We present a new cellular automata model of vehicular traffic in cities by
combining ideas borrowed from the Biham-Middleton-Levine (BML) model of city
traffic and the Nagel-Schreckenberg (NaSch) model of highway traffic. The model
exhibits a dynamical phase transition to a completely jammed phase at a
critical density which depends on the time periods of the synchronized signals.Comment: 6 pages, 5 figures, uses Springer Macros 'lncse', to appear in
  "Traffic and Granular Flow '99: Social, Traffic, and Granular Dynamics"
  edited by D. Helbing, H. J. Herrmann, M. Schreckenberg, and D. E. Wolf
  (Springer, Berlin

Brockfeld, E.

Chowdhury, D.

Klauck, K.

Santen, L.

Schadschneider, A.

Zittartz, J.

English

arXiv

We present a new cellular automata model of vehicular traffic in cities by combining ideas borrowed from the Biham-Middleton-Levine (BML) model of city traffic and the Nagel-Schreckenberg (NaSch) model of highway traffic. The model exhibits a dynamical phase transition to a completely jammed phase at a critical density which depends on the time periods of the synchronized signals

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99 A new cellular automata model for city traffic
A. Schadschneider1, D. Chowdhury1,2, E. Brockfeld3, K. Klauck1, L.
Santen1, and J. Zittartz1
1 Institut fu¨r Theoretische Physik, Universita¨t zu Ko¨ln, 50937 Ko¨ln, Germany
2 Physics Department, I.I.T., Kanpur 208016, India
3 Institut fu¨r Umweltsystemforschung, Universita¨t Osnabru¨ck, 49076 Osnabru¨ck,
Germany
Abstract. We present a new cellular automata model of vehicular traffic in cities
by combining ideas borrowed from the Biham-Middleton-Levine (BML) model of
city traffic and the Nagel-Schreckenberg (NaSch) model of highway traffic. The
model exhibits a dynamical phase transition to a completely jammed phase at a
critical density which depends on the time periods of the synchronized signals.
1 Introduction
A one-dimensional cellular automata (CA) model of highway traffic and a
two-dimensional CA model of city traffic were developed independently by
Nagel and Schreckenberg (NaSch) [1] and Biham, Middleton and Levine
(BML) [2], respectively1. Highway traffic becomes gradually more and more
congested in the NaSch model with the increase of density. Traffic jams ap-
pear because of the intrinsic stochasticity of the dynamics but no jam persists
for ever. On the other hand, a first order phase transition takes place in the
BML model at a finite non-vanishing density, where the average velocity of
the vehicles vanishes discontinuously signalling complete jamming. In the
BML model, the randomness arises only from the random initial conditions,
as the dynamical rule for the movement of the vehicles is fully deterministic
[2].
In the BML model a square lattice models the network of the streets.
Each site of the lattice represents a crossing of an east-bound and a north-
bound street; it can be empty or occupied by a vehicle moving either to the
east or to the north. The dynamics is controlled by signals of period 1 such
that at odd time steps only the east-bound vehicles are updated whereas at
even time steps only the north-bound vehicles are updated, both in parallel.
During the updating a vehicle moves one site forward if the next site ahead
is not occupied by any other vehicle. In this simplest version of the model
lane changes (e.g. by turning) are not possible and the number of vehicles on
each street is conserved separately.
In the NaSch model a highway is represented by a one-dimensional lattice
of cells that can accomodate not more than one vehicle at a time. Each vehicle
1 For a review of the different approaches in modelling traffic flow we refer to [3]
and references therein.
is characterized by a maximum velocity vmax and a randomization parameter
p. The dynamics consists of four steps, each applied in parallel to all vehicles.
In the first step all vehicles accelerate by 1 if they have not already reached
the maximum velocity vmax. Step 2 is the interaction step. Vehicles which
have d empty cells in front and a velocity v > d reduce their velocity to v = d
in order to avoid a crash. In step 3 the velocity is reduced by one unit with
probability p. In step 4 the vehicles move forward v cells where v is the new
velocity after the randomization step 3.
2 Definition of the model
In the BML model the interplay between the vehicle dynamics and the time-
scale set by the length of the signal period can not be studied. We therefore
suggested a ”unified” model [4] combining the BML model with the rules
for the vehicle dynamics of the NaSch model. The lattice of our new model
consists (in the simplest case) of N north-bound and N east-bound streets.
The N × N crossings of these streets are arranged equidistantly. Between
two consecutive crossings on a street there are D − 1 cells, i.e. each street
has length L = ND (see Fig. 1). The signals, installed at the crossings, are
synchronized in such a way that all the signals remain green for the east-
bound vehicles (and simultaneously, red for the north-bound vehicles) for a
time interval T and then, simultaneously, all the signals turn red for the east-
bound vehicles (and green for the north-bound vehicles) for the next T time
steps before turning green again. This process is repeated so that there is a
total time interval 2T between the beginning of two successive green (or red)
phases of the signals.
As in the NaSch model the speed v of each vehicle can take one of
the vmax + 1 integer values v = 0, 1, ..., vmax. Suppose, vn is the speed
of the n-th vehicle at time t while moving either towards east or towards
north. In the initial state of the system, Nx (Ny) vehicles are distributed
among the east-bound (north-bound) streets. Here we only consider the case
Nx = Ny = Nv/2 where Nv is the total number of vehicles. Since in the ini-
tial configuration the occupation of a crossing is strictly avoided, the global
density is defined by ρ = Nv/N
2(2D−1). Also, suppose dn is the distance to
the next vehicle in front while sn denotes the distance to the nearest crossing
