TRAFFIC SIGNAL CONTROL ALGORITHM BASEDON QUEUING MODEL USING ITS SENSINGTECHNOLOGIES
Miho Asano1), Akira Nakajima2), Ryota Horiguchi3),
Hiroyuki Oneyama4) and Masao Kuwahara5)
1) Institute of Industrial Science, The University of Tokyo
Cw-504, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
Tel: +81 3 5452 6419 - Fax: +81 3 5452 6420 e-mail: asano@nishi.iis.u-tokyo.ac.jp
2) Institute of Industrial Science, The University of Tokyoe-mail: nkjm@sak.iis.u-tokyo.ac.jp
3) i - transport lab, 2-12-404 Ageba-cho, Shinjuku-ku, Tokyo 162-0824, Japane-mail: horiguchi@i-transportlab.jp
4) National Institute for Land and Infrastructure Management,
1 Asahi, Tsukuba-shi, Ibaraki 305-0804, Japane-mail: oneyama-h92bi@nilim.go.jp
5) Institute of Industrial Science, The University of Tokyoe-mail: kuwahara@nishi.iis.u-tokyo.ac.jp

SUMMARY
This study has developed a new traffic signal control algorithm with ITS sensing technologies.
These technologies can measure the travel time of individual vehicle, so this algorithm canevaluate delay and search the optimal combination of signal parameters to minimize the total
delay in a network, based on a queuing model. We examined the effect of this algorithm withsimulation experiment and plan to implement it as a real-world system for a demonstration in
the next ITS World Congress in Nagoya, 2004.

INTRODUCTION
The purpose of this research is to develop a new traffic signal control algorithm that canminimize total delay directly measured by Intelligent Transport Systems (ITS) sensing
technologies.
Technologies of traffic sensing and communications have been developing rapidly by the
progress of ITS. We may expect that the technology to identify individual vehicles by image
processing or uplink from floating cars will be much facilitated in near future. These
technologies allow us to directly measure the travel time of individual vehicle in a desired
section, which has not been utilized in the conventional signal control methods such as
SCOOT or SCAT.
The algorithm proposed here uses the travel time of each vehicle in order to calculate total
delay of each approaching link at a signalised intersection(1). For each cycle of a signalisedintersection, the algorithm updates signal parameters, i.e. split, cycle and offset, in terms of
the improvement on delay at the intersection. As the changes on parameters are discretized
into sufficiently small unit, e.g. 2 or 3 seconds, and are limited one or two units at maximum,
the algorithm can find the optimal parameter set by solving combinational optimisation1

problem. In order to reduce the calculation cost, the algorithm assumes ‘common cycle
length’ to be shared in the signalised intersection in a “sub-area”, which normally consists ofseveral intersections.
The most remarkable feature of this algorithm is adaptation. Most of the conventional signal
control methods would have some key parameters that greatly affect performance. For
instance, the program selection control used in Japan has pre-determined sets of signal
parameters, which should be carefully calibrated to achieve better performance, and selects
appropriate parameters for the traffic condition at the moment. However, after a long-run,traffic situation may be different from those for which traffic signal was calibrated, and theprogram selection control may not work well without continuous maintenance. In thisalgorithm, all key parameters are directly measured and updated at every cycle, therefore thealgorithm can maintain the expected performance for a long time.
We have a plan to implement the algorithm proposed here to a real-world system. In thisexperiment, AVI (Automatic Vehicle Identification) sensors are used by taking into accountpractical use of the algorithm, but we may expect to use cheaper sensors based on atechnology of tracing vehicle trajectories by image processing in the future(2). As ademonstration at the next ITS World Congress in Nagoya, 2004, the system will be operated
for the Nagoya-Nagakute road, which is also test bed for this experiment.
In the following chapters, we will explain how to estimate the delay from the arrivaldeparture cumulative flow diagrams (called as “cumulative curves” in this paper) and how thecontrolling algorithm proposed here will modify signal parameters cycle by cycle. The lastpart of this paper will report the performance of the algorithm through computational
experiments with traffic simulation.

DELAY ESTIMATION OVER A LINK
Data requirements and arrangement of sensors
The algorithm requires two kinds of information to figure out the cumulative curve for a
subjective link; i.e. the time profile of vehicle counts of departure traffic from the intersection
and the travel times of individual vehicles over the link. Especially for departure vehicle
counts, data of all the vehicles are required to know the amount of traffic flow. In order toobtain those information, we are using two types of sensors, ultrasonic detector and AVI
sensor, which are currently installed at the road in Japan. AVI sensors can measure the travel
time of each vehicle, and ultrasonic detectors can provide the profile of vehicle counts moreaccurately than AVI sensors.

