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Traffic jam induced by a crosscut road in a traffic-flow model
_E!!J
JQsJ 1-._
Physica A 207 (1994) 574-583
ELSEVIER
Traffic jam induced by a crosscut road in a traffic-flow model Takashi
Nagatani
College of Engineering,
and Tadachika
Shizuoka
Seno
University, Hamamatsu 432, Japan
Received 21 December 1993
Abstract
A deterministic cellular automaton model is presented to simulate the traffic jam induced by a crosscut road in a two-dimensional traffic flow. The effect of a crosscut road on the traffic flow is investigated by the use of a computer simulation. The traffic jam appears when a shock (discontinuous interface of different car densities) is formed. The condition for shock formation is derived for car densities p, and p, of the crosscut road and its crossing streets. The phase diagram and the dependence of the traffic flow on the car densities are shown. Also, we study the shock structure and the scaling of its width. The width Aw of the shock scales with the system size L as AW = L"'. We present a self-consistent mean-field theory for the traffic flow.
1. Introduction
Recently, traffic problems have attracted considerable attention [I-S]. The computer simulation of traffic flow in an entire city is a formidable task since it involves many degrees of freedom. The availability of powerful supercomputers has encouraged researchers to tackle simulations of various realistic and formidable problems. Traffic problems are included in these problems and extensively studied by using the various hydrodynamic models [1,2]. For simpler and more flexible models, cellular automaton (CA) models are increasingly used in simulations of complex physical systems. The CA models have provided some clear physical insight in many cases [9,10]. The one-dimensional (1D) exclusion model is one of the simplest examples of a driven system [11,12]. The model has been extensively studied for understanding systems of interacting particles [13,14]. The 1D exclusion models are used to investigate the microscopic structure of shocks [15,16] and are closely linked to growth processes [17,18]. The 1D asymmetric simple exclusion model can be 0378.4371/94/$07.00 0 1994 Elsevier SSDI 0378-4371(93)E0599-A
Science
B.V. All rights
reserved
T. Nagatani, T. Seno / Physica A 207 (1994) 574-583
575
formulated as traffic jam problems. Very recently, Biham et al. [3] have proposed a simple CA model to describe a traffic flow in two dimensions. The traffic-flow model is given by a three-state CA on the square lattice. The model is the extended version of the 1D asymmetric exclusion model to two dimensions. They found that a dynamical jamming transition occurs at a critical density of cars. Nagatani [6] presented the mean-field theory for the jamming transition. The two-dimensional CA model was investigated by some researchers [7,8,19,20]. Also, in order to simulate freeway traffic, Nagel and Schreckenberg [4] extended the ZD asymmetric exclusion model to take into account car velocity. They showed that a transition from laminar traffic flow to start-stop waves occurs with increasing car density as is observed in real freeway traffic. In real traffic-flow systems, the traffic jam is frequently induced by crossings when streets cross with another street (crosscut road). The crosscut road prevents cars from crossing their road. As soon as a crosscut road begins to be congested, a traffic jam spreads from the crosscut road throughout space. The occurrence of a traffic jam strongly depends on whether or not the crosscut road becomes congested. The traffic jam appears as a shock (a discontinuity of densities) which separates between the low-density traffic flow and the high-density traffic flow. Also, the geometry of the shock front gives rise to an interesting example of random interfaces. The growing interface has attracted considerable attention [21]. It gives rise to fractal landscapes. The concept of fractal landscapes has been successfully applied to the DNA sequences and the heartbeat [22-251. The scaling behaviour of the shock front will be one of currently interesting problems. The mechanism of shock formation in the 1D asymmetric simple exclusion model was examined by Janowsky and Lebowitz [15]. They studied a shock structure when the translation invariance is broken by the insertion of a blockage. The model corresponds to the traffic-jam problem induced by a car accident. The mechanism of shock formation induced by the car accident will be different from that by the crosscut road. However, the shock formation and the traffic jam induced by a crosscut road have not been studied until now. In this paper, we present a deterministic cellular automaton model to simulate the traffic jam induced by a crosscut road in a two-dimensional traffic flow. We study the effect of a crosscut road on the traffic flow by using a computer simulation. We show the condition of the occurrence of the traffic jam for the car densities. We also study the traffic-jam structure and the scaling of its width.
