Table of Contents
Traffic Deadlock Caused By Two
Intersections
Omer Akkentli, Bogazici University Industrial Engineering Department, Istanbul
Mert Nuhodlu, Bogazici University Industrial Engineering Department, Istanbul
Contact Person:
Name: Mert Nuhoglu
Organization: Bogazici University
Address: Madalyon Sk. No:2/7, 80220, Nisantasi, Istanbul, Turkey
Phone: +90(212) 224 8486 , Mobile: +90(535) 712 0767
E-mail address: mert.nuhoglu@ isnet.net.tr
Key words: traffic flow deadlock feedback modelling “individuum vs. system”
Problem Structuring
The behavior of the vehicle drivers depends usually on the traffic conditions and some
extemal factors such as the traffic lights. Normally it is assumed that traffic flow is a
linear process that is feedback effect of some driving behavior to itself is assumed to
be negligible. The usual simulation models of the traffic flow are based on spatial
automata modeling, queuing theory or other modeling technique that doesn’t consider
or focus on the feedback dynamics.
Our work focuses on an important feedback problem in the traffic flow. The problem
is observed in Etiler - a neighborhood in Istanbul. But the structure lying under this
problem is very general. The problem is triggered by the congestion in Oven Road.
The cars coming from the Torito Road pass the Torito Junction, when the light is
green to them. But there remains no sufficient place to allow all the cars flowing into
the Oven Road. So some cars stay on the Torito Junction. These cars prevent the
outflow of the Kodak Road through the junction. Therefore the congestion of cars in
the Kodak Road increases more. This causes the cars flowing into the Kodak Road
from the Akmerkez Road to go slower or to stay in the Akmerkez Junction, when the
ih
light is green to them. When the traffic lights in Akmerkez Junction tur green to
Oven Road, the halting cars in the Akmerkez Junction prevent the outflow from this
toad. Therefore the congestion in Oven Road increases as well. This results in a
higher halting period for the cars in Torito Junction and therefore for the cars in
Akmerkez Junction. In ideal conditions, if there were no any other outflow road from
Oven and Kodak Roads, then this dynamics would lead to an absolute jam, which
would never allow the cars to move any further.
This problem resembles to a snake trying to eat its own tail. As the snake tries more to
teach to its tail, its tail goes further away. It is a vicious cycle and the snake never
reaches its goal. In our problem, the cars in the Oven Road try to move faster. As they
try more, the traffic gets more congested and their conditions get worse. So in order to
simplify the referral to the dynamic problem, we call it “snake tail” problem.
Problem Definition
In this work we focus on a unique problem that helps us understand snake tail
problem better. We focus on the load of the traffic. We will answer the problem, how
much loading does start the snake tail problem?
Time Horizon and Perspective
The time horizon relevant to this work is the period, when the people go home from
their work. This is the time period, when the congestion in the roads exceed a certain
level, which results the flow in the junctions be stopped up. The variables are
measured in time units of minutes, which is relevant for a time horizon of a few hours.
Behavior over Time Graphs
There is no detailed observed data available. But the hypothetical dynamics of
number of cars in the Oven and Kodak Roads and the outflows from these roads are
shown in Figure 1 and Figure 2, respectively.
Lf
a
Vehicles in Oven
Road
Number of Cars
Vehicles in Kodak
Road
Time
Figure 1 Hypothetical Dynamic Behaviour of the Number of Cars in the Roads
The cars accumulate in the road firstly in a linear manner, assuming that there is
enough space for the new coming cars and inflow rate is higher than the outflow rate
at the beginning. These assumptions are valid for the concemed period. The inflow
rate into the road is very high due to the people going to their home at evening. The
outflow rate is lower than the inflow rate due to the already congested roads which the
Oven and Kodak Roads are connected to.
The linear accumulation of the cars occurs until a certain point, when the remaining
capacity in the Oven and Kodak Roads constrain the inflow rate. Until this point, the
inflow rate was constrained by the duration of the traffic lights. But from this point
on, the new coming cars cannot flow through the junction into the Kodak or Oven
Road as rapid as they want, because of the short distance to the next car.
From then on, the accumulation of the cars occurs in an s-shaped manner. The cars
flow into the Kodak and Oven Road, until their capacity is fully utilized. The capacity
of a road is fully utilised, when there is no more space left for the newcomers.
\ —— Outflow from Oven
Road
WA |——Outflow from Kodak
~ Road
Number of Cars/Period
Time
Figure 2 Hypothetical Outflow Behaviour
The outflows are initially at a high rate. But because of the closings of the junctions,
the outflows decrease one after another. The decrease in one outflow causes the
decrease of the other via the accumulation of the cars in the junction from where the
cars of the other road outflow.
The oscillations in the behavior patterns are caused by the traffic lights.
Causal Loop Diagram
The causal loop diagram showing the relationship between the critical variables is
shown in Figure 3. The loop that triggers the problem is the central loop. The two
small loops function as regulatory loops.