A

D

Cumulative
number of vehicles

Ultrasonicdetector

AVI

Free flow travel time from A to D
Passing time at A
Passing time at D

Delay
time

Travel time from A to D

Figure 1 Cumulative curves to estimate delay and arrangement of sensors

2

The required arrangement of sensors is shown as Figure 1. To exclude the influence of thequeue extended from the downstream intersection, these detectors are put the upstream ofeach link. AVIs are set on both point A and D. Data obtained from AVI is the passing timeand ID number of each vehicle. Matched by the ID numbers between two points, the travel
time of the vehicles can be measured. Ultrasonic detectors are put on point D to get the
number of vehicles which passed the intersection.

Drawing cumulative curves from the observed data
This algorithm measures the delay to optimize signal parameters using cumulative curves.
The cumulative curves of vehicles which passed the point A and D are drawn as right part of
Figure 1. The horizontal distance between these curves shows the travel time from A to D ofeach vehicle. If the curve of passing time at A is shifted to right as much as the free flow
travel time from A to D, the delay caused by signal controller can be estimated as the graycolored area.
The throughput volume at the upstream end of each out-flow link of an intersection ismeasured by ultrasonic detector. Synchronizing the cumulative throughput with the signal
phase of the intersection, the cumulative departure curve of each approach can be obtained(the D-curve in Figure 2).

Cumulative numberof vehicles

Then, cumulative arrival curves of each approach of an intersection are estimated. Suppose
each link in the subjective area has the AVI sensors to identify individual vehicle at its
upstream end as shown in Figure 1, the travel time of a sample vehicle can be measured overthe link. By subtracting the free flow travel time of the link, the delay of an identified vehicle
is obtained.
Individual delay from A to D

A-curve(estimated)= travel time from A to D

- free flow travel time from A to D
Passing order at D

D-curve(measured)
Total delay F

Passing time at Dt

Figure 2 Cumulative curves

As shown in Figure 2, the arrival time of each identified vehicle can be estimated by shifting
corresponding point on D-curve to the left (= past) as much as the delay of the vehicle. SinceFirst In First Out (FIFO) principle is not always kept in the real world, the estimated arrival
points of identified vehicles are to be sorted by time, and we estimate the cumulative arrival
curve (A-curve) by interpolating those arrival points. The total delay per cycle F at thisstream is represented as the sum of the delay of each vehicle, which means the area enclosedby A-curve and D-curve.
The free flow travel time of each links is estimated in order to take out the delay from travel
time. From the AVI data, the travel time of each vehicle from point A to point D in Figure 1
can be measured. The histogram of the travel time in Figure 3 can be divided into two parts;
one shows travel times of vehicles which can pass this link without stopping, and the othershows the travel times of vehicles which stop by red light. The free flow travel time TF isdetermined as the mode of the vehicles which can pass this link without stopping. The
difference between travel time and free flow travel time is the delay caused by traffic signals.
3

TF

90
10
0
11
0
12
0

70
80

50
60

30
40

0
10
20

Number of vehicles

12
10
8
6
4
2
0

Travel time (sec.)