2. Model and simulation We describe the deterministic CA model showing a traffic jam induced by a crosscut road in two dimensions. The model is defined on the square lattice of L X L sites with periodic boundary conditions. We set the position of the crosscut road on the vertical line passing through the center of the square lattice. The cars
516
T. Nagatani, T. Seno I Physica A 207 (1994) 574-583
I I
-+----= Lelf== Y
Crosscut
road
Fig. 1. Schematic illustration of the cellular automaton model for the traffic jam induced by the crosscut road. The arrows pointing up represent cars moving upwards. The arrows pointing to the right represent the cars moving to the right. The crosscut road is positioned on the vertical line passing through the center of the square lattice. The cars move ahead from left to right on the horizontal streets. The cars on the crosscut road move upwards.
on the crosscut road move upwards. The cars move ahead from left to right on the
L streets except the crosscut road. Fig. 1 shows the schematic illustration of the CA model. Each site except the crosscut road (vertical central line) contains either an arrow pointing to the right or is empty. Each site on the crosscut road contains either an arrow pointing upwards, an arrow pointing to the right, or is empty. The arrow pointing upwards represents the car moving up. The arrow pointing to the right represents the car moving to the right. The traffic-flow model is a three-state CA. The density of cars moving to the right is p, and the density of cars moving upwards on the crosscut road is py. It is hard for cars pointing to the right to cross through the crosscut road since the cars on the crosscut road prevent other cars from crossing through the crosscut road. When the number of cars on the crosscut road becomes larger than the critical value, a traffic jam occurs and it spreads from the crosscut road throughout space. We shall investigate the effect of the crosscut road on the traffic flow. The traffic flow is described in terms of the same CA model as that proposed by Biham et al. [3]. For an arbitrary configuration, one update of the system consists of the two steps: the first step is performed in parallel for all the cars on the horizontal streets and the second step is performed in parallel for all the cars on the crosscut road. The move or stop of cars in each step is the same as the CA model by Biham et al. Each arrow moves forward one step unless the forward nearest-neighbour site is occupied by another arrow. If an arrow is blocked ahead by another arrow, it does not move even if the blocking arrow moves out of the site during the same time step. In this model, the total number of cars is conserved. The traffic problem on L horizontal streets connected by a crosscut road is reduced to its simplest form. The essential features are maintained. These features include the simultaneous flow in two perpendicular directions of cars
T. Nagatani,
T. Seno
I Physica
A 207 (1994) 574-583
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which cannot overlap. Furthermore, this model possesses the property that it is hard for cars on horizontal streets to cross through the crosscut road. We perform the computer simulation of the traffic-flow model. We study the effect of the crosscut road on the traffic flow. We show the condition of the occurrence of the traffic jam. Initially, cars pointing upwards are randomly distributed at the sites on the central crosscut road with the density pY and cars pointing to the right are randomly distributed at the sites on the L horizontal streets with the density p,. Due to the periodic boundary conditions, the total number of up arrows on the central crosscut road and the total number of right arrows in each row are conserved, giving rise to L + 1 conservation rules. Starting the initial random configuration of cars, each car is moved or stopped according to the CA rule described above.
3. Simulation result We have performed simulations of the CA model starting with an ensemble of random initial conditions where the system size is L = 50-1000, and the initial densities of cars are pY = 0.0-1.0 and p, = 0.0-1.0. Each run is calculated until 10000 time steps. We show the patterns of cars for the traffic flow. Fig. 2a shows the typical pattern of configuration of cars for the densities p, = 0.2 and pY = 0.8 after 5000 time steps where the system size is 50 x 50. The cars moving upwards are indicated by the vertical bars. The cars moving to the right are indicated by
-_-_
_ -_ _____, ________,
_ -_
_____,
-__-_ _ -_-___ -_ _-_ -_ ,-_ _____ _ - _-_-_ _ _________, _ -_ _
(a)
(b
Fig. 2. The typical patterns of configurations of cars for the car density p, = 0.2 on the horizontal streets after 5000 time steps where the system size is 50 x 50. The cars moving upwards are indicated by the vertical bars. The cars moving to the right are indicated by the horizontal bars. (a) The pattern obtained for the car density p, = 0.8 on the crosscut road. Cars on the horizontal streets are blocked by the cars on the crosscut road and a traffic jam is induced by the crosscut road. (b) The pattern obtained for p, = 0.3. A traffic jam does not occur.
T. Nagatani, T. &no
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I Physica A 207 (1994) 574-583
the horizontal bars. Cars on the horizontal streets are blocked by the cars on the central crosscut road and a traffic jam induced by the crosscut road occurs. The traffic
jam
appears
as a discontinuous
interface
between
the low-density
flow and the high-density traffic flow. In the region of the low density, cars move with the maximum value u = 1, indicating that the arrow blocked.
The
numbers
N, = 471, and the mean
NY and
TX of up
velocities
are (u,)
and
right
= 0.152 and
arrows
are
(u,)
= 0.514.
