An information-based traffic control in a public conveyance system:
reduced clustering and enhanced efficiency
Akiyasu Tomoeda and Katsuhiro Nishinari
Department of Aeronautics and Astronautics, Faculty of Engineering,
University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan.
Debashish Chowdhury
Department of Physics, Indian Institute of Technology, Kanpur 208016, India.
Andreas Schadschneider
Institut für Theoretische Physik, Universität zu Köln D-50937 Köln, Germany
(Dated: October 31, 2018)
A new public conveyance model applicable to buses and trains is proposed in this paper by
using stochastic cellular automaton. We have found the optimal density of vehicles, at which the
average velocity becomes maximum, significantly depends on the number of stops and passengers
behavior of getting on a vehicle at stops. The efficiency of the hail-and-ride system is also discussed
by comparing the different behavior of passengers. Moreover, we have found that a big cluster
of vehicles is divided into small clusters, by incorporating information of the number of vehicles
between successive stops.
I. INTRODUCTION
The totally asymmetric simple exclusion process [1, 2,
3] is the simplest model of non-equilibrium systems of in-
teracting self-driven particles. Various extensions of this
model have been reported in the last few years for captur-
ing the essential features of the collective spatio-temporal
organizations in wide varieties of systems, including those
in vehicular traffic [4, 5, 6, 7, 8]. Traffic of buses and bicy-
cles have also been modeled following similar approaches
[9, 10]. A simple bus route model [10] exhibits clustering
of the buses along the route and the quantitative features
of the coarsening of the clusters have strong similarities
with coarsening phenomena in many other physical sys-
tems. Under normal circumstances, such clustering of
buses is undesirable in any real bus route as the effi-
ciency of the transport system is adversely affected by
clustering. The main aim of this paper is to introduce
a traffic control system into the bus route model in such
a way that helps in suppressing this tendency of cluster-
ing of the buses. This new model exhibits a competition
between the two opposing tendencies of clustering and
de-clustering which is interesting from the point of view
of fundamental physical principles. However, the model
may also find application in developing adaptive traffic
control systems for public conveyance systems.
In some of earlier bus-route models, movement of the
buses was monitored on coarse time intervals so that the
details of the dynamics of the buses in between two suc-
cessive bus stops was not described explicitly. Instead,
the movement of the bus from one stop to the next was
captured only through probabilities of hopping from one
stop to the next; hopping takes place with the lower prob-
ability if passengers are waiting at the approaching bus
stop [10]. An alternative interpretation of the model is
as follows: the passengers could board the bus whenever
and wherever they stopped a bus by raising their hand,
this is called the hail-and-ride system.
Several possible extensions of the bus route model have
been reported in the past [11, 12, 13]. For example, in
[11], in order to elucidate the connection between the
bus route model with parallel updating and the Nagel-
Schreckenberg model, two alternative extensions of the
latter model with space-/time-dependent hopping rates
are proposed. If a bus does not stop at a bus stop, the
waiting passengers have to wait further for the next bus;
such scenarios were captured in one of the earlier bus
route models [12], using modified car-following model.
In [13], the bus capacity, as well as the number of pas-
sengers getting on and off at each stop, were introduced
to make the model more realistic. Interestingly, it has
been claimed that the distribution of the time gaps be-
tween the arrival of successive buses is described well by
the Gaussian Unitary Ensemble of random matrices [14].
In this paper, by extending the model in [10], we sug-
gest a new public conveyance model (PCM). Although
we refer to each of the public vehicles in this model as
a “bus”, the model is equally applicable to train traffic
on a given route. In this PCM we can set up arbitrary
number of bus stops on the given route. The hail-and-
ride system turns out to be a special case of the general
PCM. Moreover, in the PCM the duration of the halt
of a bus at any arbitrary bus stop depends on the num-
ber of waiting passengers. As we shall demonstrate in
this paper, the delay in the departure of the buses from
crowded bus stops leads to the tendency of the buses to
cluster on the route. Furthermore, in the PCM, we also
introduce a traffic control system that exploits the in-
formation on the number of buses in the “segments” in
between successive bus stops; this traffic control system
helps in reducing the undesirable tendency of clustering
by dispersing the buses more or less uniformly along the
http://arxiv.org/abs/0704.1555v2
route.
In this study we introduce two different quantitative
measures of the efficiency of the bus transport system,
and calculate these quantities, both numerically and an-
alytically, to determine the conditions under which the
system would operate optimally.
This paper is organized as follows, in Sec. 2 PCM is in-
troduced and we show several simulation results in Sec. 3.
The average speed and the number of waiting passengers
are studied by mean field analysis in Sec. 4, and conclu-
sions are given in Sec. 5.
II. A STOCHASTIC CA MODEL FOR PUBLIC
CONVEYANCE
In this section, we explain the PCM in detail. For the
sake of simplicity, we impose periodic boundary condi-
tions. Let us imagine that the road is partitioned into
L identical cells such that each cell can accommodate
at most one bus at a time. Moreover, a total of S
(0 ≤ S ≤ L) equispaced cells are identified in the begin-
ning as bus stops. Note that, the special case S = L cor-
responds to the hail-and-ride system. At any given time
step, a passenger arrives with probability f to the sys-
tem. Here, we assume that a given passenger is equally
likely to arrive at any one of the bus stops with a proba-
bility 1/S. Thus, the average number of passengers that
arrive at each bus stop per unit time is given by f/S.
In contrast to this model, in ref. [15, 16] the passengers
were assumed to arrive with probability f at all the bus
stops in every time step.
Model AModel A
Model BModel B
BUS BUS BUS
BUSBUS BUS
)0( =iNQ)0( =iNQ 1H 2H
)( qQ >
1 2( )Q H H> >
FIG. 1: Schematic illustration of the PCM. In the model A,
the hopping probability to the bus stop does not depend on
the number of waiting passengers. In contrast, in the model
B the hopping probability to the bus stop depends on the
number of waiting passengers. Thus if the waiting passengers
increase, the hopping probability to the bus stop is decreased.
The model A corresponds to those situations where,
because of sufficiently large number of broad doors, the
time interval during which the doors of the bus remain
open after halting at a stop, is independent of the size
of waiting crowd of passengers. In contrast, the model
B captures those situations where a bus has to halt for
a longer period to pick up a larger crowd of waiting pas-
sengers.
The symbol H is used to denote the hopping probabil-
ity of a bus entering into a cell that has been designated
as a bus stop. We consider two different forms of H in
the two versions of our model which are named as model
A and model B. In the model A we assume the form
Q no waiting passengers
q waiting passengers exist
where both Q and q (Q > q) are constants independent
of the number of waiting passengers. The form (1) was
used in the original formulation of the bus route model
by O’Loan et al. [10].
