numerical experiments deal with the
expected situations of vehicle proportions described by the CAV
penetration rate pin Eq. (15).
Accidents are a common traffic phenomenon and often cause
queues during cleanup. If an accident occurs on a freeway, a period
of time is needed for cleanup, during which vehicles upstream
of the accident location form a queue. Normal operation is recov-
ered after cleanup. Our concern here is impacts on the queue
upstream under different values of pin mixed traffic flow. As an
example, assume that arrival flow is fixed at 1,500 veh=h in mixed
traffic flow. Fifteen minutes is needed for cleanup, during which
a queue is gradually formed. Following cleanup, access to the
freeway is again available and the queue begins to dissipate until
normal operation is achieved.
According to Shladover et al. (2018), CACC vehicles can
reduce the time gap to 0.6 s, while the lowest time gap for
ACC vehicles is usually 1.1 s. Therefore, we consider Tm¼Tc¼
0.6s for CACC vehicles in Eq. (1), in which the subscript cdenotes
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the CACC vehicle class. Moreover, Tm¼Ta¼1.1s is considered
in Eq. (1) for the ACC vehicle class. In the case of human vehicles,
Tm¼Th¼1.5s is considered based on previous studies (Ahn
et al. 2004;Johansson and Rumar 2016;Levin and Boyles
2016b). The same dmin Eq. (1) is considered for the three types
of vehicles, set as 7 m as is usual (Kesting et al. 2008;Milanés and
Shladover 2014). We substitute these values into Eq. (4) and let
subscript min Eq. (4)bec,a,h, respectively, to calculate the
triangular fundamental diagram of homogenous CACC, ACC,
and human flow, respectively. The calculation results are shown
in Fig. 2. Moreover, we calculate the mixed triangular fundamental
diagram under different CAV penetration rates pby substituting
Eq. (15) into Eq. (4) as is shown in Fig. 3, which indicates that
capacity gradually increases with the increase in p, because a
smaller time gap results in higher capacity (Shladover et al. 2018).
In addition, cell length Δx¼100 m, discretized time step
Δt¼3s, and free-flow speed vf¼120 km=h in the CTM
numerical experiments, satisfying Δx¼Δt×vf. Based on the
mixed triangular fundamental diagram, we can calculate values
for maximum capacity and vehicles for each cell under different
values of p, following Eqs. (12) and (13). Then the cell occupancy
update for mixed traffic flow can be calculated based on Eqs. (11)
and (14).
The impact results for the traffic accident using macroscopic
CTM numerical experiments are shown in Fig. 4. The grayscale
bar in Fig. 4denotes traffic flow speeds in cells over time. At
the initial time, the vehicle speed is the free-flow speed under
the arrival flow of 1,500 veh=h for mixed traffic flow in the triangu-
lar fundamental diagram. Moreover, the accident occurs at 20 km,
where vehicle speed becomes zero during the 15-min cleanup time.
Then Fig. 4shows the queue caused by the traffic accident over
space and time. It can be seen that queue length and impact time
gradually decrease with the increase in p. More important, the
mixed CTM proposed in this paper can deal with the traffic prob-
lem in terms of mixed CACC–ACC–human vehicular flow with
different CAV market penetration rates p. Moreover, compared
with the original CTM (Daganzo 1994), computation of the pro-
posed mixed CTM does not increase because we use a mixed tri-
angular fundamental diagram to replace the homogenous diagram
in the original CTM (Daganzo 1994).
Microscopic Car-Following Simulations
Important in the traffic flow problem is analysis of the consis-
tency between macroscopic results and microscopic simulations.
Therefore, this section performs car-following simulations and
compares their results with the those for the macroscopic mixed
CTM numerical experiments. We use a linear car-following model
for human vehicles and a model with a constant time gap for
CACC/ACC vehicles because these models are consistent with
the triangular fundamental diagram used in CTM.
In the case of human vehicles, Newell’s lower order model
(Newell 2002) is employed. The model equation can be written
as follows:
xnðtþThÞ¼xn−1ðtÞ−dhð16Þ
where xn−1ðtÞ= position of the preceding vehicle n−1at time t;
xnðtþThÞ= position of follower nafter time Th.
For consistency with the setting in the macroscopic mixed
CTM numerical experiments, Th¼1.5s and dh¼7m in the
car-following simulations.
The CACC car-following model with a constant time gap is
written as
¨
xnðtÞ¼k0¨
xn−1ðtÞþk1ðxn−1ðtÞ−xnðtÞ−vnðtÞTc−dcÞ
þk2ðvn−1ðtÞ−vnðtÞÞ ð17Þ
where x,v, and ¨
x= vehicle position, speed, and acceleration,
respectively.
The same setting for Tcand dcis satisfied—namely, Tc¼
0.6s and dc¼7m. Based on previous studies (Talebpour
and Mahmassani 2016;van Arem et al. 2006), k0¼1.0,k1¼
0.1s−2, and k2¼0.58 s−1.
Without V2V communication, CAVs travel under ACC to
respond to the distance gap and speed difference represented by
the preceding vehicle. Then the ACC car-following model with
constant time gap is written as follows:
¨
xnðtÞ¼k1½xn−1ðtÞ−xnðtÞ−vnðtÞTa−daþk2½vn−1ðtÞ−vnðtÞ
ð18Þ
where Ta¼1.1s, da¼7m, and k1and k2¼k1and k2in Eq. (17)
in the car-following simulations.
