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Simulation study of traffic car accidents in single-lane highway
Khalid Bentaleb, Noureddine Lakouari, Rachid Marzoug, HamidEz-Zahraouy, Abdelilah Benyoussef
PII: S0378-4371(14)00581-0DOI: http://dx.doi.org/10.1016/j.physa.2014.07.014Reference: PHYSA 15385
To appear in: Physica A
Received date: 28 December 2013Revised date: 29 June 2014
Please cite this article as: K. Bentaleb, N. Lakouari, R. Marzoug, H. Ez-Zahraouy, A.Benyoussef, Simulation study of traffic car accidents in single-lane highway, Physica A(2014), http://dx.doi.org/10.1016/j.physa.2014.07.014
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Simulation study of traffic car accidents
in single-lane highway
Khalid BENTALEB1, Noureddine LAKOUARI1, Rachid MARZOUG1, Hamid EZ-ZAHRAOUY1,*,
Abdelilah BENYOUSSEF1, 2, 3
1Laboratoire de Magnétisme et de Physique des Hautes Energies, associé au CNRST (URAC
12), Département de physique, B.P. 1014, Faculté des sciences, Université Mohammed V
Agdal, Rabat, Morocco
2Institute of Nanomaterials and Nanotechnologies, INANOTECH, MACsIR, Rabat, Morocco
3Hassan II Academy of Science and Technology, Rabat, Morocco
*The corresponding Author E-mail: [email protected]
ABSTRACT
In this paper we numerically study the probability Pac of the occurrence of car accidents in the
extended Nagel-Schreckenberg (NS) model in the case of mixture of fast (Vmax1=5) and slow
vehicles (Vmax2=1) by taking also to the risky overtaking of fast vehicles. In comparison with
previous existing models, we find that accidents can occur in the free traffic phase and/or
congested one depending on the overtaking rate of fast vehicles. The effect of evacuation of
damaged vehicles from the road with probabilities Pevf and Pevs of fast and slow vehicles
respectively on the traffic flow behavior is also computed.
Keywords: Traffic flow, accident, NaSch model, overtaking, Cellular automata
I-INTRODUCTION
Recently, traffic problems have attracted much attention and various traffic models have been
proposed [1-8]. There have been two types of models, microscopic and macroscopic models.
Macroscopic traffic models represent the average characteristic of the flow, which consists of
vehicles each of which has a stochastic (or random) characteristic. These models are based on
fluid-dynamical description. Microscopic models consider the individual vehicles and the
human driving behavior. Indeed, the movement of individual vehicles is governed by the
driver’s behavior, the road topology, and the headway distribution. Each vehicle may be
described by a set of parameters that includes actual speed, position, and desired speed.
Among them, the CA models for traffic problems have been extensively studied. Compared
with other approaches, the CA models can be used very efficiently for computers to perform
real simulation, and are simple in nature [9]. Presently, there are two basic CA models that
describe single-lane traffic flow, the Fukui-Ishibashi (FI) model [10] and the Nagel-
Schreckenberg (NS) model [11]. Despite its simplicity (NS) is capable of capturing some
essential features observed in realistic traffic like density waves. After that, many CA models
have been extended to investigate real traffic systems such as road blocks and hindrances [12]
and two level-crossing [13].
Traffic jams and traffic accidents have become significant problems in a modern
society. Further source of dangerous event is sudden velocity changes and small safety gaps.
In recent years, CA models have been extended to study car accidents. [14,17]. The first CA
model including dangerous situations (accident) have been proposed by Boccara et al. [14],
simulation of the probability for car accident to occur have been offered in the NS models
with periodic or open boundaries and in the FI model [18-21]. An approximate probability for
car accidents to occur can be determined according to the following three conditions
originally proposed by Boccara et al. [14]. The first condition is that the number of empty
cells Gap1 in front of a car “i” is smaller than the speed limit (Gap1<Vmax), which means that
the position of the car ahead “i+1” can be reached by the car “i” at the next time. The second
condition is that the car ahead is moving at time t within the velocity v(i + 1, t) > 0, and the
last condition is that the moving car ahead “i+1” is suddenly stopped at the next time step
(v (i + 1, t + 1) = 0). If the above three conditions are satisfied, then it will be defined as an
accident of the car "i" at time t + 1 with a probability p. The dynamics itself would still
proceed; however, in this case the two vehicles are stopped at the place of crash and stopped
there until they get towed away.