Jp Vehicles in Akmerker Junction
Remaining Capacity in Kodak Road — »
A fo suc so
ee Ps \
(A Veit a Kodak Road at) \
Kodak Road Becomes Full Snake Bits lis Tall [ — |
\ vateis na Road
. |
Inflow to Kodak Road \
Outflow ffom Kodak Road \
Be ___ Inflow to Oven. Rowe
~Vehites in Tonto Juncio~
Remaining Capacity in Oven Road
Figure 3 Causal Loop Diagram of Snake Tail Problem
The central causal loop is active only if the traffic on the Kodak and Oven Roads are
already congested. At these times the number of vehicles in Oven Road is fairly high.
When the number of vehicles in Oven Road increases more, some vehicles flowing
into the Oven Road stay in the Torito Junction. Thus the number of vehicles in Torito
Junction increases. This prevents the outflow from Kodak Road. Which decreases the
number of vehicles in Kodak Road compared to the state, when there was no decrease
in the outflow from Kodak Road. As the number of vehicles in Kodak Road increases
the number of vehicles in Akmerkez Junctions increases, as there is no remaining
capacity for them in the Kodak Road. This stops up the outflow from the Oven Road
through the A kmerkez Junction. Thus the outflow from the Oven Road decreases and
consequently the number of vehicles in Kodak Road increases compared to the state,
when there was no decrease in the outflow from Oven Road.
In summary, the increase in the number of vehicles in Oven Road results in a further
increase in Oven Road. This is the dynamic behaviour of the central loop. This loop is
called as “snake bites its tail”, because it triggers the snake tail problem. The snake
tries to reach the goal. But this causes the goal to go further away.
The increase in the number of vehicles in Oven Road or Kodak Road is not unlimited.
They are controlled by the capacities of the roads. The limiting behaviour is modelled
by the two small loops in Figure 3. As the number of vehicles in Kodak Road
increases, the remaining capacity decreases. As it decreases, the inflow to Kodak
Road decreases as well. This results in a decrease of the vehicles in Kodak Road,
when compared to the state that there was no decrease in inflow. The same dynamics
occur in Oven Road.
Model Construction
The stock-flow model is developed in gradual steps. In this article, we will explain the
gradual improvements in the model step by step. The first model, on which the
improvements are made, is shown in Figure 4. There are two stock variables, one
showing the number of vehicles in Kodak Road, the other the one in Oven Road. The
loops shown in causal loop diagram can also be seen in the stock flow diagram. The
“snake bites its own tail” loop is the central loop in the stock-flow model. It starts
from the stock variables and extend thru the Vehicles in Torito/A kmerkez Junction,
Effect of Torito/A kmerkez Junction on Outflow Kodak/Oven and Outflow
Kodak/Oven.
()
Normal Outflow Kodak
Vehicles in Kodak Road
outfw Kodak Inflow Kodak
kez | unction on Inflow Kodak
Effect of Torito | unction of\Qutflow Kodak
Remaining Capacity in’
Capacity of Oven Road
Remaining pacity in Qyen Road
Effect of Akmerkez | ugtion on Outlow Oven
Effect of Torta Junction on Inflow
Vehities in Oven Road
flow Oven Outfow Odea,
@) es
Figure 4 Initial Stock-Flow Model of the Snake Tail Problem
Variables
Stocks:
Vehicles in Kodak Road: The number of vehicles in Kodak Road {vehicles}
Vehicles in Oven Road: The number of vehicles in Oven Road {vehicles}
Flows:
Outflow Kodak: The number of vehicles flowing out of the Kodak Road through the
Torito Junction per minute {vehicles/minute}
Inflow Kodak: The number of vehicles flowing into the Kodak Road through the
Akmerkez Junction per minute {vehicles/minute}
Outflow Oven: The number of vehicles flowing out of the Oven Road through the
Akmerkez Junction per minute {vehicles/minute}
Inflow Oven: The number of vehicles flowing into the Oven Road through the Torito
Junction per minute {vehicles/minute}
Converters:
Capacity_of Kodak Road: The capacity of the Kodak Road - excluding the
Akmerkez Junction - {vehicles}
Capacity_of_Oven_Road: The capacity of the Oven Road - excluding the Torito
Junction - {vehicles}
Normal_Inflow_Kodak: The normal inflow rate into the Kodak Road - when there is
no place limitation in the road {vehicles/minute}
Normal_Inflow_Oven: The normal inflow rate into the Oven Road - when there is no
place limitation in the road {vehicles/minute}
Normal_Outflow_Kodak: The normal outflow rate from the Kodak Road through the
Torito Junction - when there is no stopped up cars in the junction {vehicles/minute}
Normal_Outflow_Oven: The normal outflow rate from the Oven Road through the
Akmerkez Junction - when there is no stopped up cars in the junction
{vehicles/minute}
Effect_of_Akmerkez_ Junction_on_Inflow_Kodak: The effect of the remaining
capacity in the Kodak Road on the inflow rate into the Kodak Road. {no unit}
Effect_of_Akmerkez_Junction_on_ Outflow Oven: The effect of the vehicles stopped
up in the Akmerkez Junction on the outflow rate from the Oven Road. {no unit}
Effect_of_Torito_Junction_on_Inflow_Oven: The effect of the remaining capacity in
the Oven Road on the inflow rate into the Oven Road. {no unit}
Effect_of_Torito_Junction_on_Outflow_Kodak: The effect of the vehicles stopped up
in the Torito Junction on the outflow rate from the Kodak Road. {no unit}
Remaining_Capacity_in_ Kodak Road: The remaining capacity in the Kodak Road.