traffic the right is never
NY = 40 and The mean
velocities ( uX) and (v,) indicate respectively those of cars moving to the right and moving upwards. The mean velocity (uX) of cars moving to the right in a unit time interval is defined to be the number of arrows successfully moving right divided by the number of right arrows. The mean velocity (u,) of cars moving upwards in a unit time interval is defined to be the number of arrows successfully moving up divided by the number of up arrows. The velocity (u) has maximum value (u) = 1, indicating that the arrow is never blocked, while (u) = 0 means that the arrow is stopped and never moves at all. Fig. 2b shows the typical pattern of configuration of cars for the densities p, = 0.2 and pY = 0.3 after 5000 time steps where system size is 50 x 50. The numbers NY and N, of up and right arrows are NY = 12 and N, = 471, and the mean velocities are (u,) = 0.917 and ( uX) = 1.0. A discontinuous interface of densities is not induced by the crosscut road. For low density pY, the traffic jam does not appear. (J,) on the We study the mean velocities ( ux), (u,) and the traffic current horizonal streets. Fig. 3 shows the plot of the mean velocities (u,) (indicated by the circle), (u,) (indicated by the triangles) and the current (1,) (indicated by the squares) against the car density p, for the density pY = 0.3 on the crosscut road after 10000 time steps where the system size is 400 X 400. At p, =pxd = 0.31, a traffic jam occurs. Until p, = 0.31, all the cars on the horizontal streets move with
0
0.5
1
Fig. 3. The plot of the mean velocities (u,) (indicated by the circles), (u,) (indicated by the triangles) and the traffic current (J,) (indicated by the squares) against the car density p, for the density p, = 0.3 after 10000 time steps where the system size is 400 X 400. The solid lines represent the theoretical result obtained by the mean-field approach.
T. Nagatani, T. Seno
I Physica A 207(1994)574-583
579
the maximal velocity (u,) = 1. The current (.I,) on the horizontal streets increases linearly with the density p, until p, =p,,(=O.31). The current (.I,) remains at the constant value of (.7,) = 0.32 between p, = 0.31 and p, =p,, = 0.69. The traffic current (J,) is saturated when the shock (the discontinuous interface of density) is formed. The flow pattern showing the traffic jam is characterized by the constant current. For the larger density p, than p, =p,, = 0.69, the discontinuous interface of density disappears, and the current (.I,) and the velocities (u,) and ( uY) decrease with the density p, and become zero at p, = 1. We find that the dynamical jamming transition between the continuous phase and the shock phase occurs at pxd = 0.31 and p,, = 0.69. Figs. 4 and 5 show the plots of the mean velocities (u,), (u,) and the current (.I,) against the car density p, for the cases of pY = 0.5 and 0.7. For pY = 0.5, the traffic jam occurs between pxd = 0.21 and p,, = 0.77. The current (J,) remains at the constant value of (I,) -0.23 when the shock is formed. For p, = 0.7, the traffic jam appears between pxd = 0.13 and p,, = 0.84. The current (J,) is saturated at (J,) = 0.14 when the traffic jam occurs. With increasing pY, the traffic jam occurs at low density p,. The saturated current (TY) decreases with increasing pY. Fig. 6 shows the plot of the transition points pxd and p,, against pY. The transition points are indicated by the circles. The traffic jam occurs between pxd and p,,. The condition of the shock formation induced by the insertion of a blockage in 1D asymmetric exclusion model is given by P xd = r/(1 + r)
and
p,, =
l/(1 + r)
(1)
,
where r is the probability that a particle passes through the blockage. The transition point is represented by the broken line in Fig. 6 where r = 1 -pY. The 1
cv,> WY>
0.5
J.
.
\
0
0.5
I
. : . :
\
0
0.5
Fig. 4.
p,=o.7
1
Fig. 5.
Fig. 4. The plot of the mean velocities (u,), solid lines indicate the mean-field result.
(II,)
and the current
(.I,)
against
p, for py = 0.5. The
Fig. 5. The plot of the mean velocities (u,), solid lines indicate the mean-field result.
(u,)
and the current
(J,)
against
p, for p, = 0.7. The
580
T. Nagatani,
T. Seno
I Physica
A 207 (1994) 574-583
1
PY
Fig. 6. The plot the circles. The transition points exclusion model
of the transition points pxd and ,u,. against p,. The simulation result is indicated by shock (or traffic jam) appears between prd and p,,. The mean-field result (4) for the is indicated by the solid curves. The theoretical result (1) for the ID asymmetric is represented by the broken line.
transition point induced by the crosscut road is different from that in the 1D asymmetric exclusion model. We study the shock structure and the scaling of its width. Fig. 7 shows the space-time averaged density profile p, against the distance x along the horizontal street for the case densities pY = 0.7 and p, = 0.4 where the system size is 500 x 500 and the density is averaged over all the horizontal streets and 5000 time steps. The numbers NY and N, of up and right arrows are NY= 349 and N, = 99424. The crosscut road is positioned on x = 250. The shock appears at x = 81.25 on the horizontal street. The shock separates between the low density p,,,,, = 0.151 and the high density px,high = 0.849. The low density px,,ow = 0.151 on the upstream region of the shock is nearly consistent with the transition point P xd = 0.13 in Fig. 5. The high density px,hjgh = 0.849 on the downstream region of
I
I 0
X
250
Fig. 7. The density profile p, against the distance x along p, = 0.4 where the system size is 500 x 500 and the density streets and 5000 time steps.
500 the horizontal p, is averaged
street for p, = 0.7 and over all the horizontal
T. Nagatani,
T. Seno
/ Physica
102
slope: I12 .‘. ./ /’
AW
581
A 207 (1994) 574-583
/
l ’
. /
.