In this paper, we propose a car accident model which, in addition to the accident
introduced in Ref. [14], it also takes into account the accidents due also to the risky
overtaking occurring in a single lane high way of probabilistic NaSch model. The paper is
structured as follows: Section II is devoted to the car accident model. Results and discussion
are given in section III. Section IV is reserved to conclusion.
II- The car accident model
Our studies are based on the cellular automata model introduced by Nagel and
Schreckenberg [11] to describe single-lane highway traffic. The model consists of a one-
dimensional array of L cells with periodic boundary condition. Every cell can be either empty
or occupied by a car with velocity V= 0, 1, 2, . . . ,Vmax, where Vmax is the speed limit. Let
Gap1 denote the number of empty cells in front of a car. On the street, N vehicles are moving
so that their density is given by ρ = N/L. The following steps for all cars are performed in
parallel. The first rule is acceleration: if the speed of a car is lower than Vmax, the speed is
increased by one. The second rule is deceleration due to other cars: if the speed of a car is
larger than Gap1, then it is reduced to Gap1. The third rule is randomization: the speed of a
moving car is decreased randomly by one unit with a braking probability P1 for slow vehicles
and P5 for fast ones. And the fourth rule is that each car is moved forward according to its
new speed determined by the above three rules. However additional rules should be added, in
order to capture more complex situations. In this context we have formulated our model. Thus
in this paper we utilize our conditions to simulate traffic accidents. In the process of
simulations, car accidents do not really happen. However, when the three necessary
conditions as defined by Boccara [14] above are met simultaneously with a probability p,
, we consider that the accident takes place. In fact, Pd is the probability with which fast car
may cause an accident with another slow vehicle that is located just ahead, if the fast car tries
an risky overtaking; where slow and fast cars occupy the same position at time t+1. These
dangerous situations are calculated and considered as the signal of the occurrence of
accidents. Usually, the probability per car per time step for car accidents to occur is denoted
by Pac.
The model contains some basic parameters such as the stochastic braking probabilities
P5 for fast vehicles (Vmax1=5) and P1 for slow ones (Vmax2=1), the total probability Pac for an
accident to occur, the probability Pov of overtaking, and the average density ρ =N/L. The risky
overtaking probability Pd and the probability p of stopped car introduced in Ref. [14]. Note
that when the car really collides with another one ahead with probability Pd, so to avoid the
congestion in the road, we suggest that cars causing the collision will be removed from the
road with exit probabilities Pevf and Pevs for fast and slow vehicles respectively.
In this paper we will consider two types of vehicles. Type1 represents the fast cars, and
moves with high maximal speed Vmax1=5. Type2 corresponds to slow vehicles, and has a low
maximal speed Vmax2=1. To explore the effect of different maximal velocities, Ez-Zahraouy et
al. [22] proposed a CA traffic flow model in which the vehicles are filled according to their
maximal speed.
The situation of overtaking is considered only when the following rules, which deal
with the deterministic NaSch model, are satisfied:
Incentive criteria:
R1: type1 follows type2 This means that only vehicles type1 have the right to overtake types2. The overtaking
between vehicles type 1 or between types 2, is not taken into account in this model.
R2: Gap1 ≤ Vmax2+1 Safety criteria:
R3: Gap2 ≥GapC Gap2 is the number of empty sites in front of type2, while GapC=min(V (type2)+1,Vmax2) +1
is the minimal safety distance required for overtaking, as a consequence R3 ensures that at the
situation of overtaking type2 moves without any hindrance, which means that type1 estimates
the velocity of type2 before overtaking.
R4: If the rules R1 - R3 are satisfied, the fast vehicle can overtake the slow one with the
probability Pov, so velocity of overtaking is: V(type1) = min (Gapn, Vmax1), where:
Gapn= Gap1+1+Gap2 is the new gap required for overtaking, which is obtained according to
R2 and R3.
When the position of the slow (old) lead car is at position x, then the overtaking car will be
put at the position x + 1 ( fig.1b).
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the second rule by one unit with a certain probability. Moreover in our model we take into
account the accident due to overtake of fast vehicles, i.e. when a driver does not take the
desired position during the process of overtaking. At the next time fast vehicle will arrive at
the position of the slow one which is moving ahead, so car will cause an accident with
probability p. In the numerical simulations, the car accident defined as a fast car that hits the
slow one ahead when overtake state, does not actually occur. We are looking for those
dangerous situations on the road and take them as the indicator to the appearance of car
accidents. The probability per car and per time step for an accident to occur is denoted by Pac.
III-Results and discussion:
In our computational studies we have considered a chain with L=1000 sites and periodic
boundary conditions. In this case simulation begins with particles randomly distributed
around the chain according to their fraction of fast and slow vehicles (ffast and fslow
respectively). The systems run for 20000 time-steps. The sampling for the calculation of the
results is done for the last 5000 time steps.