In contrast to most of all the earlier bus route models,
we assume in the model B that the maximum number
of passengers that can get into one bus at a bus stop is
Nmax. Suppose, Ni denotes the number of passengers
waiting at the bus stop i (i = 1, · · · , S) at the instant of
time when a bus arrives there. In contrast to the form
(1) for H in model A, we assume in model B the form
min(Ni, Nmax) + 1
where min(Ni, Nmax) is the number of passengers who
can get into a bus which arrives at the bus stop i at
the instant of time when the number of passengers wait-
ing there is Ni. The form (2) is motivated by the com-
mon expectation that the time needed for the passengers
boarding a bus is proportional to their number. FIG. 1
depicts the hopping probabilities in the two models A
and B schematically.
The hopping probability of a bus to the cells that are
not designated as bus stops is Q; this is already captured
by the expressions (1) and (2) since no passenger ever
waits at those locations.
In principle, the hopping probability H for a real bus
would depend also on the number of passengers who get
off at the bus stop; in the extreme situations where no
passenger waits at a bus stop the hopping probability H
would be solely decided by the disembarking passengers.
However, in order to keep the model theoretically simple
and tractable, we ignore the latter situation and assume
that passengers get off only at those stops where waiting
passengers get into the bus and that the time taken by the
waiting passengers to get into the bus is always adequate
for the disembarking passengers to get off the bus.
Note that Nmax is the maximum boarding capacity at
each bus stop rather than themaximum carrying capacity
of each bus. The PCM model reported here can be easily
extended to incorporate an additional dynamical variable
associated with each bus to account for the instantaneous
number of passengers in it. But, for the sake of simplic-
ity, such an extension of the model is not reported here.
Instead, in the simple version of the PCMmodel reported
here, Nmax can be interpreted as the maximum carrying
capacity of each bus if we assume that all of the passen-
gers on the bus get off whenever it stops.
The model is updated according to the following rules.
In step 2 − 4, these rules are applied in parallel to all
buses and passengers, respectively:
1. Arrival of a passenger
A bus stop i (i = 1, · · · , S) is picked up randomly,
with probability 1/S, and then the corresponding
number of waiting passengers in increased by unity,
i.e. Ni → Ni+1, with probability f to account for
the arrival of a passenger at the selected bus stop.
2. Bus motion
If the cell in front of a bus is not occupied by an-
other bus, each bus hops to the next cell with the
probability H . Specifically, if passengers do not ex-
ist in the next cell in both model A and model B
hopping probability equals to Q because Ni equals
to 0. Else, if passengers exist in the next cell,
the hopping probability equals to q in the model
A, whereas in the model B the corresponding hop-
ping probability equals to Q/(min(Ni, Nmax) + 1).
Note that, when a bus is loaded with passengers to
its maximum boarding capacity Nmax, the hopping
probability in the model B equals to Q/(Nmax+1),
the smallest allowed hopping probability.
3. Boarding a bus
When a bus arrives at the i-th (i = 1, · · · , S) bus
stop cell, the corresponding number Ni of waiting
passengers is updated to max(Ni −Nmax, 0) to ac-
count for the passengers boarding the bus. Once
the door is closed, no more waiting passenger can
get into the bus at the same bus stop although the
bus may remain stranded at the same stop for a
longer period of time either because of the unavail-
ability of the next bus stop or because of the traffic
control rule explained next.
4. Bus information update
Every bus stop has information Ij (j = 1, · · · , S)
which is the number of buses in the segment of the
route between the stop j and the next stop j+1 at
that instant of time. This information is updated at
each time steps. When one bus leaves the j-th bus
stop, Ij is increased to Ij + 1. On the other hand,
when a bus leaves (j+1)-th bus stop, Ij is reduced
to Ij − 1. The desirable value of Ij is I0 = m/S,
where m is the total number of buses, for all j so
that buses are not clustered in any segment of the
route. We implement a traffic control rule based
on the information Ij : a bus remains stranded at a
stop j as long as Ij exceeds I0.
We use the average speed 〈V 〉 of the buses and the
number of the waiting passengers 〈N〉 at a bus stop as
two quantitative measures of the efficiency of the pub-
lic conveyance system under consideration; a higher 〈V 〉
and smaller 〈N〉 correspond to an efficient transportation
system.
III. COMPUTER SIMULATIONS OF PCM
In the simulations we set L = 500, Q = 0.9, q = 0.5
and Nmax = 60. The main parameters of this model,
which we varied, are the number of buses (m), the num-
ber of bus stops (S) and the probability (f) of arrival of
passengers. The number density of buses is defined by
ρ = m/L.
SPACE
SPACE
Initial stage late stage
FIG. 2: Space-time plots in the model B for the parameter
values f = 0.6, S = 5, m = 30. The upper two figures cor-
respond to the case where no traffic control system based on
the information {I} is operational. The upper left figure cor-
responds to the initial stage (from t = 1000 to t = 1500)
whereas the upper right plot corresponds to the late stages
(from t = 4000 to t = 4500). The lower figures correspond to
the case where the information ({I}) based bus-traffic control
system is operational (left figure shows data from t = 1000
to t = 1500 while the right figure corresponds to t = 4000
to t = 4500). Clearly, information-based traffic control sys-
tem disperses the buses which, in the absence of this control
system, would have a tendency to cluster.
Typical space-time plots of the model B are given in
FIG. 2. If no information-based traffic control system
exits, the buses have a tendency to cluster; this phe-
nomenon is very simular to that observed in the ant-
trail model [15, 16]. However, implementation of the
information-based traffic control system restricts the size
of such clusters to a maximum of I0 buses in a segment
of the route in between two successive bus stops. We
study the effects of this control system below by compar-
ing the characteristics of two traffic systems one of which
includes the information-based control system while the
other does not.
A. PCM without information-based traffic control
In the FIG. 3 - FIG. 8, we plot 〈V 〉 and 〈N〉 against
the density of buses for several different values of f . Note
that, the FIG. 5 and FIG. 8 corresponds to the hail-and-
ride system for models A and B, respectively.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 3: The average speed and the average number of waiting
passengers in the model A are plotted against the density for
the parameters S = 5 and f = 0.3, 0.6 and 0.9.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
rs f = 0.3
f = 0.6
f = 0.9
FIG. 4: The plot of 〈V 〉 and 〈N〉 of the model A for S = 50
and f = 0.3, 0.6 and 0.9.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 5: The plot of 〈V 〉 and 〈N〉 of the model A for S =
500(= L) and f = 0.3, 0.6 and 0.9.
These figures demonstrate that the average speed 〈V 〉,
which is a measure of the efficiency of the bus traffic
system, exhibits a maximum at around ρ = 0.2 ∼ 0.3
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 6: The plot of 〈V 〉 and 〈N〉 of the model B for S = 5
and f = 0.3, 0.6 and 0.9.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 7: The plot of 〈V 〉 and 〈N〉 of the model B for S = 50
and f = 0.3, 0.6 and 0.9.
especially in the model B (comparing FIG. 3 with FIG. 6,
it shows the model B (FIG. 6) reflects the bus bunching
more clearly than the model A (FIG. 3) especially at large
f and small ρ). The average number of waiting passengers
〈N〉, whose inverse is another measure of the efficiency of
the bus traffic system, is vanishingly small in the region
0.3 < ρ < 0.7; 〈N〉 increases with decreasing (increasing)
ρ in the regime ρ < 0.3 (ρ > 0.7).