Fig. 8. The log-log plot of the width Aw of the shock against the slope f is shown.
the system
size L. The straight
line with
the shock agrees with the transition point p,, = 0.84 in Fig. 5. The width Aw of the shock is larger than that in the 1D asymmetric exclusion model. Fig. 8 shows the log-log plot of the width Aw against the system size L. The width Aw of the shock scales as Aw=Ll”.
(2)
The value + of the scaling exponent is consistent with that obtained in the 1D asymmetric exclusion model. Its value also agrees with that in the random deposition model. The origin of the value will be due to the fluctuation of the occupation of up arrows on the crosscut road. The traffic flow on the crosscut road is not long-range fluctuation but short-range fluctuation. The traffic flow on the horizontal streets separated with a sufficiently large distance is uncorrelated. The position of the shock on the horizontal street fluctuates randomly.
4. Mean-field approach We present a simple mean-field theory for the traffic flow. The traffic flow on the horizontal street will be assumed to be approximated by that in the 1D asymmetric exclusion model since the traffic flow on the horizontal streets separated with a sufficiently long distance is uncorrelated. The probability pblock that the cars on the horizontal street cannot pass through the crosscut road is approximated by the probability that cars occupy the site on the crossing. It is By replacing r in Eq. (1) with 1 - pblock, the transition given by Pblock =Py +Pxd’ points pxd and p,, are determined: P xd =
(l -py
-Pxd)'(2
-py
-Pxd)
and i% = l/c2 -py
-Pxd)
.
(3)
Eq. (3) gives a self-consistent mean-field theory for the transition points pxd and point pxd at which the traffic jam appears is given by
P X”’ The transition
T. Nagatani, T. Seno I Physica A 207 (1994) 574-583
582
(4)
-~(3-qi)2-4(1-Py)1.
Pxd=+[3-py
The transition
point p,,
by Eq. (4) are plotted
is given by p,,
=
1 - pxd. Two transition
lines determined
by the solid lines in Fig. 6. The transition
point determined
by the mean-field theory agrees with the simulation result. For pY = 0.7, Eq. (4) gives pxd = 0.1388 and p,, = 0.8611. The values agree with p,,,,, = 0.151 and Px,high
=
0.849 obtained
in Fig. 5.
For lower density p, than the transition point pxd, right cars move with the maximal velocity and the current increases with p, since right cars are not blocked by other cars. The traffic flow is consistent with that in the 1D asymmetric where the shock is formed, the current (J,) is exclusion model. For pxd < p,
=p,
(1,)
=Pxd
(J,)
= 1 -P,
and (u,) and
for OSp,
= 1
h>
for
%d’Px
and (u,> = (1 -P,>/P,
Pxd
6pxd,
cP,,
(5)
)
forp,,
The mean current (J,) and the mean velocity (II,) given by Eq. (5) are indicated by the solid curves in Figs. 3, 4 and 5. The theoretical result obtained by the mean-field theory is consistent with the simulation result.
5. Summary We presented the deterministic cellular automaton model to simulate the traffic jam induced by a crosscut road in a two-dimensional traffic flow. We investigated the condition of the occurrence of the traffic jam. We studied the shock structure and the scaling of its width. We also presented a simple mean-field theory for the traffic flow. The pattern of the traffic jam gives rise to a random surface. The rough surface is a currently interesting problem.
Acknowledgement We would
like to thank
Professor
H.E.
Stanley
for stimulating
suggestions.
References [l] N.H. Gatner 1987).
and
N.H.M.
Wilson,
Transportation
and
Traffic
Theory
(Elsevier,
New York,
T. Nagatani, T. Seno I Physica A 207 (1994) 574-583 [2] [3] [4] [5] [6] [7] [S] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
583
W. Leutzbach, Introduction to the Theory of Traffic Flow (Springer, Berlin, 1988). 0. Biham, A.A. Middleton and D. Levine, Phys. Rev. A 46 (1992) R6124. K. Nagel and M. Schreckenberg, J. Phys. 1 (France) 2 (1992) 2221. T. Nagatani, J. Phys. Sot. Jpn. 62 (1993) 1085. T. Nagatani, J. Phys. Sot. Jpn. 62 (1993) 2625. M. Fukui and Y. Ishibashi, J. Phys. Sot. Jpn. 62 (1993) 3841. S. Tadaki and M. Kikuchi, preprint. S. Wolfram, Theory and Applications of Cellular Automata (World Scientific, Singapore, 1986). K. Kaneko, in: Formation, Dynamics and Statistics of Patterns, vol. 1, K. Kawasaki, M. Suzuki and A. Onuki, eds. (World Scientific, Singapore, 1990) p. 1. T.M. Liggett, Interacting Particle Systems (Springer, New York, 1985). H. Spohn, Large Scale Dynamics of Interacting Particles (Springer, Berlin, 1991). B. Derrida, M.R. Evans, V Hakim and V. Pasquier, J. Phys. A 26 (1993) 1493. L.H. Gwa and H. Spohn, Phys. Rev. Lett. 68 (1992) 725. S.A. Janowsky and J.L. Lebowitz, Phys. Rev. A 45 (1992) 618. M. Bramson, J. Stat. Phys. 51 (1988) 863. P. Meakin, P. Ramanlal, L.M. Sander and R.C. Ball, Phys. Rev. A 34 (1986) 5091. J. Krug and H. Spohn, Phys. Rev. A 38 (1988) 4271. T. Nagatani, Physica A 198 (1993) 108. T. Nagatani, Phys. Rev. E 48 (1993) 3290. F. Family and T. Vicsek, Dynamics of Fractal Surfacaes (World Scientific, Singapore, 1991). C.K. Peng, S.V. Buldyrev, A.L. Goldberger, S. Havlin, F. Sciortino, M. Simons and H.E. Stanley, Nature 356 (1992) 168. S.V. Buldyrev, A.L. Goldberger, S. Havlin, C.K. Peng, M. Simons and H.E. Stanley, Phys. Rev. E 47 (1993) 4514. C.K. Peng, J. Mietus, J.M. Hausdorff, S. Havlin, H.E. Stanley and A.L. Goldberger, Phys. Rev. Lett. 70 (1993) 1343. H.E. Stanley, Physica A 186 (1992) 1.