In the following of this paper the simulation was done in the case of mixture of two
types of vehicles (fast ones with Vmax1= 5 and slow ones with Vmax2=1) the fraction of fast
vehicles is ffast=0.9 whereas the fraction of slow ones is fslow=0.1
In the absence of overtaking process and for mixture of fast (Vmax1 = 5) and slow
vehicles (Vmax2 = 1), Figure 2 shows the behavior of accident probability Pac as a function of
density ρ for various values of the probability p. It is clear that there is a critical density below
which no car accident occurs, over which the probability for a car accident Pac increases with
the increase of ρ, and reaches a maximum, but decreases with further increase of ρ. Moreover,
when the probability p increases, the accident probability Pac increases but the critical density
remains unchanged. These results are easy to understand, as the appearance of the car
accident is directly tied to the number of stopped cars, and beneath the critical density (critical
density of slow vehicles (ρ = 0.5) because there is no overtaking so both types of vehicles run
with the same average velocity), there is no car stopped, so no car accident occurs. This is in
good agreement with results obtained in several previous studies [14, 16].
Now we will take into account the overtaking effect of fast vehicles on the traffic
accident behavior especially on the accident probability Pac for fixed fractions of fast and slow
vehicles. Fig.3 displays the accident probability Pac as a function of density ρ for various
values of Pd. Apparently; effects of overtaking at the accident probability Pac are tall. As
shown, the probability Pd influences on the probability Pac: with the increase of the probability
Pd, the accident probability Pac increase correspondingly. Moreover the probability of a car
accident increases with the density, passes through a maximum, and then decreases with
further density. However, this maximum is ever situated in the region of low-density. So
accidents also occur in the free traffic phase. When the density is low, the velocity is high and
fast vehicles overtake slow ones without hesitation. Risky overtaking would easily cause an
accident with probability Pd. However, in excess of the critical density we found that no
accident happens because the overtaking becomes more difficult due to the lack of empty
space.
In fact there are several parameters that can affect traffic (overtaking probability Pov,
braking probability P1 and P5). Hence, the variation of the probability of accident Pac as a
function of density for different value of overtaking probability Pov is given in Fig.4. As
shown in this figure, the relation of the probability Pac to the probability of overtaking Pov, if
the value of Pov is very small the probability Pac is too small; because when Pov is small the
overtaking decreases so accident due to overtaking also decreases as result Pac is small, but
when Pov increases Pac increase too.
The behavior of the probability of accident Pac caused by both risky overtaking with
probability Pd and the stopped car conditions, with probability p, proposed in Ref.[14], as a
function of density is given in Fig.5. It is clear that the accident occur in two regions; first, at
low density so with the increase of the density the value of the Pac increases to a maximum
value which corresponds to the critical density of fast types and then decreases due to
overtaking. The second region is observed at high density, from critical density of slow
vehicles this is corresponds to the case of Ref.[14], where car accident will not happen until
the density reaches critical value. Increasing the density leads to the increase of the
probability of accident, reaches a maximum, and then decreases.
As was said there are several parameters that can affect traffic. Fig.6 shows the relations
of the probability Pac to the density ρ with different values of the Pd in the case of mixture of
fast (Vmax1 = 5) and slow vehicles (Vmax2 = 1) with overtaking. With fixing braking
probability P1 = 0.2 and P5=0.2. It appears that there are great effects of the probability of
breaking P1 and P5 on the accident probability Pac. As shown in Fig.6, the braking probability
P1 and P5 impacts on the value of the probability Pac. In fact, the increase in the braking
probability leads to the increases of the probability Pac. However, the accident probability Pac
indicates the approximate plateau around a certain density region. The accident probability Pac
increases in the low-density region with the increase of the density. But it decreases with the
increase of the density in the high-density region.
So now we study the behavior of the final density ρf according to the initial density ρi,
the evacuation probability of particles Pevf and Pevs and Pd. We denote by ρf the final density
of particles at the steady state when all damaged ones are evacuated with a probability Pevf
or/and Pevs.
However, Fig.7a gives the behavior of the final density ρf as a function of the initial
density ρi for several values of evacuation probability Pevf (evacuation of fast vehicles) and a
fixed Pd. It is clear that the final density ρf is dependent on the value of the Pd for ρi smaller
than de critical density ρc. This is caused by the collisions between particles in the free flow
density region, which lead to the evacuation of some particles (fast ones) off the chain.