The same traffic conditions for the traffic accident in the mac-
roscopic CTM experiments are considered in the microscopic
Fig. 2. Homogenous triangular fundamental diagram.
Fig. 3. Mixed triangular fundamental diagram for different CAV
rates p.
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simulations. The microscopic simulations are performed in
MATLAB using the three car-following models under different
CAV penetration rates p. The randomness of vehicle order in
the mixed traffic flow is considered for each simulation. Then
the CAVs randomly become CACC or ACC depending on the pre-
ceding vehicle type in each case, based on the mixed traffic flow
definition in this paper. The car-following simulation step time is
0.1 s, the maximum acceleration is 4m=s2, and the emergency
deceleration is −6m=s2(Ni et al. 2016). Heat maps of vehicle
speed over all time and space are calculated, in which the time
interval is 1 s and the space interval is 100 m, as shown in Fig. 5.
Because our concern is the impacts of the traffic accident on flow
dynamics, only the vehicles that travel upstream of the traffic
accident location, when the accident occurs, are considered in
calculating the simulation results in Fig. 5. This means that only
vehicles affected by the accident are considered. A comparison of
Figs. 4and 5shows that the impact results for the macroscopic
numerical experiments with the proposed mixed CTM are almost
consistent with those of the microscopic car-following simulations.
In order to illustrate wave speed changes for each vehicle, we
provide time-space vehicle simulation trajectories based on car-
following simulations. Fig. 6shows the wave speed changes for
vehicles with respect to the simulation time. Specifically, it shows
that vehicles slow and finally stop in front of the accident location
because of the cleanup. During this time, the queue appears. After
that, vehicles accelerate and traffic finally returns to normal oper-
ation. We should also mention that the vehicles in Fig. 6are
sampled at a 10-vehicle interval in order to better illustrate the
features of individual vehicles. The results of the CTM numerical
experiments and car-following simulations reveal the usefulness of
Fig. 4. Impact results for traffic accident in macroscopic mixed CTM numerical experiment: (a) p¼0; (b) p¼0.2; (c) p¼0.4; (d) p¼0.6;
(e) p¼0.8; and (f) p¼1.
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the mixed CTM proposed in this paper for solving the traffic
accident problem, from a macroscopic perspective, under different
CAV penetration rates.
Conclusions
Some studies (e.g., Qian et al. 2017;Tiaprasert et al. 2017) have
investigated multiclass CTMs taking into consideration complex
vehicle behaviors such as lane changing and overtaking. However,
a macroscopic perspective takes into consideration vehicular flow
evolution as a whole, regardless of local vehicle behavior. There-
fore, it needs a simple and practical CTM for mixed traffic flow
with different CAV penetration rates. This paper describes a pro-
posed simplified mixed CTM that extends the original CTM
(Daganzo 1994). The mixed CTM is based on the mixed triangular
fundamental diagram, which defines the proportions of different
vehicle types. It deals with various scenarios of mixed traffic flow
with different proportions of CAV and human vehicles.
To validate the usefulness of the proposed mixed CTM, an
example application was conducted to solve traffic accident prob-
lem. The mixed traffic flow in the application was defined as
random mixed CACC–ACC–human vehicular flow determined
by the CAV market penetration rate. The mixed CTM numerical
experiments provided macroscopic impacts of traffic accidents
on a vehicle queue over space and time for different CAV penetra-
tion rates. Results show that the queue created by a traffic accident
decreases with increasing CAV penetration rate. Moreover, the
microscopic car-following simulations showed consistency with re-
sults for the macroscopic CTM experiments. As illustrated by the
example application, the proposed mixed CTM is a simple and
practical way to solve traffic problems with less computation.
Fig. 5. Heat maps of speed over time and space in microscopic car-following simulations: (a) p¼0; (b) p¼0.2; (c) p¼0.4; (d) p¼0.6;
(e) p¼0.8; and (f) p¼1.
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We are aware that the proposed mixed CTM has limitations.
For example, it cannot track traffic flow dynamics accurately if
the proportion of each vehicle class in each cell fluctuates greatly,
because class-specific density is assumed to be uniformly distrib-
uted throughout the cells (Levin and Boyles 2016a). Moreover, the
proposed mixed CTM should be further validated using available
experimental data given that, at present, large-scale experimental
CAV data are not available. Only small-scale experiments with
several CAVs have been conducted; however, CAV equilibrium
spacing-speed can be determined based on them and then the
CTM proposed in this paper can be used to analyze realistic
dynamics of mixed CAV flows.
Acknowledgments
This work was supported by the National Natural Science Foun-
dation of China (Grant Nos. 51478113 and 51878161); the Scien-
tific Research Foundation of the Graduate School of Southeast
University (Grant No. YBJJ1792), and the Fundamental Re-
search Funds for the Central Universities and the Postgraduate
Research & Practice Innovation Program of Jiangsu Province
(Grant No. KYCX17_0146).
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Fig. 6. Simulation trajectories for vehicles over time and space in microscopic car-following simulations: (a) p¼0; (b) p¼0.2; (c) p¼0.4;
(d) p¼0.6; (e) p¼0.8; and (f) p¼1.
© ASCE 04019014-7 J. Transp. Eng., Part A: Syst.
J. Transp. Eng., Part A: Systems, 2019, 145(5): 04019014
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