Figure 3 An example of Travel time statistics to estimate TF

PARAMETER MODIFICATION TO MINIMIZE DELAY
Outlines of parameter modification
In this chapter, the signal parameters to minimize the total delay will be determined. Now wewill represent the splits and offsets in second in this paper while they are usually represented
as the percentage of cycle time.
In this algorithm, changes on parameters are discretized by small intervals. We search theparameters from the neighbourhood of the present parameter set, estimating the change ondelay in each case. It is result in combinational search of the signal control parameters; split
and cycle times of each intersection and offsets of each link. The intervals of each parameterare ∆s, ∆c and ∆o. We have 3 patterns to choose split of one intersection, and same as offset
and cycle times, which are the cases that parameter increases, decreases and keeps the present
value. However, the number of choice set is too large to search at this time. To reduce thenumber of combination, we have some assumptions.
The first assumption is that the traffic conditions in the next cycle will be the same as in thelatest cycle. This assumption makes us possible to use the cumulative curves drawn in theprevious chapter as the curve of the next cycle if the signal parameters keep the same value as
the latest cycle. The changes on delay when signal parameters change are estimated using the
cumulative curves of the latest cycle.
The second, third and fourth assumptions are to reduce the number of combination of searcharea. The second is that we consider a group of intersections which share the common cycle
time. The group is called as “sub-area” in this paper.
The third is that the search of offset is done only when the common cycle time keeps thepresent value. That’s because the offsets naturally change when cycle time changes and so
we only consider the influence of changes on offset caused by change of cycle time.
According to the third assumption, the change of offset with cycle increase can be decided.
Here, we only consider ‘positive’ offset values; i.e. when the offset of one direction is x [sec.],the other direction must be (C-x) [sec.], where C is a cycle length. Suppose the common
cycle length will be increased by ∆c as shown in Figure 4, the offset values will be x and (Cx+ ∆c), if they are untouched. However, it is quite unfair to impose a burden on one direction.
Therefore, the changes on offset will be shared for both directions as (x/C) ∆c and ((C-x)/C) ∆c.
Remember the sum of both changes must be always ∆c.

4

B

A

C+∆c

C
A
x
B

C-x

C-x+∆cx

Figure 4 Changes on ‘positive’ offset values when the cycle length changes

The fourth assumption is that the splits are decided first, with no relation to change on offsets
and cycle time. Offsets and cycle time will be determined after choosing splits.
For instance, suppose there is a network consists of N intersections and N-1 links. Since theparameters can take 3 patterns; no change, increase and decrease, the number of parameter
combination is 33N-1. If the whole network is considered as one sub-area, the number of
combination with these assumptions can be reduced to 4 * 3N-1 + 2.

Estimation of changes in delay
Applying the first assumption, we may evaluate the improvement on the delay of each stream
when the signal parameters change. Figure 5 explains that the case when the split of one
green phase of an intersection changes by ±∆s. The shape of arrival and departure curves
were borrowed from the latest observation. If the split increases ∆s to the current value, the Dcurve will shift to the upper dotted line, because the green phase of the next cycle starts withthe lag ∆s.
The case when the offset of a link changes by ±∆o is similar to this. The estimated curves areshown in Figure 6. Since the number of combination of the changes on split and offset is 3 x
3, the 3 x 3 types of the curves and the changes on delay are estimated. The changes on delay
example: when only split changes
Split = S + ∆s
Split = S - ∆s
In the present
Split = S∆s∆s
Figure 5 Estimating the changes on delay when split S is modified
example: when only offset changes
Offset = O - ∆oOffset = O + ∆oIn the present
Offset = O

∆o

∆o

∆o ∆o

Figure 6 Estimating the changes on delay when offset O is modified

5

Table 1 Matrix of changes on delay by the change of split and offset

Change of offset−

 ∂F ∂F 
 −, − ∂s ∂o  ∂F 
 0, −  ∂o 
 ∂F ∂F 
 −, + ∂s ∂o −

Changeof split

0
+

0

+

 ∂F 
 − ,0  ∂s ∂F ∂F 
 +, − ∂s ∂o  ∂F 
 0, +  ∂o 
 ∂F ∂F 
 +, + ∂s ∂o 0 ∂F 
 + ,0  ∂s

of each stream can be formed as a matrix like Table 1. Here, F is the total delay of the latest
cycle at this stream. This table shows, for instance, the estimated value of the change ondelay when the split and offset increase ∆s and ∆s is represented as  ∂F , ∂F  ⋅  ∆ s  , where ∂s+∂o +   ∆ o  ∂F ∂F  is put in the cells that both row and column are “+”. From the matrices of each +, + ∂s ∂o 

stream, the parameters to minimize the total delay of the sub-area are decided.

Modification of parameters without changing common cycle length
Modification of splits
Assume that the offsets and the common cycle time do not change. In the case that the splitof Stream 1 and 3 in Figure 7 is increased, the change of the delay at this intersection is the− ∆s 
 of Stream 2 and 4. The  0 sum of the  ∂F ,0  ⋅  ∆ s  of Stream 1 and 3 and of the  ∂F− ,0  ⋅ 
 ∂s

+

 ∂s

  0 opposite is in the case that the spilt of Stream 1 and 3 is decreased. If the split doesn’t change,
the change on delay is equal to 0. Split is chosen from these 3 cases which minimize thedelay of the intersection.
Stream 4

+∆s

-∆s

0

-∆s

0

+ ∆s

Stream 1

- ∆s

+ ∆s
Stream 3

0
0

+∆s

- ∆s

Stream 2

Figure 7 Three patterns of split change at an intersection

+∆o

0

-∆o

-∆o

0

+∆o

Figure 8 Three patterns of offset change at a link

Modification of offsets
For each link, there are 3 patterns to be chosen as Figure 8; offset to east increases ordecreases, and offset doesn’t change. In this case, change on split in each intersection is6

obtained from the previous process, modification of splits. From the matrices of changes ondelay, choose one pattern to minimize the delay in the link.