The average velocity of the model A becomes smaller as
S increases in the low density region (see FIG. 3, FIG. 4
and FIG. 5). In contrast, in the model B (FIG. 7 and
FIG. 8) we observe that there is no significant differ-
ence in the average velocity. Note that the number of
waiting passengers is calculated by (total waiting pas-
sengers)/(number of bus stops). The total number of
waiting passengers in this system is almost the same un-
der the case S = 50 and hail-and-ride system S = L in
both models. When the parameter S is small (comparing
FIG. 3 and FIG. 6), in the model B the waiting passen-
gers are larger and the average velocity is smaller than
in the model A, since the effect of the delay in getting on
a bus is taken into account. In the model B (comparing
FIG. 6, FIG. 7 and FIG. 8), the case S = 50 is more effi-
cient than S = 5, i.e. the system is likely to become more
efficient, as S increases. However, we do not find any sig-
nificant variation between S = 50 and S = 500. When S
is small, the system becomes more efficient by increasing
the number of bus stops. If the number of bus stops in-
crease beyond 50, then there is little further variation of
the efficiency as S is increased up to the maximum value
From FIG. 9, the distribution of 〈N〉 over all the bus
stops in the system is shown. We see that the distribution
does not show the Zipf’s law, which is sometimes seen in
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 8: The plot of 〈V 〉 and 〈N〉 of the model B for S =
500(= L) and f = 0.3, 0.6 and 0.9.
0 10 20 30 40 50
Ranking
FIG. 9: The distribution of waiting passengers is plotted
against all bus stops for the parameters f = 0.6, B = 50,
S = 50. The horizontal line means the ranking, where we
arrange the bus stops according to the descending order of
natural and social phenomena; frequency of used words
[17], population of a city [18], the number of the access to
a web site [19], and intervals between successive transit
times of the cars of traffic flow [20].
Next, we investigate the optimal density of buses at
which the average velocity becomes maximum. The op-
timal density depends on Q and is ρ = 0.3 for Q = 0.8
(FIG. 10, see also FIG. 11). In FIG. 10, it is shown
that the density corresponding to the maximum veloc-
ity shifts to higher values as Q becomes larger. FIG. 11
shows the optimal density of buses in the model B with-
out information-based control system. From this figure,
we find that the optimal density, for case S = 50, is
smaller than that for S = 5. Moreover, for given S, the
optimal density decreases with decreasing f . However,
for both S = 5 and S = 50, the optimal density corre-
sponding to Q = 1.0 is higher for f = 0.6 than that for
f = 0.9.
What is more effective way of increasing the efficiency
of the public conveyance system on a given route by in-
creasing the number of buses without increasing the car-
rying capacity of each bus, or by increasing the carrying
capacity of each bus without recruiting more buses? Or,
are these two prescriptions for enhancing efficiency of the
public conveyance system equally effective? In order to
address these questions, we make a comparative study
of two situations on the same route: for example, in the
0 0.2 0.4 0.6 0.8 1
Density
Q = 0.8
Q = 1.0
0 0.2 0.4 0.6 0.8 1
Density
rs Q = 0.8
Q = 1.0
FIG. 10: The average speed and the average number of wait-
ing passengers in the model B are plotted against the density
for the parameters f = 0.9, S = 50; the hopping parameters
are Q = 0.8 and Q = 1.0.
0.6 0.7 0.8 0.9 1
f = 0.9, S = 5
f = 0.6, S = 5
f = 0.9, S = 50
f = 0.6, S = 50
FIG. 11: The optimal density of buses in the model B is
plotted against Q. The parameters are f = 0.9, S = 5 (normal
line),f = 0.6, S = 5 (finer broken line), f = 0.9, S = 50 (bold
broken line), f = 0.6, S = 50 (longer broken line).
first situation the number of buses is 10 and each has a
capacity of 60, whereas in the second the number of buses
is 5 and each has a capacity of 120. Note that the total
carrying capacity of all the buses together is 600 (60×10
and 120× 5 in the two situations), i.e., same in both the
situations. But, the number density of the buses in the
second situation is just half of that in the first as the
length of the bus route is same in both the situations. In
FIG. 12, the results for these two cases are plotted; the
different scales of density used along the X-axis arises
from the differences in the number densities mentioned
above.
From FIG. 12, we conclude that, at sufficiently low
densities, the average velocity is higher for Nmax = 60
compared to those for Nmax = 120. But, in the same
regime of the number density of buses, larger number of
passengers wait at bus stops when the bus capacity is
smaller. Thus, in the region ρ < 0.05, system adminis-
trators face a dilemma: if they give priority to the aver-
age velocity and decide to choose buses with Nmax = 60,
the number of passengers waiting at the bus stops in-
creases. On the other hand if they decide to make the
passengers happy by reducing their waiting time at the
bus stops and, therefore, choose buses with Nmax = 120,
the travel time of the passengers after boarding a bus
becomes longer.
However, at densities ρ > 0.05, the system administra-
Density
(0.05) (0.1) (0.15) (0.2) (0.25)
Density
(0.05) (0.1) (0.15) (0.2) (0.25)
Density
(0.05) (0.1) (0.15) (0.2) (0.25)
Density
(0.05) (0.1) (0.15) (0.2) (0.25)
FIG. 12: Comparison between the case of bus capacity 60
with bus capacity 120. The parameters are Q = 0.9, S = 10,
f = 0.6 in the model B without information. The top figure
shows the average velocity, the center figure shows waiting
passengers and the bottom figure shows the number of con-
veyed passengers per unit bus, i.e. this number is calculated
by (total number of on-boarding passengers on all buses)/(the
number of buses), against the bus density up to 0.5. In each
figure, the horizontal axis shows the density; the numbers
without parentheses denote the number densities in the case
Nmax = 60, whereas the numbers in the parentheses denote
the number densities in the case Nmax = 120.
tors can satisfy both the criteria, namely, fewer wait-
ing passengers and shorter travel times, by one sin-
gle choice. In this region of density, the public con-
veyance system with Nmax = 60 is more efficient than
that with Nmax = 120 because the average velocity is
higher and the number of waiting passengers is smaller
for Nmax = 60 than for Nmax = 120. Thus, in this regime
of bus density, efficiency of the system is enhanced by re-
ducing the capacity of individual buses and increasing
their number on the same bus route.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
rs f = 0.3
f = 0.6
f = 0.9
FIG. 13: The plot of 〈V 〉 and 〈N〉 of the model B with infor-
mation (S = 5 and f = 0.3, 0.6 and 0.9)
0 0.2 0.4 0.6 0.8 1
Density
without info
with info
0 0.2 0.4 0.6 0.8 1
Density
rs without info
with info
FIG. 14: The model B with S = 5 and f = 0.9. The left
vertical dash line is ρ = 0.28 and the right is ρ = 0.73 in the
two figures.
B. PCM with information-based traffic control
The results for the PCM with information-based traf-
fic control system is shown in FIG. 13 and FIG. 14. In
the FIG. 13 we plot 〈V 〉 and 〈N〉 against the density of
buses for the parameter S = 5. The density correspond-
ing to the peak of the average velocity shifts to lower
values when the information-based traffic control system
is switched on.