JQsJ 1-._
Physica A 207 (1994) 574-583
ELSEVIER
Traffic jam induced by a crosscut road in a traffic-flow model Takashi
Nagatani
College of Engineering,
and Tadachika
Shizuoka
Seno
University, Hamamatsu 432, Japan
Received 21 December 1993
Abstract
A deterministic cellular automaton model is presented to simulate the traffic jam induced by a crosscut road in a two-dimensional traffic flow. The effect of a crosscut road on the traffic flow is investigated by the use of a computer simulation. The traffic jam appears when a shock (discontinuous interface of different car densities) is formed. The condition for shock formation is derived for car densities p, and p, of the crosscut road and its crossing streets. The phase diagram and the dependence of the traffic flow on the car densities are shown. Also, we study the shock structure and the scaling of its width. The width Aw of the shock scales with the system size L as AW = L"'. We present a self-consistent mean-field theory for the traffic flow.
1. Introduction
Recently, traffic problems have attracted considerable attention [I-S]. The computer simulation of traffic flow in an entire city is a formidable task since it involves many degrees of freedom. The availability of powerful supercomputers has encouraged researchers to tackle simulations of various realistic and formidable problems. Traffic problems are included in these problems and extensively studied by using the various hydrodynamic models [1,2]. For simpler and more flexible models, cellular automaton (CA) models are increasingly used in simulations of complex physical systems. The CA models have provided some clear physical insight in many cases [9,10]. The one-dimensional (1D) exclusion model is one of the simplest examples of a driven system [11,12]. The model has been extensively studied for understanding systems of interacting particles [13,14]. The 1D exclusion models are used to investigate the microscopic structure of shocks [15,16] and are closely linked to growth processes [17,18]. The 1D asymmetric simple exclusion model can be 0378.4371/94/$07.00 0 1994 Elsevier SSDI 0378-4371(93)E0599-A
Science
B.V. All rights
reserved
T. Nagatani, T. Seno / Physica A 207 (1994) 574-583
575
formulated as traffic jam problems. Very recently, Biham et al. [3] have proposed a simple CA model to describe a traffic flow in two dimensions. The traffic-flow model is given by a three-state CA on the square lattice. The model is the extended version of the 1D asymmetric exclusion model to two dimensions. They found that a dynamical jamming transition occurs at a critical density of cars. Nagatani [6] presented the mean-field theory for the jamming transition. The two-dimensional CA model was investigated by some researchers [7,8,19,20]. Also, in order to simulate freeway traffic, Nagel and Schreckenberg [4] extended the ZD asymmetric exclusion model to take into account car velocity. They showed that a transition from laminar traffic flow to start-stop waves occurs with increasing car density as is observed in real freeway traffic. In real traffic-flow systems, the traffic jam is frequently induced by crossings when streets cross with another street (crosscut road). The crosscut road prevents cars from crossing their road. As soon as a crosscut road begins to be congested, a traffic jam spreads from the crosscut road throughout space. The occurrence of a traffic jam strongly depends on whether or not the crosscut road becomes congested. The traffic jam appears as a shock (a discontinuity of densities) which separates between the low-density traffic flow and the high-density traffic flow. Also, the geometry of the shock front gives rise to an interesting example of random interfaces. The growing interface has attracted considerable attention [21]. It gives rise to fractal landscapes. The concept of fractal landscapes has been successfully applied to the DNA sequences and the heartbeat [22-251. The scaling behaviour of the shock front will be one of currently interesting problems. The mechanism of shock formation in the 1D asymmetric simple exclusion model was examined by Janowsky and Lebowitz [15]. They studied a shock structure when the translation invariance is broken by the insertion of a blockage. The model corresponds to the traffic-jam problem induced by a car accident. The mechanism of shock formation induced by the car accident will be different from that by the crosscut road. However, the shock formation and the traffic jam induced by a crosscut road have not been studied until now. In this paper, we present a deterministic cellular automaton model to simulate the traffic jam induced by a crosscut road in a two-dimensional traffic flow. We study the effect of a crosscut road on the traffic flow by using a computer simulation. We show the condition of the occurrence of the traffic jam for the car densities. We also study the traffic-jam structure and the scaling of its width.