Consequently, the final density at the steady state becomes lower than the initial one. Whereas
for ρf> ρc the overtaking in this region becomes more difficult due to the lack of empty space
so no collision between particles occurs. As result the final density ρf is independent on the
value of Pd (ρf= ρi). In this region the final density increases with increasing the initial one
until ρf = 1 for very high density.
Indeed, in Fig.7b. is given the behavior of the final density ρf as a function of the initial
one ρi for different values of evacuation probability Pevf and Pevs (evacuation of fast and slow
vehicles) and a fixed Pd. As we have shown will there is collision between particles in the free
flow density region, hence the final density is shut to the initial one because the evacuation of
particles out of the chain is very important (evacuation of fast and slow ones). Consequently
ρf = ρi, by increasing the density, the evacuation of the particles is fewer important compared
with free flow region, that whitens the increase of the final density as a non linear function
with the initial one.
Fig.8 presents the variation of the current J as a function of the initial density for
different values of Pd and for a fixed value of Pevf (Pevf = 0.1). Nevertheless, according to the
values of Pevf and/or Pd, the current versus the initial density presents two different behaviors.
The current increases as a non linear function with ρi for ρi < ρc and for any value of Pd.
whereas, for ρi>ρc there are two different situations. The current increase linearly with density
until it reaches a maximum than it decreases linearly with ρi.
Finally, in order to inquire how the accident and evacuation probability influences the
final density at the steady state in the chain Fig.9 exhibit the space-time structure, for Pd = 0.1
and for different values of Pevf and Pevs. Every horizontal line of dots represents the
momentary position of cars moving right (red color represents the positions of fast vehicles
while positions of slow ones are presented by black color) while the successive lines (yellow
color) indicate the position of accidents at successive time steps, The empty space between
vehicles is represented by the white color; First we focus on the case in which there is no
evacuation (Fig.9a), thus vehicles cannot move at their desirable speed due to the accident car
(yellow color) which retards the movement of other cars behind; eventually the flow
decreases, this leads to a bad situation in the road. As result jams appear in the traffic. So to
overcome this situation the cars involved in collision due to overtaking and which are
corrupted, must be removed from the chain. So in Fig.9b and Fig.9c we give the case of
evacuation only of fast vehicles with different probabilities, certain vehicles are still blocked
and more cars are overcrowding behind them at the rear end of the jam. When accidents really
happen other vehicles behind cannot move with their desired speed. While, for a very
important evacuation of both types of vehicles (Fig.9d), cars can move freely without forming
jam.
I V-CONCLUSION
We have studied the behavior of car accident in the one-lane traffic model described by
the well-known Nagel-Schreckenberg model, taking into account the accident due to risky
overtaking and to the stopped cars. Depending on the fraction of fast and slow vehicles in the
road, the car accident occurs in both low and intermediate density regions. In the free flow
region, the probability of a car accident increases with the density, reaches a maximum and
then decreases with further density before vanishing. The same scenario is repeated in the
congestion and high density regions in good agreement with Ref. [14]
Figure captions:
Fig.2: Probability Pac as a function of density ρ for various value of p.
Fig.3: Probability Pac as a function of density ρ for various Pd.
Fig.4: Probability Pac as a function of density ρ for various Pov and fixed Pd =0.1.
Fig.5: Probability Pac as a function of density ρ for various Pd and p.
Fig.6: Probability Pac as a function of density ρ for Vmax1=5, Vmax2=1 for fixed value of P1=0.2
and P5=0.2 and for different value of Pd.
Fig.7a: the variation of final density, ρf, versus the initial one, ρi, for different values of the
accident probability Pd and for an intermediate exit probability Pevf = 0.1.
Fig.7b: the variation of final density, ρf, versus the initial one, ρi, for different values of the
accident probability Pd and for an intermediate exit probability Pevf = 0.1, and Pevs = 0.1.
Fig.8: the variation of average current J as a function of initial density for different values of
the accident probability Pd and for an intermediate exit probability Pevf = 0.1.
Fig.9: The space-time structure for Pd = 0.1 and in for following cases: (a) Pevf = 0, Pevs = 0
and (b) Pevf = 0.1, Pevs = 0. (c) Pevf = 0.2, Pevs = 0. (d) Pevf = 0.2, Pevs = 0.2.