The data shown in FIG. 14 establish that implemen-
tation of the information-based traffic control system
does not necessarily always improve the efficiency of
the public conveyance system. In fact, in the region
0.3 < ρ < 0.7, the average velocity of the buses is
higher if the information-based control system is switched
off. Comparing 〈V 〉 and 〈N〉 in FIG. 14, we find that
information-based traffic control system can improves the
efficiency by reducing the crowd of waiting passengers.
But, in the absence of waiting passengers, introduction
of the information-based control system adversely affects
the efficiency of the public conveyance system by holding
up the buses at bus stops when the number of buses in
the next segment of the route exceeds I0.
0 10 20 30 40 50
Ranking
without information
with information
FIG. 15: Distribution of headway distance for S = 10, m =
50, f = 0.9 in model B. This figure shows the plot of headway
distance against the ranking.
Finally, FIG. 15 shows the distribution of headway dis-
tance against the ranking, where we arrange the order of
magnitude according to the headway distance of buses
in descending order. From this figure it is found that
the headway distribution is dispersed by the effect of
the information. The average headway distance with the
information-based traffic control is equal to 8.34, in con-
trast to a much shorter value of 0.66 when that control
system is switched off. Thus we confirm that the avail-
ability of the information Ij and implementation of the
traffic control system based on this information, signifi-
cantly reduces the undesirable clustering of buses.
IV. MEAN FIELD ANALYSIS
Let us estimate 〈V 〉 theoretically in the low density
limit ρ → 0. Suppose, T is the average time taken by a
bus to complete one circuit of the route. In the model
A, the number of hops made by a bus with probability
q during the time T is S, i.e. the total number of bus
stops. Therefore the average period T for a bus in the
model A is well approximated by
and hence,
〈V 〉 =
q(L− S) +QS
. (4)
In model B, in the low density limit where m buses run
practically unhindered and are distributed uniformly in
the system without correlations, the average number of
passengersN waiting at a bus stop, just before the arrival
of the next bus, is
. (5)
The first factor f/S on the right hand side of the equation
(5) is the probability of arrival of passengers per unit
time. The second factor on the right hand side of (5)
is an estimate of the average time taken by a bus to
traverse one segment of the route, i.e. the part of the
route between successive bus stops. The last factor in the
same equation is the average number of segments of the
route in between two successive buses on the same route.
Instead of the constant q used in (4) for the evaluation
of 〈V 〉 in the model A, we use
N + 1
in eq. (4) and eq. (5) for the model B. Then, for the
model B, the hopping probability Q is estimated self-
consistently solving
〈V 〉 = Q−
, (7)
(4) and (6) simultaneously.
We also obtain, for the model B, the average number
of passengers 〈N〉 waiting at a bus stop in the ρ → 0
limit. The average time for moving from one bus stop to
the next is ∆t = (L/S − 1)/Q + 1/q̄ and, therefore, we
〈N〉 = (f/S) · (∆t+ 2∆t+ · · ·+ (S − 1)∆t)/S
f(S − 1)(q̄(L− S) + SQ)
2S2Qq̄
. (8)
As long as the number of waiting passengers does not
exceed Nmax, we have observed reasonably good agree-
ment between the analytical estimates (4), (8) and the
corresponding numerical data obtained from computer
simulations. For example, in the model A, we get the es-
timates 〈V 〉 = 0.85 and 〈N〉 = 1.71 from the approximate
mean field theory for the parameter set S = 50, m = 1,
Q = 0.9, q = 0.5, f = 0.3. The corresponding numbers
obtained from direct computer simulations of the model
A version of PCM are 0.84 and 1.78, respectively. Simi-
larly, in the model B under the same conditions, we get
〈V 〉 = 0.60 and 〈N〉 = 2.45 from the mean field theory,
while the corresponding numerical values are 0.60 and
2.51, respectively. If we take sufficiently small f ’s, then
the mean-field estimates agree almost perfectly with the
corresponding simulation data. However, our mean field
analysis breaks down when a bus can not pick up all the
passengers waiting at a bus stop.
V. CONCLUDING DISCUSSIONS
In this paper, we have proposed a public conveyance
model (PCM) by using stochastic CA. In our PCM, some
realistic elements are introduced: e.g., the carrying ca-
pacity of a bus, the arbitrary number of bus stops, the
halt time of a bus that depends on the number of waiting
passengers, and an information-based bus traffic control
system which reduces clustering of the buses on the given
route.
We have obtained quantitative results by using both
computer simulations and analytical calculations. In par-
ticular, we have introduced two different quantitative
measures of the efficiency of the public conveyance sys-
tem. We have found that the bus system works efficiently
in a region of moderate number density of buses; too
many or too few buses drastically reduce the efficiency
of the bus-transport system. If the density of the buses
is lower than optimal, not only large number of passen-
gers are kept waiting at the stops for longer duration, but
also the passengers in the buses get a slow ride as buses
run slowly because they are slowed down at each stop to
pick up the waiting passengers. On the other hand, if
the density of the buses is higher than optimal, the mu-
tual hindrance created by the buses in the overcrowded
route also lowers the efficiency of the transport system.
Moreover, we have found that the average velocity in-
creases, and the number of waiting passengers decreases,
when the information-based bus traffic control system is
switched on. However, this enhancement of efficiency of
the conveyance system takes place only over a particular
range of density; the information-based bus traffic con-
trol system does not necessarily improve the efficiency of
the system in all possible situations.
We have compared two situations where the second
situation is obtained from the first one by doubling the
carrying capacity of each bus and reducing their number
to half the original number on the same route. In the
density region ρ > 0.05 the system of Nmax = 60 is more
efficient than that with Nmax = 120. However, at small
densities (ρ < 0.05), although the average velocity in-
creases, the number of waiting passengers also increases,
by doubling the carrying capacity from Nmax = 60 to
Nmax = 120. Hence, bus-transport system administra-
tors would face a dilemma in this region of small density.
Finally, in our PCM, the effect of the disembarking
passengers on the halt time of the buses has not been
captured explicitly. Moreover, this study is restricted to
periodic boundary conditions. The clustering of particles
occurs not only in a ring-like bus route, but also in shuttle
services of buses and trains. Thus it would be interesting
to investigate the effects of the information-based traffic
control system also on such public transport systems. In
a future work, we intend to report the results of our in-
vestigations of the model under non-periodic boundary
conditions. We hope our model will help in understand-
ing the mechanism of congestion in public conveyance
system and will provide insight as to the possible ways
to reduce undesirable clustering of the vehicles.
Acknowledgments: Work of one of the authors (DC)
has been supported, in part, by the Council of Scientific
and Industrial Research (CSIR), government of India.
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  A new public conveyance model applicable to buses and trains is proposed in
this paper by using stochastic cellular automaton. We have found the optimal
density of vehicles, at which the average velocity becomes maximum,
significantly depends on the number of stops and passengers behavior of getting
on a vehicle at stops. The efficiency of the hail-and-ride system is also
discussed by comparing the different behavior of passengers. Moreover, we have
found that a big cluster of vehicles is divided into small clusters, by
incorporating information of the number of vehicles between successive stops.