The variable can have positive or negative values. If it is positive, then it defines the
space left in the Kodak Road. If it is negative, then Kodak Road is overloaded, i.e.
some vehicles stop up in the Akmerkez Junction. {vehicles}
Remaining_Capacity_in_Oven_Road: The remaining capacity in the Oven Road. The
variable can have positive or negative values. If it is positive, then it defines the space
left in the Oven Road. If it is negative, then Oven Road is overloaded, i.e. some
vehicles stop up in the Torito Junction. {vehicles}
In this model, the names of the variables “Remaining Capacity_in Kodak Road” and
“Remaining Capacity_in Oven Road” are partially misleading. They do not show
only the remaining capacity. They also show the number of vehicles stopped up in the
Akmerkez and Torito Junction, respectively. If the variables become negative, then
the roads are overloaded and the new coming cars stop up in the junctions. When the
variables are positive, it is the remaining capacities of the roads.
Mathematical Relationships
The mathematical equations showing the relationships between the variables are the
following ones:
Vehicles_in_Kodak_Road(t) =Vehicles_in_Kodak_Road(t- dt) +(Inflow_Kodak -
Outflow_Kodak) * dt
INIT Vehicles_in_Kodak_Road =5
INFLOWS:
Inflow_Kodak = if(mod(4*TIME,8)<6) then 0 else
Effect_of_Akmerkez_J unction_on_Inflow_Kodak*Normal_Inflow_Kodak
OUTFLOWS:
Outflow_Kodak = if(mod(4*TIME,8)<5) then 0 else
Normal_Outflow_Kodak*€ ffect_of_Torito_J unction_on_Outflow_Kodak
Vehicles_in_Oven_Road(t) =Vehicles_in_Oven_Road(t - dt) + (Inflow_Oven -
Outflow_Oven) * dt
INIT Vehicles_in_Oven_Road =5
INFLOWS:
Inflow_Oven = if(mod(4*TIME,8)<3) then 0 else
Effect_of_Torito_J unction_on_Inflow_Oven*Normal_Inflow_Oven
OUTFLOWS:
Outflow_Oven = if(mod(4*TIME ,8)<2) then 0 else
Normal_Outflow_Oven*e ffect_of_Akmerkez_] unction_on_Outflow_Oven
Capacity_of_Kodak_Road =38
Capacity_of_Oven_Road =38
Normal_Inflow_Kodak =18
Normal_Inflow_Oven =18
Normal_Outflow_Kodak =4
Normal_Outflow_Oven =4
Remaining_Capacity_in_Kodak_Road =
Capacity_of_Kodak_Road-Vehicles_in_Kodak_Road
Remaining_Capacity_in_Oven_Road =Capacity_of_Oven_Road-
Vehicles_in_Oven_Road
Effect_of_Akmerkez_} unction_on_Inflow_Kodak =
GRAPH(Remaining_Capacity_in_Kodak_Road)
(-3.00, 0.00), (-0.7, 0.175), (1.60, 0.295), (3.90, 0.575), (6.20, 0.705), (8.50, 0.81),
(10.8, 0.875), (13.1, 0.93), (15.4, 0.965), (17.7, 0.985), (20.0, 1.00)
Effect_of_Akmerkez_J unction_on_Outflow_Oven =
GRAPH(Remaining_Capacity_in_Kodak_Road)
(-3.00, 0.00), (-2.70, 0.015), (-2.40, 0.1), (-2.10, 0.39), (-1.80, 0.655), (-1.50, 0.84), (-
1.20, 0.93), (-0.9, 0.965), (-0.6, 0.98), (-0.3, 0.995), (-3.33e-016, 1.00)
Effect_of_Torito_J unction_on_Inflow_Oven =
GRAPH(Remaining_Capacity_in_Oven_Road)
(-1.50, 0.00), (0.65, 0.175), (2.80, 0.295), (4.95, 0.575), (7.10, 0.705), (9.25, 0.81),
(11.4, 0.875), (13.6, 0.93), (15.7, 0.965), (17.9, 0.985), (20.0, 1.00)
Effect_of_Torito_J unction_on_Outflow_Kodak =
GRAPH(Remaining_Capacity_in_Oven_Road)
(-1.50, 0.00), (-1.35, 0.015), (-1.20, 0.1), (-1.05, 0.39), (-0.9, 0.655), (-0.75, 0.84), (-
0.6, 0.93), (-0.45, 0.965), (-0.3, 0.98), (-0.15, 0.995), (-1.67e-016, 1.00)
The graphical functions of the Effect of Akmerkez/Torito Junction on Inflow
Kodak/Oven are shown in Appendix B, Figure 36 and Figure 37 respectively.