2. Model and simulation We describe the deterministic CA model showing a traffic jam induced by a crosscut road in two dimensions. The model is defined on the square lattice of L X L sites with periodic boundary conditions. We set the position of the crosscut road on the vertical line passing through the center of the square lattice. The cars
516
T. Nagatani, T. Seno I Physica A 207 (1994) 574-583
I I
-+----= Lelf== Y
Crosscut
road
Fig. 1. Schematic illustration of the cellular automaton model for the traffic jam induced by the crosscut road. The arrows pointing up represent cars moving upwards. The arrows pointing to the right represent the cars moving to the right. The crosscut road is positioned on the vertical line passing through the center of the square lattice. The cars move ahead from left to right on the horizontal streets. The cars on the crosscut road move upwards.
on the crosscut road move upwards. The cars move ahead from left to right on the
L streets except the crosscut road. Fig. 1 shows the schematic illustration of the CA model. Each site except the crosscut road (vertical central line) contains either an arrow pointing to the right or is empty. Each site on the crosscut road contains either an arrow pointing upwards, an arrow pointing to the right, or is empty. The arrow pointing upwards represents the car moving up. The arrow pointing to the right represents the car moving to the right. The traffic-flow model is a three-state CA. The density of cars moving to the right is p, and the density of cars moving upwards on the crosscut road is py. It is hard for cars pointing to the right to cross through the crosscut road since the cars on the crosscut road prevent other cars from crossing through the crosscut road. When the number of cars on the crosscut road becomes larger than the critical value, a traffic jam occurs and it spreads from the crosscut road throughout space. We shall investigate the effect of the crosscut road on the traffic flow. The traffic flow is described in terms of the same CA model as that proposed by Biham et al. [3]. For an arbitrary configuration, one update of the system consists of the two steps: the first step is performed in parallel for all the cars on the horizontal streets and the second step is performed in parallel for all the cars on the crosscut road. The move or stop of cars in each step is the same as the CA model by Biham et al. Each arrow moves forward one step unless the forward nearest-neighbour site is occupied by another arrow. If an arrow is blocked ahead by another arrow, it does not move even if the blocking arrow moves out of the site during the same time step. In this model, the total number of cars is conserved. The traffic problem on L horizontal streets connected by a crosscut road is reduced to its simplest form. The essential features are maintained. These features include the simultaneous flow in two perpendicular directions of cars
T. Nagatani,
T. Seno
I Physica
A 207 (1994) 574-583
577
which cannot overlap. Furthermore, this model possesses the property that it is hard for cars on horizontal streets to cross through the crosscut road. We perform the computer simulation of the traffic-flow model. We study the effect of the crosscut road on the traffic flow. We show the condition of the occurrence of the traffic jam. Initially, cars pointing upwards are randomly distributed at the sites on the central crosscut road with the density pY and cars pointing to the right are randomly distributed at the sites on the L horizontal streets with the density p,. Due to the periodic boundary conditions, the total number of up arrows on the central crosscut road and the total number of right arrows in each row are conserved, giving rise to L + 1 conservation rules. Starting the initial random configuration of cars, each car is moved or stopped according to the CA rule described above.
3. Simulation result We have performed simulations of the CA model starting with an ensemble of random initial conditions where the system size is L = 50-1000, and the initial densities of cars are pY = 0.0-1.0 and p, = 0.0-1.0. Each run is calculated until 10000 time steps. We show the patterns of cars for the traffic flow. Fig. 2a shows the typical pattern of configuration of cars for the densities p, = 0.2 and pY = 0.8 after 5000 time steps where the system size is 50 x 50. The cars moving upwards are indicated by the vertical bars. The cars moving to the right are indicated by
-_-_
_ -_ _____, ________,
_ -_
_____,
-__-_ _ -_-___ -_ _-_ -_ ,-_ _____ _ - _-_-_ _ _________, _ -_ _
(a)
(b
Fig. 2. The typical patterns of configurations of cars for the car density p, = 0.2 on the horizontal streets after 5000 time steps where the system size is 50 x 50. The cars moving upwards are indicated by the vertical bars. The cars moving to the right are indicated by the horizontal bars. (a) The pattern obtained for the car density p, = 0.8 on the crosscut road. Cars on the horizontal streets are blocked by the cars on the crosscut road and a traffic jam is induced by the crosscut road. (b) The pattern obtained for p, = 0.3. A traffic jam does not occur.