An information-based traffic control in a public conveyance system:
reduced clustering and enhanced efficiency
Akiyasu Tomoeda and Katsuhiro Nishinari
Department of Aeronautics and Astronautics, Faculty of Engineering,
University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan.
Debashish Chowdhury
Department of Physics, Indian Institute of Technology, Kanpur 208016, India.
Andreas Schadschneider
Institut für Theoretische Physik, Universität zu Köln D-50937 Köln, Germany
(Dated: October 31, 2018)
A new public conveyance model applicable to buses and trains is proposed in this paper by
using stochastic cellular automaton. We have found the optimal density of vehicles, at which the
average velocity becomes maximum, significantly depends on the number of stops and passengers
behavior of getting on a vehicle at stops. The efficiency of the hail-and-ride system is also discussed
by comparing the different behavior of passengers. Moreover, we have found that a big cluster
of vehicles is divided into small clusters, by incorporating information of the number of vehicles
between successive stops.
I. INTRODUCTION
The totally asymmetric simple exclusion process [1, 2,
3] is the simplest model of non-equilibrium systems of in-
teracting self-driven particles. Various extensions of this
model have been reported in the last few years for captur-
ing the essential features of the collective spatio-temporal
organizations in wide varieties of systems, including those
in vehicular traffic [4, 5, 6, 7, 8]. Traffic of buses and bicy-
cles have also been modeled following similar approaches
[9, 10]. A simple bus route model [10] exhibits clustering
of the buses along the route and the quantitative features
of the coarsening of the clusters have strong similarities
with coarsening phenomena in many other physical sys-
tems. Under normal circumstances, such clustering of
buses is undesirable in any real bus route as the effi-
ciency of the transport system is adversely affected by
clustering. The main aim of this paper is to introduce
a traffic control system into the bus route model in such
a way that helps in suppressing this tendency of cluster-
ing of the buses. This new model exhibits a competition
between the two opposing tendencies of clustering and
de-clustering which is interesting from the point of view
of fundamental physical principles. However, the model
may also find application in developing adaptive traffic
control systems for public conveyance systems.
In some of earlier bus-route models, movement of the
buses was monitored on coarse time intervals so that the
details of the dynamics of the buses in between two suc-
cessive bus stops was not described explicitly. Instead,
the movement of the bus from one stop to the next was
captured only through probabilities of hopping from one
stop to the next; hopping takes place with the lower prob-
ability if passengers are waiting at the approaching bus
stop [10]. An alternative interpretation of the model is
as follows: the passengers could board the bus whenever
and wherever they stopped a bus by raising their hand,
this is called the hail-and-ride system.
Several possible extensions of the bus route model have
been reported in the past [11, 12, 13]. For example, in
[11], in order to elucidate the connection between the
bus route model with parallel updating and the Nagel-
Schreckenberg model, two alternative extensions of the
latter model with space-/time-dependent hopping rates
are proposed. If a bus does not stop at a bus stop, the
waiting passengers have to wait further for the next bus;
such scenarios were captured in one of the earlier bus
route models [12], using modified car-following model.
In [13], the bus capacity, as well as the number of pas-
sengers getting on and off at each stop, were introduced
to make the model more realistic. Interestingly, it has
been claimed that the distribution of the time gaps be-
tween the arrival of successive buses is described well by
the Gaussian Unitary Ensemble of random matrices [14].
In this paper, by extending the model in [10], we sug-
gest a new public conveyance model (PCM). Although
we refer to each of the public vehicles in this model as
a “bus”, the model is equally applicable to train traffic
on a given route. In this PCM we can set up arbitrary
number of bus stops on the given route. The hail-and-
ride system turns out to be a special case of the general
PCM. Moreover, in the PCM the duration of the halt
of a bus at any arbitrary bus stop depends on the num-
ber of waiting passengers. As we shall demonstrate in
this paper, the delay in the departure of the buses from
crowded bus stops leads to the tendency of the buses to
cluster on the route. Furthermore, in the PCM, we also
introduce a traffic control system that exploits the in-
formation on the number of buses in the “segments” in
between successive bus stops; this traffic control system
helps in reducing the undesirable tendency of clustering
by dispersing the buses more or less uniformly along the
http://arxiv.org/abs/0704.1555v2
route.
In this study we introduce two different quantitative
measures of the efficiency of the bus transport system,
and calculate these quantities, both numerically and an-
alytically, to determine the conditions under which the
system would operate optimally.
This paper is organized as follows, in Sec. 2 PCM is in-
troduced and we show several simulation results in Sec. 3.
The average speed and the number of waiting passengers
are studied by mean field analysis in Sec. 4, and conclu-
sions are given in Sec. 5.
II. A STOCHASTIC CA MODEL FOR PUBLIC
CONVEYANCE
In this section, we explain the PCM in detail. For the
sake of simplicity, we impose periodic boundary condi-
tions. Let us imagine that the road is partitioned into
L identical cells such that each cell can accommodate
at most one bus at a time. Moreover, a total of S
(0 ≤ S ≤ L) equispaced cells are identified in the begin-
ning as bus stops. Note that, the special case S = L cor-
responds to the hail-and-ride system. At any given time
step, a passenger arrives with probability f to the sys-
tem. Here, we assume that a given passenger is equally
likely to arrive at any one of the bus stops with a proba-
bility 1/S. Thus, the average number of passengers that
arrive at each bus stop per unit time is given by f/S.
In contrast to this model, in ref. [15, 16] the passengers
were assumed to arrive with probability f at all the bus
stops in every time step.
Model AModel A
Model BModel B
BUS BUS BUS
BUSBUS BUS
)0( =iNQ)0( =iNQ 1H 2H
)( qQ >
1 2( )Q H H> >
FIG. 1: Schematic illustration of the PCM. In the model A,
the hopping probability to the bus stop does not depend on
the number of waiting passengers. In contrast, in the model
B the hopping probability to the bus stop depends on the
number of waiting passengers. Thus if the waiting passengers
increase, the hopping probability to the bus stop is decreased.
The model A corresponds to those situations where,
because of sufficiently large number of broad doors, the
time interval during which the doors of the bus remain
open after halting at a stop, is independent of the size
of waiting crowd of passengers. In contrast, the model
B captures those situations where a bus has to halt for
a longer period to pick up a larger crowd of waiting pas-
sengers.
The symbol H is used to denote the hopping probabil-
ity of a bus entering into a cell that has been designated
as a bus stop. We consider two different forms of H in
the two versions of our model which are named as model
A and model B. In the model A we assume the form
Q no waiting passengers
q waiting passengers exist
where both Q and q (Q > q) are constants independent
of the number of waiting passengers. The form (1) was
used in the original formulation of the bus route model
by O’Loan et al. [10].