Similarly the graphical functions of Effect of Akmerkez/Torito Junction on
Oven/Kodak Outflow are shown in Figure 38 and Figure 39, respectively.
The capacities of Oven Road and Kodak Road are measured as 38 vehicles. This
number doesn’t include the excess capacity of the road, when the Torito Junction is
congested. Therefore the stock variable Vehicles in Oven Road may exceed the
Capacity of Oven Road by the capacity of the Torito Junction. The same is also valid
for the Kodak Road. The capacities of the Torito and Akmerkez Junctions are 1.5 and
3 vehicles, respectively. Thus the maximum allowable level of the stock variables
Vehicles in Oven and Kodak Road are 39.5 and 41, respectively.
All the graphical functions defining the effect variables are selected as s-shape
functions. The effect function of the vehicles in Torito Junction on Kodak Outflow -
Figure 39- states that if there is no vehicle in Torito Junction, then it doesn’t effect the
Kodak Outflow, if the junction is full of cars, then the Kodak Outflow is totally
prevented. Between the extreme values we assumed the relationship to be s-shaped. If
there is only 0.5 cars on the junction, then the cars may pass through the junction by
some intricate manevoire, which is happening in these junctions, in real life. The
effect of partial load on the junction is the slower movement of the cars through the
junction, which results in a lower outflow rate. The decelaration is at an increasing
rate at first until some threshold and then it occurs at a decreasing rate, which justifies
the s-shape of the effect function. The same is also valid for the effect of Akmerkez
Junction on the Oven Outflow.
The effect of the remaining capacity in the Oven Road -which is reflected by the
variable Vehicles in Torito Junction - on the inflow rate into Oven Road is similar.
First of all, as the remaining capacity decrease, the inflow rate must decrease. This
occurs because of the flow dynamics of the vehicles. The vehicle drivers determine
their speed according to the spacing between their vehicle and the next vehicle and the
speed of the next vehicle. The speed of the next vehicle is determined in the same
way. From the statistical point of view, the normal speed of the cars on a road
depends only on the spacing between the consecutive cars. When the spacing between
the consecutive cars is higher than a threshold value, the speed of the cars are limited
10
by their normal speed on the road. When the spacing is zeroing the speed is zero as
well. When the spacing between the cars is a little lower than the threshold, their
speed is affected in a very low amount. But as the spacing decreases further, the effect
on the speed is stronger. Thus starting from zero point, the effect of the spacing on the
speed is at an increasing rate until some inflection point. Then it occurs at a
decreasing rate until the normal speed. The remaining capacity on the road is directly
proportional with the spacing between the consecutive cars. So the s-shape
relationship between the remaining capacity and the inflow rate is justified.
Parameters
There are certain constant parameters in the model, Table 1. The parameters
Normal_Outflow_Kodak, Normal_Outflow_Oven, Capacity of Akmerkez Junction,
Capacity of Torito Junction, Capacity_of Kodak Road, Capacity_of_Oven_Road are
measured parameters. The outflow rates are measured at a time, when there was no
junction congestion problem and the roads were loaded with sufficient cars to supply
the normal outflow rate. The normal inflow rates depend highly on the remaining
capacity. Therefore a measurement for the normal rate is not possible. But we
estimated them to be slightly higher than the normal outflow rate into the Kodak
Road, because the normal outflow rate from the Kodak Road is measured at a time,
when there was a balance between the inflow and outflow. But since the number of
cars flowing into the roads is higher at evening, they are taken slightly higher.
Table 1 Parameters of the Model
DT 0.1 minute
Capacity_of Kodak Road 38 vehicles
Capacity_of_Oven Road 38 vehicles
Normal_Inflow_Kodak 18 vehicles/minute
Normal_Inflow_Oven 18 vehicles/minute
Normal_Outflow_Kodak 4 vehicles/minute
Normal_Outflow_Oven 4 vehicles/minute
Capacity of Akmerkez Junction 3 vehicles
Capacity of Torito Junction 1.5 vehicles
11
The initial values for the stocks are taken near to their capacities, which is the normal
case during the after work hours.
Verification
The Extreme Condition Tests:
When the normal inflow rate into the Oven Road is zero, after some while there will
remain certainly no other car in the Oven Road. The simulation model gave the
correct behaviour, Figure 5.
Untitled
Figure 5 Normal Inflow Oven = 0
The second extreme condition test is the reverse. The outflow from the Kodak Road is
taken as zero. In this case, the inflow into the Kodak Road will cause the vehicles in
the road to increase upto the maximum allowable capacity, which is 41. A fter that the
inflow cannot push the stock to increase any further.
The simulation run is performed and we found that the inflow to the Kodak Road
doesn’t continue forever. The capacity of the road except the junction is 38. The
capacity of the junction is 3. Therefore the total number of the vehicles in Akmerkez
road cannot exceed 41. The number of vehicles in Akmerkez road vs. time is
displayed in Figure 6. The steady state is achieved at 41 which is the capacity of the
12
road including the junction. The inflow to the Kodak Road goes to zero as time
passes.