T. Nagatani, T. &no
578
I Physica A 207 (1994) 574-583
the horizontal bars. Cars on the horizontal streets are blocked by the cars on the central crosscut road and a traffic jam induced by the crosscut road occurs. The traffic
jam
appears
as a discontinuous
interface
between
the low-density
flow and the high-density traffic flow. In the region of the low density, cars move with the maximum value u = 1, indicating that the arrow blocked.
The
numbers
N, = 471, and the mean
NY and
TX of up
velocities
are (u,)
and
right
= 0.152 and
arrows
are
(u,)
= 0.514.
traffic the right is never
NY = 40 and The mean
velocities ( uX) and (v,) indicate respectively those of cars moving to the right and moving upwards. The mean velocity (uX) of cars moving to the right in a unit time interval is defined to be the number of arrows successfully moving right divided by the number of right arrows. The mean velocity (u,) of cars moving upwards in a unit time interval is defined to be the number of arrows successfully moving up divided by the number of up arrows. The velocity (u) has maximum value (u) = 1, indicating that the arrow is never blocked, while (u) = 0 means that the arrow is stopped and never moves at all. Fig. 2b shows the typical pattern of configuration of cars for the densities p, = 0.2 and pY = 0.3 after 5000 time steps where system size is 50 x 50. The numbers NY and N, of up and right arrows are NY = 12 and N, = 471, and the mean velocities are (u,) = 0.917 and ( uX) = 1.0. A discontinuous interface of densities is not induced by the crosscut road. For low density pY, the traffic jam does not appear. (J,) on the We study the mean velocities ( ux), (u,) and the traffic current horizonal streets. Fig. 3 shows the plot of the mean velocities (u,) (indicated by the circle), (u,) (indicated by the triangles) and the current (1,) (indicated by the squares) against the car density p, for the density pY = 0.3 on the crosscut road after 10000 time steps where the system size is 400 X 400. At p, =pxd = 0.31, a traffic jam occurs. Until p, = 0.31, all the cars on the horizontal streets move with
0
0.5
1
Fig. 3. The plot of the mean velocities (u,) (indicated by the circles), (u,) (indicated by the triangles) and the traffic current (J,) (indicated by the squares) against the car density p, for the density p, = 0.3 after 10000 time steps where the system size is 400 X 400. The solid lines represent the theoretical result obtained by the mean-field approach.
T. Nagatani, T. Seno
I Physica A 207(1994)574-583
579
the maximal velocity (u,) = 1. The current (.I,) on the horizontal streets increases linearly with the density p, until p, =p,,(=O.31). The current (.I,) remains at the constant value of (.7,) = 0.32 between p, = 0.31 and p, =p,, = 0.69. The traffic current (J,) is saturated when the shock (the discontinuous interface of density) is formed. The flow pattern showing the traffic jam is characterized by the constant current. For the larger density p, than p, =p,, = 0.69, the discontinuous interface of density disappears, and the current (.I,) and the velocities (u,) and ( uY) decrease with the density p, and become zero at p, = 1. We find that the dynamical jamming transition between the continuous phase and the shock phase occurs at pxd = 0.31 and p,, = 0.69. Figs. 4 and 5 show the plots of the mean velocities (u,), (u,) and the current (.I,) against the car density p, for the cases of pY = 0.5 and 0.7. For pY = 0.5, the traffic jam occurs between pxd = 0.21 and p,, = 0.77. The current (J,) remains at the constant value of (I,) -0.23 when the shock is formed. For p, = 0.7, the traffic jam appears between pxd = 0.13 and p,, = 0.84. The current (J,) is saturated at (J,) = 0.14 when the traffic jam occurs. With increasing pY, the traffic jam occurs at low density p,. The saturated current (TY) decreases with increasing pY. Fig. 6 shows the plot of the transition points pxd and p,, against pY. The transition points are indicated by the circles. The traffic jam occurs between pxd and p,,. The condition of the shock formation induced by the insertion of a blockage in 1D asymmetric exclusion model is given by P xd = r/(1 + r)
and
p,, =
l/(1 + r)
(1)
,
where r is the probability that a particle passes through the blockage. The transition point is represented by the broken line in Fig. 6 where r = 1 -pY. The 1
cv,> WY>
0.5
J.
.
\
0
0.5
I
. :
\
0
0.5
Fig. 4.
p,=o.7
1
Fig. 5.
Fig. 4. The plot of the mean velocities (u,), solid lines indicate the mean-field result.
(II,)
and the current
(.I,)
against
p, for py = 0.5. The
Fig. 5. The plot of the mean velocities (u,), solid lines indicate the mean-field result.