Modification of parameters with changing common cycle length
Modification of splits
The way of modification of split is exactly the same as in the case without changing common
cycle time.
Modification of offsets and common cycle time(1) Basic change on delay
Offset and common cycle time are decided simultaneously. Suppose the subjective
intersection and the upstream intersection share the same cycle time. The shape of the arrival
curve of the subjective stream is affected by the shape of the departure curve at the upstream
intersection. Since the departure curve at the upstream intersection may change by changing
the cycle time, we assume that the arrival curve at subjective intersection will also change inaccordance with change in departure curve of upstream intersection, as common cycle timechanges by ∆c as shown in Figure 9. In Figure 9, G, R and L show the effective green lengthof the subjective stream, sum of the effective green length of other streams and loss time of
the intersection respectively. If departure curve at subjective intersection also changes similar
to the arrival, the changes on delay in the whole sub-area per unit time are shown as follows.
2

F∆
1
1
 C + ∆c 

⋅
F


∑i  i  C  ⋅ C + ∆ − Fi ⋅ C  = ∑i Ci 2 cc(Eq. 1)

Here, Fi is the total delay in stream i per cycle.
In fact, departure curves don’t change exactly similarly. There are three properties whichcorrect the departure curve. The first is “correction of offset” which occurs because of thechanges on offsets as the common cycle time changes. The second property is “correction ofloss time” which occurs because the loss time does not change with cycle time. The last one
is “correction of saturation flow rate decline.” If the cycle time gets larger, it is known thatsaturation flow rate may decline at a certain point because the right turners spill over the
exclusive right turn lane and block another lane, or because queue from the downstream mayreach the intersection.
Cum. # of
Present cumulative
vehicles Enlarged curve

Cum. # of vehicle
Enlarged departure
curve in similarity

similarly

L+R
 C + ∆c 
( L + R) × 

 C 

G

B

C

Actual departure
curve when cycle time
= C + ∆c

time

time

 C + ∆c 
G×

 C 

A

Figure 9 Enlarged cumulative curves
as common cycle time increases

Figure 10 Correction of saturation flow rate
decline(2) Correction of offset and loss time
Consider the difference of delay between the basic curve and the curve which corrected taking
into account of the property of offset and loss time. Since the loss time is constant, the green

7length does not change similarly by cycle. Moreover, the offset changes byx∆ c from theCassumption. Combining these properties, the correction of offset and loss time is obtained.
The proof is in the appendix.
  ∂F ∂F  ∆   ∆ (C + ∆ c ) 
GL
x 
+


Coro _ l = ∑∑   + , +  s   ⋅ c( Eq. 2)+22C∂o  ∆ o   jij   ∂s (C − L)∆ s ∆ o 
Where i and j show the j th stream at i th intersection.
(3) Correction of saturation flow rate decline
When departure curve is enlarged similarly, the curve will be drawn as AB in Figure 10.
However, the actual departure curve is drawn as AC, because the saturation flow rate may
decrease as the green length gets longer. The gray area in Figure 10 should be the correctionof saturation flow rate decline.

Choice of the best parameter set
Now the combination of signal parameters which best decreases the total delay in the wholesub-area is chosen. At first, splits of each intersection are decided. The total change on delaycaused by change on offset and common cycle time is obtained as follows.
In the case without changing cycle time, the total change on delay is the sum of the change ondelay of each link whose parameters are chosen to minimize the delay. In the case withchanging cycle time, the total changes on delay are expediently represented as the sum of thebasic changes and corrections of each stream. Comparing these three cases of no change,
increase and decrease, the combination of common cycle time and offsets that minimize thetotal delay is chosen as the parameter of the next cycle.