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0 ,0 0 ,1 0 ,2 0 ,3 0 ,4 0 ,5 0 ,6 0 ,7 0 ,8 0 ,9 1 ,00 ,0 0 0
0 ,0 0 2
0 ,0 0 4
0 ,0 0 6
0 ,0 0 8
0 ,0 1 0
0 ,0 1 2
0 ,0 1 4
0 ,0 1 6
0 ,0 1 8
0 ,0 2 0
0 ,0 2 2
0 ,0 2 4
0 ,0 2 6
P ac
ρ
p = 0 .0 1 p = 0 .1 p = 0 .7
F ig .2
0 ,0 0 ,1 0 ,2 0 ,3 0 ,4 0 ,5 0 ,6 0 ,7 0 ,8 0 ,9 1 ,00 ,0 0 0 0
0 ,0 0 0 5
0 ,0 0 1 0
0 ,0 0 1 5
0 ,0 0 2 0
0 ,0 0 2 5
0 ,0 0 3 0
0 ,0 0 3 5
0 ,0 0 4 0
0 ,0 0 4 5
0 ,0 0 5 0
0 ,0 0 5 5
0 ,0 0 6 0
0 ,0 0 6 5
0 ,0 0 7 0
P ac
ρ
P d = 0 .0 5
P d = 0 .4
P d = 0 .7
F ig .3
0 ,0 0 ,1 0 ,2 0 ,3 0 ,4 0 ,5 0 ,6 0 ,7 0 ,8 0 ,9 1 ,00 ,0 0 0 0
0 ,0 0 0 1
0 ,0 0 0 2
0 ,0 0 0 3
0 ,0 0 0 4
0 ,0 0 0 5
0 ,0 0 0 6
0 ,0 0 0 7
0 ,0 0 0 8
0 ,0 0 0 9
0 ,0 0 1 0
P
ac
ρ
P o v = 0 .1
P o v = 0 .5
P o v = 1
F ig .4
0 ,0 0 ,1 0 ,2 0 ,3 0 ,4 0 ,5 0 ,6 0 ,7 0 ,8 0 ,9 1 ,00 ,0 0
0 ,0 1
0 ,0 2
0 ,0 3
0 ,0 4
0 ,0 5
0 ,0 6
0 ,0 7
0 ,0 8
0 ,0 9
0 ,1 0
0 ,1 1
0 ,1 2
0 ,0 0 ,1 0 ,2 0 ,3 0 ,40 ,0 0 0
0 ,0 0 1
0 ,0 0 2
0 ,0 0 3
0 ,0 0 4
0 ,0 0 5
0 ,0 0 6
p = 0 .7 , P d = 0 .7
p = 0 .1 , P d = 0 .1
P ac
ρ
F ig .5
P ac
ρ
0 ,0 0 ,1 0 ,2 0 ,3 0 ,4 0 ,5 0 ,6 0 ,7 0 ,8 0 ,9 1 ,00 ,0 0 0
0 ,0 0 1
0 ,0 0 2
0 ,0 0 3
0 ,0 0 4
0 ,0 0 5
0 ,0 0 6
0 ,0 0 7
P
ac
ρ
P d = 0 .1 P d = 0 .5 P d = 0 .7
F ig .6
0 ,0 0 ,1 0 ,2 0 ,3 0 ,4 0 ,5 0 ,6 0 ,7 0 ,8 0 ,9 1 ,00 ,0
0 ,1
0 ,2
0 ,3
0 ,4
0 ,5
0 ,6
0 ,7
0 ,8
0 ,9
1 ,0
ρ f
ρ i
P d = 0 .0 1 P d = 0 .0 5 P d = 0 .1
F ig .7 .a
0 ,0 0 ,1 0 ,2 0 ,3 0 ,4 0 ,5 0 ,6 0 ,7 0 ,8 0 ,9 1 ,00 ,0
0 ,1
0 ,2
0 ,3
0 ,4
0 ,5
0 ,6
0 ,7
0 ,8
0 ,9
1 ,0
ρ f
ρ i
P d = 0 .0 1 P d = 0 .0 5 P d = 0 .1
F ig .7 .b
0 ,0 0 ,1 0 ,2 0 ,3 0 ,4 0 ,5 0 ,6 0 ,7 0 ,8 0 ,9 1 ,00 ,0
0 ,1
0 ,2
0 ,3
0 ,4
0 ,5
0 ,6
0 ,7
J
ρ
P d = 0 .0 1 P d = 0 .0 5 P d = 0 .1
F ig .8
Fig.9a Pevf=0 & Pevs=0 Fig.9b Pevf=0.1 & Pevs=0
Fig.9c Pevf=0.2 & Pevs=0 Fig.9d Pevf=0.2 & Pevs=0.2
Highlights
Realistic car accident model Risky Overtaking of fast vehicles Occurrence of accident even in free flow traffic