Also after the Kodak Road has reached its capacity of 41 , the Oven Road’s outflow
decreases up to zero , and it also reaches its capacity .
S® 1 venices in Kodak Road 2: Inflow Kodak
1 50,004
2 10,00
1 1 1
0.00 30.00 60.00 90.00 120.00
aeaF Graph 1 (Untitled) Time 10:16 05 Oca 2002 Cum
Figure 6 Outflow Kodak =0
Partial Tests
The next test is done by activating both the input and the outflow from the Kodak
Road, see Appendix B Figure 40. The outflow rate is taken constant as 5 cars per
minute. Equilibrium in Vehicles in Kodak Road is attained at 32 cars. The inflow
decreased to the level of the outflow and remained stable, because of the limiting
influence of Effect of Vehicles in Akmerkez Junction, Appendix B Figure 41.
The same verification runs are made for the Oven Road. The results are reasonable,
Figure 7.
13
BD 1: Vehicles in Oven Road 2: Inflow Oven 3: Outflow Oven
1 40,00-7——2.
2] 12,00
3
1 1
di 0.00 |
5)
3 0,00
0.00 5.00 10:00 15.00 20.00
aeaF Graph 2 (Untitled) Time 11:06 05 Oca 2002 Cum
Figure 7 Partial Run (Inflow vs. Constant Outflow) - Oven Alone
Another simulation run is done in order to verify that the effect formulation works
properly. The effect of the number of vehicles in Torito Junction - in the model, this
number is defined in the variable Remaining Capacity in Oven Road - on Kodak
Outflow is activated - see Appendix B Figure 39 for the relationship between the
variables. The outflow from Oven Road is taken as zero. This caused the number of
vehicles in the Oven Road to increase up to the full capacity 39.5. After that the
Torito Junction is congested. And this caused the outflow from Kodak Road to
decrease down to zero.
Next we verified the effect of Torito Junction on Kodak Outflow, when there is no car
in Torito Junction, Appendix B Figure 42 and Figure 43. To do this we allowed the
outflow from Kodak Road to be a positive number. Thus the inflow into the Kodak
Road didn’t increase the number of vehicles in Kodak Road such that its capacity is
not fully utilized. The result is displayed in Figure 8. In this case, the number of
vehicles in Kodak Road is independent of the number of vehicles in Oven Road.
14
& 2 vebicies in Kodak Road 2: Vehicles in Oven Road 3: Vehicles in Torito J unction
1
2:
3
=,
16.00 32:00 48.00 64.00
aeaF Graph 1 (Untitled) Time 17:50 05 Oca 2002 Cum
Figure 8 Partial Run (Effect of Torito Junction on Kodak Outflow Activated) -
Outflow from Oven Road is taken as six
Validation
Behaviour Validity
During the after work hours, sometimes the normal outflow rates from the Oven
decreases further below the values measured by our observations. This is due to the
congestion in the Zincirlikuyu Road. When this road becomes congested, the outflow
from the Oven Road is down-limited by the remaining capacity in the Zincirlikuyu
Road. According to our trials the normal outflow/inflow ratio should be 1/20. Only
after this ratio, the model reproduces the hypothetical dynamic behaviour. The
simulation outputs and hypothetical dynamic behaviour graphs are shown in Figure 9,
Figure 10 and Figure 11. The simulation runs regenerate the general pattern in the
hypothetical dynamic behaviour graphs.
15
& \: ountiow oven
1
2
4.00-4-1—2-
1.00
0.00
0.00:
2: Outflow Kodak
0.00
aaF
280
Graph 1 (Untitled)
7.50
16:24
10.00
03 Mar 2002 Paz
Figure 9 Simulated Dynamic Behaviour of the Outflow Rates
iod
Number of Cars,
|——Outflow from Oven
Road
|—— Outflow from Kodak
Road
Time
Figure 10 Hypothetical Behaviour of the Outflow Rates
16
Ss 1: Mehices in Kodak Road 2: Vehicles ir Oven Road
1 5.004 a2
a]
1 73 1 -
4 ns04 ' ' {
1
| 2.00 + r + 1
0.00 3.96 7.50 11.25 16.00
Figure 11 Simulated Dynamic Behaviour of the Stocks
—
Lf
Py,
Vehicles in Oven
Road
Number of Cars
Vehicles in Kodak
Road
Time
Figure 12 Hypothetical Behaviour of the Number of Vehicles
The above results are appropriate for the model without traffic lights. However when
traffic lights are included in the model, an oscillatory behaviour is observed. That is
because the traffic lights stop the flows for some durations of time, and stocks are also
effected by this property.
17
Figure 14 Dynamic Behaviour of the Stocks
18
Experiments
Model 1
This model does not include traffic lights and delays .It is ran for several values of
inflows and outflows and the following results are obtained.