in front of it. At each discrete time step t→ t+1, the arrangement of vehicles
is updated in parallel according to the following ”rules”:
– Step 1 (Acceleration):
vn → min(vn + 1, vmax)
– Step 2 (Deceleration due to other vehicles or signals):
Case I: The signal is red for the n-th vehicle under consideration:
vn → min(vn, dn − 1, sn − 1)
Fig. 1. Structure of the underlying lattice for N = 4 and D = 10. Shown is
a typical jammed configuration of the vehicles. The east-bound and north-
bound vehicles are represented by the blue and black symbols, respectively.
Case II: The signal is green for the n-th vehicle under consideration:
If the signal is going to turn to red in the next timestep then
vn → min(vn, dn − 1, sn − 1)
else vn → min(vn, dn − 1).
– Step 3 (Randomization):
vn → max(vn − 1, 0) with probability p
– Step 4 (Movement):
xn → xn + vn.
Note that we have simplified Case II of Step 2 in comparison to [4]. This
simplification does not change the overall behaviour of the model [5].
These rules are not merely a combination of the BML and the NaSch
rules but also involve some modifications. For example, unlike all the earlier
BML-type models, a vehicle approaching a crossing can keep moving, even
when the signal is red, until it reaches a site immediately in front of which
there is either a halting vehicle or a crossing. Moreover, if p = 0 every east-
bound (north-bound) vehicle can adjust speed in the deceleration stage so as
not to block the north-bound (east-bound) traffic when the signal is red for
the east-bound (north-bound) vehicles.
3 Results
The variations of 〈vx〉 and 〈vy〉 with time (see Fig. 2) as well as with c, D,
T and p in the flowing phase are certainly more realistic than in the BML
model [4]. In the case of ρ < ρc and for vmax = 1 the dynamics of the system
can be described accurately by treating a single street with an improved
version of the 2-cluster approximation [6], where the 2-cluster probabilities
are equipped with a time and space dependence [5] (see Fig. 3) .
The fundamental diagram (Fig. 4) also shows a rather complex behaviour,
at least for finite systems. E.g. the density corresponding to the maximum flux
shifts to smaller densities with the decrease of T . Furthermore, the maximum
throughput is a non-monotonic function of T in the ”free-flowing” phase;
this result may be of practical use in traffic engineering for maximizing the
throughput.
A phase transition from the ”free-flowing” dynamical phase to the com-
pletely ”jammed” phase takes place in this model at a vehicle density ρc. The
intrinsic stochasticity of the dynamics, which triggers the onset of jamming,
is similar to that in the NaSch model, while the phenomenon of complete
jamming through self-organization as well as the final jammed configurations
(see Fig. 1) are similar to those in the BML model.
Due to the importance of finite-size and finite-time corrections it is not
clear up to now how the critical density ρc(D) depends on the dynamical
parameters vmax, p and T . It is possible that in the thermodynamic limit
N → ∞ the density ρc is completely determined by the structure of the
underlying lattice, i.e. by D, as long as p > 0 which is necessary for the
jamming transition to occur. In that case ρc would be independent of vmax,
p and T and the transition would be of ’geometrical’ nature similar to the
percolation transition. On the other hand, the transition could also be truly
dynamical with ρc depending also on vmax, p or T . The data obtained so far
from the computer simulations (see Fig. 5) do not conclusively rule out either
of these two possible scenarios.
The ”unified” model has been formulated intentionally to keep it as simple
as possible and at the same time capture some of the interesting features of
the NaSch model as well as the BML model. We believe that this model can
be generalized (i) to allow traffic flow in both ways on each street which may
consist of more than one lane, (ii) to make more realistic rules for the right-
of-the-way at the crossings and turning of the vehicles, (iii) to implement
different types of synchronization or staggering of traffic lights, e.g. green-
waves [7].
Acknowledgment
Part of this work has been supported by the SFB 341 (Ko¨ln-Aachen-Ju¨lich).
References
1. K. Nagel and M. Schreckenberg: J. Physique I, 2, 2221 (1992)
2. O. Biham, A.A. Middleton and D. Levine: Phys. Rev. A 46, R6124 (1992)
3. D. Chowdhury, L. Santen and A. Schadschneider: to be published in Phys. Rep.
4. D. Chowdhury and A. Schadschneider: Phys. Rev. E 59, R1311 (1999)
5. D. Chowdhury, K. Klauck, L. Santen, A. Schadschneider and J. Zittartz: to be
published
6. A. Schadschneider, in M. Schreckenberg and D.E. Wolf (Eds.): Traffic and
Granular Flow ’97. Springer, Singapore (1998)
7. P.M. Simon and K. Nagel: Phys. Rev. E 58, 1286 (1998)
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5
0 50 100 150 200
Av
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Time
Fig. 2. Time-dependence of average speeds of vehicles. The symbols +, ×,
∗ and 2 correspond, respectively, to the average speeds 〈vx〉, 〈vy〉, and the
fractions of vehicles with instantaneous speed V = 0, fx0 and fy0, respec-
tively. The common parameters are vmax = 5, p = 0.1, D = 100, T = 100
and c = 0.1. The continuous line has been obtained from heuristic arguments
given in [4].
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Fig. 3. Comparison between MC data and 2-cluster results. The common
paramters are vmax = 1, D = 25, T = 100 and N = 4. The solid lines
correspond to 2-cluster results. The symbols +,×, ∗,⊡ correspond to the ρ/p
MC data sets 0.05/0.1, 0.25/0.1, 0.05/0.5, 0.25/0.5 respectively.
00.1
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Fig. 4. Fundamental diagram for vmax = 5, p = 0.5, L = 100, and
D = 20. The symbols ⋄,+,,× and △ correspond, respectively, to T =
100, 50, 20, 10, 4.
Fig. 5. Critical density ρc for different parameter combinations as function
of the number of streets N . The common parameters are D = 20 and T = 5.


A new cellular automata model for city traffic

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