APPLICATION USING TRAFFIC SIMULATION “AVENUE”
Study Area
The performance of this algorithm is examined using the traffic simulation model AVENUE(3).
The study network is selected along the Nagakute line in the Nagoya central district. The
network has a tree shape and consists of 10 links and 11 intersections. The places of thesensors are shown in Figure 11. We basically referred to the actual place of sensors andadded sensors on the arterial road if they are not in the link. The delays of the minor roadwithout any sensors are regarded as 0.
The simulation time is 2 hours and 10 minutes. The first 10 min. is used for warming up thesimulation to distribute vehicles throughout the study area, and after that the algorithm started.
We have three cases to examine whether parameters can follow the change of traffic demand.
In case 1 the degrees of saturation of intersections stay between 0.5-0.7, and 0.5-0.95 in case2 as shown in Figure 12. In case 3, the degrees of saturation are the same as case 1, and thesaturation flow rate decreases 20% in the latter half of the simulation time. The initialization
values of signal control parameters are as follows. Common cycle time is 120 sec, and splitsof arterial road are set between 53% and 62%. The offset of every link are set to 0%.
In this experiment, ∆c is set as 4 sec, and ∆s and ∆o are set as 2 sec. As restrictions, theallowable range of the common cycle time is between 80 and 160 sec, and the minimum

8green lengths of each stream are 30 sec. We regarded the whole study area as one sub-area.
The simulation results are evaluated in terms of total delay of each intersection and history of
the signal parameters as evaluation indices.

200m

2.9km
(2)
140m

(1)200m

580m(3)450m(4)255m

3
2

Uchikoshi

4

Hoshigaoka

300m

1

(5)390m(6)150m(7)280m

400m

6

7

8

5

10
470m

610m

9

11
：AVI
：Ultrasonic detector

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

8:50

8:40

8:30

8:20

8:10

8:00

7:50

7:40

7:30

7:20

7:10

7:00

case 1,3
case 2
6:50

degree of saturation

Figure 11 The study network along the Nagakute line, Nagoya

time

Figure 12 Degree of saturation at Uchikoshi intersection

Results of the simulation experiment
Total delay
Figure 13- 15 show the total delay of the major road in the whole area and of the minor roadat Uchikoshi and Hoshigaoka intersection in each case. In those figures, “without algorithm”
means the case that the signal parameters keep the initialization value in the whole simulation.
70

without algorithm(major road)with algorithm(major road)without algorithm(major road)delay (veh.*hr.)

60
50
40

with algorithm(minor road)30

without algorithm(minor road)with algorithm(minor road)

20
10

without algorithm(minor road)with algorithm(major road)

6:
50
7:
00
7:
10
7:
20
7:
30
7:
40
7:
50
8:
00
8:
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9:
00

0

6:

50
7:
00
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00
8:
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30
8:
40
8:
50
9:
00

delay (veh.*hr.)

18
16
14
12
10
8
6
4
2
0

Figure 13 Total delay in the Case 1

Figure 14 Total delay in the Case 2

9without algorithm(major road)

25
delay (veh.*hr.)

20
15
10

with algorithm(major road)without algorithm(minor road)with algorithm(minor road)

5

40

50

00

10

20

30

40

50

00

7:

7:

8:

8:

8:

8:

8:

8:

9:

20

30

10
7:

7:

00
7:

7:

50
6:

0

Figure 15 Total delay in the Case 3

Here, the delays decrease in every case. Especially in the case 3 (Figure 15), although thedelay without this algorithm increases compared to the case 1, the delay is almost the same as
the case 1 with the system. It follows from this that the algorithm can control considering thedecrease of the saturation flow rate by measuring not only the arrival flow but also thedeparture flow.
Signal parameters
Figure 16 - 21 show the changes on signal parameters in each case. The offset means theoffset to the east. In Figure 16, the cycle time gradually decreases in spite of demand change.
It suggests that the initialization value of the cycle time was too large. It also appears thatcycle time swings in every cycle. This is because the parameters are affected by the
randomness of demand, and because cycle time is required to modify to update the offsets.
In the case 2 and 3, the cycle time increases even after the traffic demand decreased. That is
because the queue occurs even if the cycle time increases to the maximum value.
There are some links whose offset constantly increases / decreases. The reason is that ∆o is
not enough large to be settled in the simulation time. To apply this algorithm to actual field, it
is better that the initialization value of offset needs to be near the optimum value to someextent or that ∆s, ∆o and ∆c should increase to follow the traffic condition earlier.
[sec.]
140