Run 1
STOCKS
Vehicles In Kodak 5
Vehicles In Oven 5
CONVERTERS
Normal Inflow Kodak 18
Normal Inflow Oven 18
Normal Outflow Oven
Normal Outflow Kodak
Results
W
Figure 15 Dynamic Behaviors of Outflows
19
N
Figure 16 Dynamic Behaviour of Stocks
The above graph shows the fact that the inflows to the roads are not high enough to
cause a traffic paralysis. Namely, outflow/inflow ratios for both of the stocks are high
and the system comes to an equilibrium where the inflows of the stocks equal the
outflows. Although inflows are higher than the outflows initially, as the stocks of
Kodak and Oven begin to fill, inflows decrease to the level of outflows and the system
comes to a dynamic equilibrium.
Run 2
STOCKS
Vehicles In Kodak 5
Vehicles In Oven 5
CONVERTERS
Normal Inflow Kodak 18
Normal Inflow Oven 18
Normal Outflow Oven 1
Normal Outflow Kodak 1
20
1: Vehicles in |
Figure 18 Dynamic Behaviour of Stocks
RESULTS
When the above resluts are analyzed, it is observed that the Kodak and Oven Road
stocks begin to increase gradually. Since inflow never becomes lower than the
outflow until the stocks are full, stocks increase up to their capacity level and when.
that level is reached both inflow and outflow becomes zero. This event is called
“snake eating its tail “. When we analyze more; the previous simulation did not lead
to such a behavior, because outflow rates were higher in that run. So traffic was
flowing at a low rate, but it did not stop. But when we decrease the outflows, the
junction points of Oven & Kodak begin to fill up with cars. Eventually the cars in the
21
junctions block both inflows and outflows and the “ snake eating its tail “ behavior
occurs.
MODEL 2
This model includes the traffic lights.
They operate according to following schedule. (Durations are estimated values)
Flow Name Duration
Kodak Outflow 45 seconds
Oven Inflow 75 seconds
Kodak Inflow 30 seconds
Oven Outflow 90 seconds
Model Description:
This model is different from the previous model in the respect that it schedules the
flow of the roads according to the real life traffic light schedule .To achieve this,
TIME function of STELLA is used.
For the flows the below equations are updated.
Inflow_Kodak = if(mod(4*TIME,8)<6) then 0 else
Effect_of_Akmerkez_Junction_on_Inflow_Kodak*Nommal_Inflow_Kodak
Outflow_Kodak = if(mod(4*TIME,8)<5) then 0 else
Normal_Outflow_Kodak*Effect_of_Torito_Junction_on_Outflow_Kodak
Inflow_Oven = if(mod(4*TIME,8)>4) then 0 else
Effect_of_Torito_Junction_on_Inflow_Oven*Normal_Inflow_Oven
Outflow_Oven = if(mod(4*TIME,8)>5) then 0 else
Normal_Outflow_Oven*Effect_of_Akmerkez_Junction_on_Outflow_Oven
RUN1
STOCKS
22
Vehicles In KODAK 5,00
Vehicles In Oven 5,00
CONVERTERS
Normal Inflow Kodak 72,00
Normal Inflow Oven 24,00
Normal Outflow Oven 4,00
Normal Outflow Kodak 2,66
These values of inflows and outflows from the stocks are adjusted to be the same as in
the previous run Since we impede the flows for some time period , average flow of
the system decreases if we do not change the values of converters . For example
inflow from Kodak Road is allowed for only 30 seconds / 2 minutes, so its average
flow is 18 vehicles/minute again if we assign 72 to converter.
B® 1: \ehictes in Kodak Road 2: Vehicles in Oven Road
4] 45.00-4°
3] 22.504
1] 0.00.
0.00 3.95 7.50 11.28 16.00
i aaF Graph 1 (Untitled) Time 17:22 03 Mar 2002 Paz
Figure 19 # of Vehicles in Kodak and Oven Roads
The other graphs related to this experiment are shown in Figure 44 and Figure 45,
Appendix B.
23
Experiments on Driver Types
Driver Type 1: Advanced and | mpatient Driver
We call this type of driver as advanced, because he/she can pass through the roads
having very narrow blanks, which are partially blocked by other cars. Such a behavior
is added to the model by modifying the effect graphs. As seen in the below graph such
a driver is not stopped completely even if the road is partially blocked.
Figure 20 Effect of Remaining Capacity on Inflow-Graph of an Advanced Driver
24
Figure 21 Effect of Remaining Capacity on Outflow-Graph of an Advanced
Driver
Interpretation of Graphs:
The most basic characteristic of the above graphs is; the effect coefficient of the
remaining capacity in the road is not reduced very much even if the road is partially
full. As a result of this fact outflow and inflow values do not decrease as soon as the
capacity of the road decreases. Consequently flows in the roads do not end even in the
crowded situations and the traffic flow goes on.
Run (Hesitated Drivers):
STOCKS
Vehicles In KODAK 5,00
Vehicles In Oven 5,00
CONVERTERS
Normal Inflow Kodak 32.5
Normal Inflow Oven 32
Normal Outflow Oven 14
Normal Outflow Kodak 1.8
25
Result:
Figure 23 Dynamic Behaviour of the Flows-Advanced (Hesitated) Driver
Interpretation of Results:
As we can observe from the above graphs, inflow and outflow values go down to zero
at time point 8.The “snake eating its tail Behaviour occurs because the hesitated
drivers do not stop coming into the roads inspire of observing the crowd in the road,
so as they come the junctions also become filled, consequently no other car can move
any further.