In contrast to most of all the earlier bus route models,
we assume in the model B that the maximum number
of passengers that can get into one bus at a bus stop is
Nmax. Suppose, Ni denotes the number of passengers
waiting at the bus stop i (i = 1, · · · , S) at the instant of
time when a bus arrives there. In contrast to the form
(1) for H in model A, we assume in model B the form
min(Ni, Nmax) + 1
where min(Ni, Nmax) is the number of passengers who
can get into a bus which arrives at the bus stop i at
the instant of time when the number of passengers wait-
ing there is Ni. The form (2) is motivated by the com-
mon expectation that the time needed for the passengers
boarding a bus is proportional to their number. FIG. 1
depicts the hopping probabilities in the two models A
and B schematically.
The hopping probability of a bus to the cells that are
not designated as bus stops is Q; this is already captured
by the expressions (1) and (2) since no passenger ever
waits at those locations.
In principle, the hopping probability H for a real bus
would depend also on the number of passengers who get
off at the bus stop; in the extreme situations where no
passenger waits at a bus stop the hopping probability H
would be solely decided by the disembarking passengers.
However, in order to keep the model theoretically simple
and tractable, we ignore the latter situation and assume
that passengers get off only at those stops where waiting
passengers get into the bus and that the time taken by the
waiting passengers to get into the bus is always adequate
for the disembarking passengers to get off the bus.
Note that Nmax is the maximum boarding capacity at
each bus stop rather than themaximum carrying capacity
of each bus. The PCM model reported here can be easily
extended to incorporate an additional dynamical variable
associated with each bus to account for the instantaneous
number of passengers in it. But, for the sake of simplic-
ity, such an extension of the model is not reported here.
Instead, in the simple version of the PCMmodel reported
here, Nmax can be interpreted as the maximum carrying
capacity of each bus if we assume that all of the passen-
gers on the bus get off whenever it stops.
The model is updated according to the following rules.
In step 2 − 4, these rules are applied in parallel to all
buses and passengers, respectively:
1. Arrival of a passenger
A bus stop i (i = 1, · · · , S) is picked up randomly,
with probability 1/S, and then the corresponding
number of waiting passengers in increased by unity,
i.e. Ni → Ni+1, with probability f to account for
the arrival of a passenger at the selected bus stop.
2. Bus motion
If the cell in front of a bus is not occupied by an-
other bus, each bus hops to the next cell with the
probability H . Specifically, if passengers do not ex-
ist in the next cell in both model A and model B
hopping probability equals to Q because Ni equals
to 0. Else, if passengers exist in the next cell,
the hopping probability equals to q in the model
A, whereas in the model B the corresponding hop-
ping probability equals to Q/(min(Ni, Nmax) + 1).
Note that, when a bus is loaded with passengers to
its maximum boarding capacity Nmax, the hopping
probability in the model B equals to Q/(Nmax+1),
the smallest allowed hopping probability.
3. Boarding a bus
When a bus arrives at the i-th (i = 1, · · · , S) bus
stop cell, the corresponding number Ni of waiting
passengers is updated to max(Ni −Nmax, 0) to ac-
count for the passengers boarding the bus. Once
the door is closed, no more waiting passenger can
get into the bus at the same bus stop although the
bus may remain stranded at the same stop for a
longer period of time either because of the unavail-
ability of the next bus stop or because of the traffic
control rule explained next.
4. Bus information update
Every bus stop has information Ij (j = 1, · · · , S)
which is the number of buses in the segment of the
route between the stop j and the next stop j+1 at
that instant of time. This information is updated at
each time steps. When one bus leaves the j-th bus
stop, Ij is increased to Ij + 1. On the other hand,
when a bus leaves (j+1)-th bus stop, Ij is reduced
to Ij − 1. The desirable value of Ij is I0 = m/S,
where m is the total number of buses, for all j so
that buses are not clustered in any segment of the
route. We implement a traffic control rule based
on the information Ij : a bus remains stranded at a
stop j as long as Ij exceeds I0.
We use the average speed 〈V 〉 of the buses and the
number of the waiting passengers 〈N〉 at a bus stop as
two quantitative measures of the efficiency of the pub-
lic conveyance system under consideration; a higher 〈V 〉
and smaller 〈N〉 correspond to an efficient transportation
system.
III. COMPUTER SIMULATIONS OF PCM
In the simulations we set L = 500, Q = 0.9, q = 0.5
and Nmax = 60. The main parameters of this model,
which we varied, are the number of buses (m), the num-
ber of bus stops (S) and the probability (f) of arrival of
passengers. The number density of buses is defined by
ρ = m/L.
SPACE
SPACE
Initial stage late stage
FIG. 2: Space-time plots in the model B for the parameter
values f = 0.6, S = 5, m = 30. The upper two figures cor-
respond to the case where no traffic control system based on
the information {I} is operational. The upper left figure cor-
responds to the initial stage (from t = 1000 to t = 1500)
whereas the upper right plot corresponds to the late stages
(from t = 4000 to t = 4500). The lower figures correspond to
the case where the information ({I}) based bus-traffic control
system is operational (left figure shows data from t = 1000
to t = 1500 while the right figure corresponds to t = 4000
to t = 4500). Clearly, information-based traffic control sys-
tem disperses the buses which, in the absence of this control
system, would have a tendency to cluster.
Typical space-time plots of the model B are given in
FIG. 2. If no information-based traffic control system
exits, the buses have a tendency to cluster; this phe-
nomenon is very simular to that observed in the ant-
trail model [15, 16]. However, implementation of the
information-based traffic control system restricts the size
of such clusters to a maximum of I0 buses in a segment
of the route in between two successive bus stops. We
study the effects of this control system below by compar-
ing the characteristics of two traffic systems one of which
includes the information-based control system while the
other does not.
A. PCM without information-based traffic control
In the FIG. 3 - FIG. 8, we plot 〈V 〉 and 〈N〉 against
the density of buses for several different values of f . Note
that, the FIG. 5 and FIG. 8 corresponds to the hail-and-
ride system for models A and B, respectively.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 3: The average speed and the average number of waiting
passengers in the model A are plotted against the density for
the parameters S = 5 and f = 0.3, 0.6 and 0.9.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
rs f = 0.3
f = 0.6
f = 0.9
FIG. 4: The plot of 〈V 〉 and 〈N〉 of the model A for S = 50
and f = 0.3, 0.6 and 0.9.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 5: The plot of 〈V 〉 and 〈N〉 of the model A for S =
500(= L) and f = 0.3, 0.6 and 0.9.
These figures demonstrate that the average speed 〈V 〉,
which is a measure of the efficiency of the bus traffic
system, exhibits a maximum at around ρ = 0.2 ∼ 0.3
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 6: The plot of 〈V 〉 and 〈N〉 of the model B for S = 5
and f = 0.3, 0.6 and 0.9.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 7: The plot of 〈V 〉 and 〈N〉 of the model B for S = 50
and f = 0.3, 0.6 and 0.9.
especially in the model B (comparing FIG. 3 with FIG. 6,
it shows the model B (FIG. 6) reflects the bus bunching
more clearly than the model A (FIG. 3) especially at large
f and small ρ). The average number of waiting passengers
〈N〉, whose inverse is another measure of the efficiency of
the bus traffic system, is vanishingly small in the region
0.3 < ρ < 0.7; 〈N〉 increases with decreasing (increasing)
ρ in the regime ρ < 0.3 (ρ > 0.7).