(u,)
and the current
(J,)
against
p, for p, = 0.7. The
580
T. Nagatani,
T. Seno
I Physica
A 207 (1994) 574-583
1
PY
Fig. 6. The plot the circles. The transition points exclusion model
of the transition points pxd and ,u,. against p,. The simulation result is indicated by shock (or traffic jam) appears between prd and p,,. The mean-field result (4) for the is indicated by the solid curves. The theoretical result (1) for the ID asymmetric is represented by the broken line.
transition point induced by the crosscut road is different from that in the 1D asymmetric exclusion model. We study the shock structure and the scaling of its width. Fig. 7 shows the space-time averaged density profile p, against the distance x along the horizontal street for the case densities pY = 0.7 and p, = 0.4 where the system size is 500 x 500 and the density is averaged over all the horizontal streets and 5000 time steps. The numbers NY and N, of up and right arrows are NY= 349 and N, = 99424. The crosscut road is positioned on x = 250. The shock appears at x = 81.25 on the horizontal street. The shock separates between the low density p,,,,, = 0.151 and the high density px,high = 0.849. The low density px,,ow = 0.151 on the upstream region of the shock is nearly consistent with the transition point P xd = 0.13 in Fig. 5. The high density px,hjgh = 0.849 on the downstream region of
I
I 0
X
250
Fig. 7. The density profile p, against the distance x along p, = 0.4 where the system size is 500 x 500 and the density streets and 5000 time steps.
500 the horizontal p, is averaged
street for p, = 0.7 and over all the horizontal
T. Nagatani,
T. Seno
/ Physica
102
slope: I12 .‘. ./ /’
AW
581
A 207 (1994) 574-583
/
l ’
. /
.
Fig. 8. The log-log plot of the width Aw of the shock against the slope f is shown.
the system
size L. The straight
line with
the shock agrees with the transition point p,, = 0.84 in Fig. 5. The width Aw of the shock is larger than that in the 1D asymmetric exclusion model. Fig. 8 shows the log-log plot of the width Aw against the system size L. The width Aw of the shock scales as Aw=Ll”.
(2)
The value + of the scaling exponent is consistent with that obtained in the 1D asymmetric exclusion model. Its value also agrees with that in the random deposition model. The origin of the value will be due to the fluctuation of the occupation of up arrows on the crosscut road. The traffic flow on the crosscut road is not long-range fluctuation but short-range fluctuation. The traffic flow on the horizontal streets separated with a sufficiently large distance is uncorrelated. The position of the shock on the horizontal street fluctuates randomly.
4. Mean-field approach We present a simple mean-field theory for the traffic flow. The traffic flow on the horizontal street will be assumed to be approximated by that in the 1D asymmetric exclusion model since the traffic flow on the horizontal streets separated with a sufficiently long distance is uncorrelated. The probability pblock that the cars on the horizontal street cannot pass through the crosscut road is approximated by the probability that cars occupy the site on the crossing. It is By replacing r in Eq. (1) with 1 - pblock, the transition given by Pblock =Py +Pxd’ points pxd and p,, are determined: P xd =
(l -py
-Pxd)'(2
-py
-Pxd)
and i% = l/c2 -py
-Pxd)
.
(3)
Eq. (3) gives a self-consistent mean-field theory for the transition points pxd and point pxd at which the traffic jam appears is given by
P X”’ The transition
T. Nagatani, T. Seno I Physica A 207 (1994) 574-583
582
(4)
-~(3-qi)2-4(1-Py)1.
Pxd=+[3-py
The transition
point p,,
by Eq. (4) are plotted
is given by p,,
=
1 - pxd. Two transition
lines determined
by the solid lines in Fig. 6. The transition
point determined
by the mean-field theory agrees with the simulation result. For pY = 0.7, Eq. (4) gives pxd = 0.1388 and p,, = 0.8611. The values agree with p,,,,, = 0.151 and Px,high
=
0.849 obtained
in Fig. 5.
For lower density p, than the transition point pxd, right cars move with the maximal velocity and the current increases with p, since right cars are not blocked by other cars. The traffic flow is consistent with that in the 1D asymmetric where the shock is formed, the current (J,) is exclusion model. For pxd < p,
=p,
(1,)
=Pxd
(J,)
= 1 -P,
and (u,) and
for OSp,
= 1
h>
for
%d’Px
and (u,> = (1 -P,>/P,
Pxd
6pxd,
cP,,
(5)
)
forp,,
The mean current (J,) and the mean velocity (II,) given by Eq. (5) are indicated by the solid curves in Figs. 3, 4 and 5. The theoretical result obtained by the mean-field theory is consistent with the simulation result.
5. Summary We presented the deterministic cellular automaton model to simulate the traffic jam induced by a crosscut road in a two-dimensional traffic flow. We investigated the condition of the occurrence of the traffic jam. We studied the shock structure and the scaling of its width. We also presented a simple mean-field theory for the traffic flow. The pattern of the traffic jam gives rise to a random surface. The rough surface is a currently interesting problem.
Acknowledgement We would
like to thank
Professor
H.E.
Stanley
for stimulating
suggestions.
References [l] N.H. Gatner 1987).
and
N.H.M.
Wilson,
Transportation
and
Traffic
Theory
(Elsevier,
New York,
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583
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