26
Driver Type 2: Normal Driver
We call this type of driver as normal, because he/she drives more patiently than the
hesitated drivers but more dangerously than the cautious ones. Such a behavior is
added to the model by modifying the effect graphs.
froo0 Input Output
oe 3.000 0.000
E 0.700 0.060
f 1.600 0.190
f 3.900 0.475
A 6.200 0.635
c 8.500 0715
t 10.80 0.800
13.10 0.880
0 15.40 0.935
' 17.70 0.970
” 20.00 7,000
0.000 :
[iy al Data Points: il
-3.000 [20.00
ovenwa!Cavaciy'y1 Kodek Head nee = 2
ToEquation _| DeleteGraph_ | __Cancel__| OK |
Figure 24 Effect of Remaining Capacity on Inflow-Graph of a Normal Driver
Input
“3.000
-2.700
-2.400
-2.100
-1.800
“1.500
“1.200
-0.900
“0.600
-0.300
0.000
ia} al Data Points:
a |-3.000 [o.c00
Remaining_Capacity_in_Kodak Road SEE as
ToEquation _| Delete Graph | __ Cancel
as
Figure 25 Effect of Remaining Capacity on Outflow-G raph of a Normal Driver
27
Interpretation of Graphs:
V2)
0,8 a
+
& 06 —+— Normal
. V2
0,4 ri; —aHesitated
Drivers
0 ee
123 45 6 # 8 9 10 11
Remaining Capacity in the Road
Figure 26 Remaining C apacity vs. Effect on Inflow Graph
When we analyze the above graph, it is observed that the hesitated driver has a higher
effect value at the same level of remaining capacity in the road. Since the flow is
effect*Normal Flow, the actual flow value is more in the hesitated drivers case. As
mentioned before this condition leads to an early filling of the road and the traffic
flow ends before the Normal Driver's case.
Run (Normal Drivers Case)
STOCKS
Vehicles In KODAK 5,00
Vehicles In Oven 5,00
CONVERTERS
Normal Inflow Kodak 32.5
Normal Inflow Oven 32
Normal Outflow Oven 1.4
Normal Outflow Kodak 1.8
28
Result:
Figure 27 Dynamic Behaviour of the Stocks-Normal Driver
1: Outflow Kodak
Figure 28 Dynamic Behaviour of the Flows-Normal Driver
Interpretation of Results:
Normal Driver case does not lead to a “snake eating its tail Behaviour” when the
previous experiment is repeated with Normal Driver case effect graphs. System
reaches equilibrium at the point of
Vehicles in Oven Road 35
Vehicles in Kodak Road 36.3
29
Consequently, Normal Driver case reaches its equilibrium point and the traffic flow
does not end.
Driver Type 3: Cautious Driver
We call this type of driver as Causal, because he/she drives more patiently than the
Normal Drivers. That is to say whenever there exists a crowd in the junctions of the
roads, they immediately decrease in speed and their flow stops.
Figure 29 Effect of Remaining Capacity on Inflow-Graph of a Cautious Driver
30
Figure 30 Effect of Remaining Capacity on Outflow-G raph of a Cautious Driver
Interpretation of Graphs:
Effect
—e— Normal
—aHesitated
Drivers
hag 25 6 7 8 9 mH m | ages
Remaining Capacity in the road
Figure 31 Remaining C apacity vs. Effect on Inflow Graph
When we analyze the above graph, it is observed that the Hesitated driver has a higher
effect value at the same level of remaining capacity in the road than the Normal
Driver, and Normal Driver has a higher effect value than the Cautious Driver. Since
the flow is effect* Normal Flow, the actual flow value is the most in the Hesitated
Drivers Case and the least in the Cautious Drivers Case.
31
Run (Cautious Drivers Case)
STOCKS
Vehicles In KODAK 5,00
Vehicles In Oven 5,00
CONVERTERS
Normal Inflow Kodak 32:5:
Normal Inflow Oven 32
Normal Outflow Oven 14
Normal Outflow Kodak 1.8
Result:
Figure 32 Dynamic Behaviour of the Stocks-Normal Driver
g
1 0,00
2 0
et 12.00 16.00
Nr =7 a it 04:13 01 Oca 1998 Per
Figure 33 Dynamic Behaviour of the Flows-Normal Driver
Interpretation of Results:
Cautious Driver case does not lead to a “snake eating its tail Behaviour” when the
previous experiment is repeated with Cautious Driver case effect graphs. System
reaches equilibrium at the point of
Vehicles in Oven Road 35
Vehicles in Kodak Road 36.3
Consequently, Cautious Driver case reaches its equilibrium point and the traffic flow
does not end.
Conclusions:
Our local system has an important implication about the systems view. That is the
disorder caused by the drivers’ personal behaviors. For a system to maintain its
operations correctly, organizational Behaviour is a must. In the above system we
observed that drivers are trying to optimize their traveling times. In spite of observing
that the road that they will enter is filled with cars, they enter the road. Since the road
is full of capacity, the last entering car stays in the middle of the junction point. By
doing so, the driver causes the other road to be blocked by his car. Since the other is
also blocked, the outflow of this road becomes blocked and none can move anywhere.