The average velocity of the model A becomes smaller as
S increases in the low density region (see FIG. 3, FIG. 4
and FIG. 5). In contrast, in the model B (FIG. 7 and
FIG. 8) we observe that there is no significant differ-
ence in the average velocity. Note that the number of
waiting passengers is calculated by (total waiting pas-
sengers)/(number of bus stops). The total number of
waiting passengers in this system is almost the same un-
der the case S = 50 and hail-and-ride system S = L in
both models. When the parameter S is small (comparing
FIG. 3 and FIG. 6), in the model B the waiting passen-
gers are larger and the average velocity is smaller than
in the model A, since the effect of the delay in getting on
a bus is taken into account. In the model B (comparing
FIG. 6, FIG. 7 and FIG. 8), the case S = 50 is more effi-
cient than S = 5, i.e. the system is likely to become more
efficient, as S increases. However, we do not find any sig-
nificant variation between S = 50 and S = 500. When S
is small, the system becomes more efficient by increasing
the number of bus stops. If the number of bus stops in-
crease beyond 50, then there is little further variation of
the efficiency as S is increased up to the maximum value
From FIG. 9, the distribution of 〈N〉 over all the bus
stops in the system is shown. We see that the distribution
does not show the Zipf’s law, which is sometimes seen in
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
FIG. 8: The plot of 〈V 〉 and 〈N〉 of the model B for S =
500(= L) and f = 0.3, 0.6 and 0.9.
0 10 20 30 40 50
Ranking
FIG. 9: The distribution of waiting passengers is plotted
against all bus stops for the parameters f = 0.6, B = 50,
S = 50. The horizontal line means the ranking, where we
arrange the bus stops according to the descending order of
natural and social phenomena; frequency of used words
[17], population of a city [18], the number of the access to
a web site [19], and intervals between successive transit
times of the cars of traffic flow [20].
Next, we investigate the optimal density of buses at
which the average velocity becomes maximum. The op-
timal density depends on Q and is ρ = 0.3 for Q = 0.8
(FIG. 10, see also FIG. 11). In FIG. 10, it is shown
that the density corresponding to the maximum veloc-
ity shifts to higher values as Q becomes larger. FIG. 11
shows the optimal density of buses in the model B with-
out information-based control system. From this figure,
we find that the optimal density, for case S = 50, is
smaller than that for S = 5. Moreover, for given S, the
optimal density decreases with decreasing f . However,
for both S = 5 and S = 50, the optimal density corre-
sponding to Q = 1.0 is higher for f = 0.6 than that for
f = 0.9.
What is more effective way of increasing the efficiency
of the public conveyance system on a given route by in-
creasing the number of buses without increasing the car-
rying capacity of each bus, or by increasing the carrying
capacity of each bus without recruiting more buses? Or,
are these two prescriptions for enhancing efficiency of the
public conveyance system equally effective? In order to
address these questions, we make a comparative study
of two situations on the same route: for example, in the
0 0.2 0.4 0.6 0.8 1
Density
Q = 0.8
Q = 1.0
0 0.2 0.4 0.6 0.8 1
Density
rs Q = 0.8
Q = 1.0
FIG. 10: The average speed and the average number of wait-
ing passengers in the model B are plotted against the density
for the parameters f = 0.9, S = 50; the hopping parameters
are Q = 0.8 and Q = 1.0.
0.6 0.7 0.8 0.9 1
f = 0.9, S = 5
f = 0.6, S = 5
f = 0.9, S = 50
f = 0.6, S = 50
FIG. 11: The optimal density of buses in the model B is
plotted against Q. The parameters are f = 0.9, S = 5 (normal
line),f = 0.6, S = 5 (finer broken line), f = 0.9, S = 50 (bold
broken line), f = 0.6, S = 50 (longer broken line).
first situation the number of buses is 10 and each has a
capacity of 60, whereas in the second the number of buses
is 5 and each has a capacity of 120. Note that the total
carrying capacity of all the buses together is 600 (60×10
and 120× 5 in the two situations), i.e., same in both the
situations. But, the number density of the buses in the
second situation is just half of that in the first as the
length of the bus route is same in both the situations. In
FIG. 12, the results for these two cases are plotted; the
different scales of density used along the X-axis arises
from the differences in the number densities mentioned
above.
From FIG. 12, we conclude that, at sufficiently low
densities, the average velocity is higher for Nmax = 60
compared to those for Nmax = 120. But, in the same
regime of the number density of buses, larger number of
passengers wait at bus stops when the bus capacity is
smaller. Thus, in the region ρ < 0.05, system adminis-
trators face a dilemma: if they give priority to the aver-
age velocity and decide to choose buses with Nmax = 60,
the number of passengers waiting at the bus stops in-
creases. On the other hand if they decide to make the
passengers happy by reducing their waiting time at the
bus stops and, therefore, choose buses with Nmax = 120,
the travel time of the passengers after boarding a bus
becomes longer.
However, at densities ρ > 0.05, the system administra-
Density
(0.05) (0.1) (0.15) (0.2) (0.25)
Density
(0.05) (0.1) (0.15) (0.2) (0.25)
Density
(0.05) (0.1) (0.15) (0.2) (0.25)
Density
(0.05) (0.1) (0.15) (0.2) (0.25)
FIG. 12: Comparison between the case of bus capacity 60
with bus capacity 120. The parameters are Q = 0.9, S = 10,
f = 0.6 in the model B without information. The top figure
shows the average velocity, the center figure shows waiting
passengers and the bottom figure shows the number of con-
veyed passengers per unit bus, i.e. this number is calculated
by (total number of on-boarding passengers on all buses)/(the
number of buses), against the bus density up to 0.5. In each
figure, the horizontal axis shows the density; the numbers
without parentheses denote the number densities in the case
Nmax = 60, whereas the numbers in the parentheses denote
the number densities in the case Nmax = 120.
tors can satisfy both the criteria, namely, fewer wait-
ing passengers and shorter travel times, by one sin-
gle choice. In this region of density, the public con-
veyance system with Nmax = 60 is more efficient than
that with Nmax = 120 because the average velocity is
higher and the number of waiting passengers is smaller
for Nmax = 60 than for Nmax = 120. Thus, in this regime
of bus density, efficiency of the system is enhanced by re-
ducing the capacity of individual buses and increasing
their number on the same bus route.
0 0.2 0.4 0.6 0.8 1
Density
f = 0.3
f = 0.6
f = 0.9
0 0.2 0.4 0.6 0.8 1
Density
rs f = 0.3
f = 0.6
f = 0.9
FIG. 13: The plot of 〈V 〉 and 〈N〉 of the model B with infor-
mation (S = 5 and f = 0.3, 0.6 and 0.9)
0 0.2 0.4 0.6 0.8 1
Density
without info
with info
0 0.2 0.4 0.6 0.8 1
Density
rs without info
with info
FIG. 14: The model B with S = 5 and f = 0.9. The left
vertical dash line is ρ = 0.28 and the right is ρ = 0.73 in the
two figures.
B. PCM with information-based traffic control
The results for the PCM with information-based traf-
fic control system is shown in FIG. 13 and FIG. 14. In
the FIG. 13 we plot 〈V 〉 and 〈N〉 against the density of
buses for the parameter S = 5. The density correspond-
ing to the peak of the average velocity shifts to lower
values when the information-based traffic control system
is switched on.
The data shown in FIG. 14 establish that implemen-
tation of the information-based traffic control system
does not necessarily always improve the efficiency of
the public conveyance system. In fact, in the region
0.3 < ρ < 0.7, the average velocity of the buses is
higher if the information-based control system is switched
off. Comparing 〈V 〉 and 〈N〉 in FIG. 14, we find that
information-based traffic control system can improves the
efficiency by reducing the crowd of waiting passengers.
But, in the absence of waiting passengers, introduction
of the information-based control system adversely affects
the efficiency of the public conveyance system by holding
up the buses at bus stops when the number of buses in
the next segment of the route exceeds I0.
0 10 20 30 40 50
Ranking
without information
with information
FIG. 15: Distribution of headway distance for S = 10, m =
50, f = 0.9 in model B. This figure shows the plot of headway
distance against the ranking.
Finally, FIG. 15 shows the distribution of headway dis-
tance against the ranking, where we arrange the order of
magnitude according to the headway distance of buses
in descending order. From this figure it is found that
the headway distribution is dispersed by the effect of
the information. The average headway distance with the
information-based traffic control is equal to 8.34, in con-
trast to a much shorter value of 0.66 when that control
system is switched off. Thus we confirm that the avail-
ability of the information Ij and implementation of the
traffic control system based on this information, signifi-
cantly reduces the undesirable clustering of buses.
IV. MEAN FIELD ANALYSIS
Let us estimate 〈V 〉 theoretically in the low density
limit ρ → 0. Suppose, T is the average time taken by a
bus to complete one circuit of the route. In the model
A, the number of hops made by a bus with probability
q during the time T is S, i.e. the total number of bus
stops. Therefore the average period T for a bus in the
model A is well approximated by
and hence,
〈V 〉 =
q(L− S) +QS
. (4)
In model B, in the low density limit where m buses run
practically unhindered and are distributed uniformly in
the system without correlations, the average number of
passengersN waiting at a bus stop, just before the arrival
of the next bus, is
. (5)
The first factor f/S on the right hand side of the equation
(5) is the probability of arrival of passengers per unit
time. The second factor on the right hand side of (5)
is an estimate of the average time taken by a bus to
traverse one segment of the route, i.e. the part of the
route between successive bus stops. The last factor in the
same equation is the average number of segments of the
route in between two successive buses on the same route.
Instead of the constant q used in (4) for the evaluation
of 〈V 〉 in the model A, we use
N + 1
in eq. (4) and eq. (5) for the model B. Then, for the
model B, the hopping probability Q is estimated self-
consistently solving
〈V 〉 = Q−
, (7)
(4) and (6) simultaneously.
We also obtain, for the model B, the average number
of passengers 〈N〉 waiting at a bus stop in the ρ → 0
limit. The average time for moving from one bus stop to
the next is ∆t = (L/S − 1)/Q + 1/q̄ and, therefore, we
〈N〉 = (f/S) · (∆t+ 2∆t+ · · ·+ (S − 1)∆t)/S
f(S − 1)(q̄(L− S) + SQ)
2S2Qq̄
. (8)
As long as the number of waiting passengers does not
exceed Nmax, we have observed reasonably good agree-
ment between the analytical estimates (4), (8) and the
corresponding numerical data obtained from computer
simulations. For example, in the model A, we get the es-
timates 〈V 〉 = 0.85 and 〈N〉 = 1.71 from the approximate
mean field theory for the parameter set S = 50, m = 1,
Q = 0.9, q = 0.5, f = 0.3. The corresponding numbers
obtained from direct computer simulations of the model
A version of PCM are 0.84 and 1.78, respectively. Simi-
larly, in the model B under the same conditions, we get
〈V 〉 = 0.60 and 〈N〉 = 2.45 from the mean field theory,
while the corresponding numerical values are 0.60 and
2.51, respectively. If we take sufficiently small f ’s, then
the mean-field estimates agree almost perfectly with the
corresponding simulation data. However, our mean field
analysis breaks down when a bus can not pick up all the
passengers waiting at a bus stop.
V. CONCLUDING DISCUSSIONS
In this paper, we have proposed a public conveyance
model (PCM) by using stochastic CA. In our PCM, some
realistic elements are introduced: e.g., the carrying ca-
pacity of a bus, the arbitrary number of bus stops, the
halt time of a bus that depends on the number of waiting
passengers, and an information-based bus traffic control
system which reduces clustering of the buses on the given
route.
We have obtained quantitative results by using both
computer simulations and analytical calculations. In par-
ticular, we have introduced two different quantitative
measures of the efficiency of the public conveyance sys-
tem. We have found that the bus system works efficiently
in a region of moderate number density of buses; too
many or too few buses drastically reduce the efficiency
of the bus-transport system. If the density of the buses
is lower than optimal, not only large number of passen-
gers are kept waiting at the stops for longer duration, but
also the passengers in the buses get a slow ride as buses
run slowly because they are slowed down at each stop to
pick up the waiting passengers. On the other hand, if
the density of the buses is higher than optimal, the mu-
tual hindrance created by the buses in the overcrowded
route also lowers the efficiency of the transport system.
Moreover, we have found that the average velocity in-
creases, and the number of waiting passengers decreases,
when the information-based bus traffic control system is
switched on. However, this enhancement of efficiency of
the conveyance system takes place only over a particular
range of density; the information-based bus traffic con-
trol system does not necessarily improve the efficiency of
the system in all possible situations.
We have compared two situations where the second
situation is obtained from the first one by doubling the
carrying capacity of each bus and reducing their number
to half the original number on the same route. In the
density region ρ > 0.05 the system of Nmax = 60 is more
efficient than that with Nmax = 120. However, at small
densities (ρ < 0.05), although the average velocity in-
creases, the number of waiting passengers also increases,
by doubling the carrying capacity from Nmax = 60 to
Nmax = 120. Hence, bus-transport system administra-
tors would face a dilemma in this region of small density.
Finally, in our PCM, the effect of the disembarking
passengers on the halt time of the buses has not been
captured explicitly. Moreover, this study is restricted to
periodic boundary conditions. The clustering of particles
occurs not only in a ring-like bus route, but also in shuttle
services of buses and trains. Thus it would be interesting
to investigate the effects of the information-based traffic
control system also on such public transport systems. In
a future work, we intend to report the results of our in-
vestigations of the model under non-periodic boundary
conditions. We hope our model will help in understand-
ing the mechanism of congestion in public conveyance
system and will provide insight as to the possible ways
to reduce undesirable clustering of the vehicles.
Acknowledgments: Work of one of the authors (DC)
has been supported, in part, by the Council of Scientific
and Industrial Research (CSIR), government of India.
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