However if the driver hadn't entered the road, he wouldn't cause the junction to be
blocked, and after a while his road would also become empty. Namely, by trying to
33
optimize his traveling time, he blocked the whole system. In the view of this
explanation, we can conclude that, this model is supporting evidence that individual
movements (movements which do not take care about the system structure and
condition) lead to system crash.
Future Work:
In the future, the roads (such as Zincirlikuyu and TRT Ortakoy Roads) that are
feeding the Etiler Region should be analyzed. The effects of blocking Behaviour in
Etiler Region on those roads are worth investigating.
Also in the current model inflow and outflows are assumed to include no random
Behaviour. However that is not the case in real life. Namely, to imitate the real life
situations in a more realistic manner, stochastic should be added to flows.
34
Appendix A
<onnneennee Kodak Fload Zincitikuyu Road-——->
/ Oven Road
>
&
ze
3
a
ra
El Torito :
sl
— 5
Torito Road---------> :
— —
poy Foye LHL
Figure 34 The Bird’s Eye View of Etiler Region
El Torito |
Figure 35 Etiler Region Direction of Traffic Flow
36
Appendix B
Figure 37 Graphical Function of Effect of Vehicles in Torito J unction on Inflow
37
Figure 39 Graphical Function of Effect of Torito J unction on Kodak Outflow
38
Normal Outflow Koda}
&
Vehicles in Kodak Road
Outgbw Kodak Inflow Kodak
kez J unction on Inflow Kodak
Effect of Torte | unction on Outflow Kodak
Remaining Capacity in KO of Kodak Road
Figure 40 Partial Model (Outflow and Inflow Kodak Active)
§ 2: vebicies in Kodak Road 2: Inflow Kodak 3: Outflow Kodak
1 40,00
2) 10,00
1 1 1
30.00 60.00 90.00 120.00
Graph 1 (Untitled) Time 10:41 05 Oca 2002 Cum
32
Figure 41 Partial Run (Inflow vs. C onstant Outflow) - Kodak Road Alone
Effect of Torito J unction on\Qutfiow Kodak
Remaining Capacity in Oven Road
Figure 42 Effect of Torito J unction on Kodak Outflow
39
Hy
37.60 760.00 112.60 160.00
Graph 2 (Untitled) Time 17:26 05 Oca 2002 Cum py
Figure 43 Partial Run (Effect of Torito Junction on Kodak Outflow Activated) - Outflow from
Oven Road is taken as zero
B 1: tost0w Kodak 2: Outflow Kodak
1: 72.004 hee
2 3.00
VAL, |
11.26 16.00
Graph 1 (Untitled) Time
17:25 03 Mar 2002 Paz Vy
Figure 44 Inflow and Outflow of the Kodak Road
40
& |: outtiow oven 2: Inflow Oven
1 Kohn Harts
2 2388
1 2.00
2 1194
1 0.00 {
2 0.00. + a
0.00 3.76 7.50 19.28 16.00
NeeFé Graph 1 (Untitled) Time
Figure 45 Inflow and Outflows of the Oven Road
17:24 03 Mar 2002 Paz |
41
Figure 1 Hypothetical Dynamic Behaviour of the Number of Cars in the Roads
Figure 2 Hypothetical Outflow Behaviour
Figure 4 Initial Stock-Flow Model of the Snake Tail Problem __
Figure 5 Normal InflowOven=0 __
Figure 6 Outflow Kodak = 0
Figure 12 Partial Run ( Effect of Torito Junction on Kodak Outflow Activated) - Outflow from Oven
Road is taken as six 15
Figure 28 Dynamic Behaviour of the Stocks-Advanced (Hesitated) Driver___
Figure 29 Dynamic Behaviour of the Flows-Advanced (Hesitated) Driver _
Figure 35 Effect of Remaining Capacity on Inflow-Graph of a Cautious Driver__
Figure 36 Effect of Remaining Capacity on Outflow-Graph ofa Cautious Driver
Figure 40 The Bird’s Eye View of Etiler Region 35
Figure 41 Etiler Region Direction of Traffic Flow 36
Figure 42 Graphical Function of Effect of Vehicles in Akmerkez J unction on Inflow Kodak 37
Figure 44 Graphical Function of Effect of Vehicles in Torito Junction on Inflow
Figure 46 Graphical Function of Effect of Akmerkez Junction on Oven Outflow _
Figure 48 Graphical Function of Effect of Torito Junction on Kodak Outflow
Figure 8 Partial Run ( Inflow vs. Constant Outflow)- KodakRoadAlone 39
Figure 10 Effect of Torito JunctiononKodakOutflow 39
Figure 11 Partial Run ( Effect of Torito Junction on Kodak Outflow Activated) - Outflow from Oven
Road is taken as zero 40
Figure 24 Inflow and Outflow of the Kodak Road__
Figure 25 Inflow and Outflows of the Oven Road
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43