40.0%

120

30.0%

Link (3)

100

20.0%

Offset

80
10.0%

60
40

-10.0%

8:52

8:41

8:30

8:20

8:09

7:58

7:46

7:35

7:24

7:13

7:02

-20.0%

6:50

0

Time

0.0%
6:
50
7:
02
7:
13
7:
24
7:
35
7:
46
7:
58
8:
09
8:
20
8:
30
8:
41
8:
52

effective green lengthof primary direction
cycle time

20

Link (2)

Link (1)
Link (4)-30.0%

Figure 16 Cycle time and green time lengthof the main direction at Uchikoshi
intersection in the Case 1

Figure 17 Offset of each link in the Case 11030.0%
20.0%

Link (2)

6:
50
7:
00
7:
09
7:
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7:
29
7:
38
7:
49
7:
59
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8:
49

Offset-30.0%-40.0%

8:47

8:31

8:15

8:01

7:49

7:36

7:25

7:13

time

-10.0%

Link (1)

Link (4)-50.0%

Figure 19 Offset of each link in the Case 2
25.0%
20.0%

Link (3)
Link (2)

15.0%
5.0%

6:
50
7:
00
7:
09
7:
19
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29
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38
7:
48
7:
57
8:
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54

-5.0%
-15.0%

8:54

8:38-20.0%

8:24-10.0%

8:11

effective green length ofprimary direction
cycle time
7:59

Time

0.0%

7:48

7:36

7:25

7:13

Offset

10.0%

7:02

[sec.]
180
160
140
120
100
80
60
40
20
0

0.0%

-20.0%

effective green lengthof the primary direcrion
cycle time

Figure 18 Cycle time and green time lengthof the main direction at Uchikoshi
intersection in the Case 2

6:50

Link (3)

10.0%

7:02

6:50

[sec.]
180
160
140
120
100
80
60
40
20
0

Figure 20 Cycle time and green time lengthof the main direction at Uchikoshi
intersection in the Case 3

Link (4)
Link (1)Figure 21 Offset of each link in the Case 3

CONCLUSIONSThis study has developed a signal control algorithm based on a traffic model that can directly
evaluate total delay measured by sensors. The result of the simulation experiment shows that
this algorithm can really reduce the delay of the whole network in each case without any
parameter tuning.
For future study, we plan to add an optimization system for case of saturated traffic condition.
To follow the traffic condition earlier, we will improve the system applying the forecast ofarrival flow. Now we are discussing for practical use through field experiment at this network.
We are also planning to have a demonstration of this system in ITS World Congress inNagoya, 2004.

REFERENCES(1) M. Koshi, et al.; “A Traffic Signal Algorithm using ITS Sensing Technologies”,
Proceedings of Infrastructure Planning, Vol.25, 2002 (in Japanese)(2) S. Kamijo, et al.; “Occlusion Robust and Illumination Invariant Vehicle Tracking for
Acquiring Detailed Statics from Traffic Images”, IEICE Trans. on Information and
System, Vol. E-85-D, No.11, November 200211

(3) R. Horiguchi, et al.; “A Network Simulation Model for Impact Studies of Traffic
Management ‘AVENUE Ver.2’ ”, Proceedings of the Third Annual World Congress onIntelligent Transport Systems, ITS America, 1996

APPENDIX---CORRECTION OF OFFSET AND LOSS TIME
Vector φ = {s, o} is defined in which s is the split and o is the offset of the subjective stream.
 ∂F ∂F  ∂F ′ ∂F ′ ∇F = 
,,, ∇F ′ =  ∂s ∂o  ∂s ∂o (Eq. 3)F ′ is the delay of the curve which is enlarged similarly by (C + ∆c) / C.
Because of the similarity assumption,
∂F
∂F ′
=A
,
∂s
∂s

∂F
∂F ′
=A
∂o
∂o

∴ ∇F ′ = ∇F ⋅ A where A =C + ∆cC

If the curve is simply enlarged, the green length would begreen length is(Eq. 4)C + ∆cG . However, the actualC

C + ∆c − L
G because the loss time is independent of cycle. The correction
C−L

value of split ∆s’ isGL∆ c′  C + ∆c − L C + ∆c 
∆s = −G =C C (C − L) C−L(Eq. 5)

Then the cumulative departure curve should be shifted like Figure 22. The gray-colored area
is shown as the correction of loss time.
The actual change on offset ∆o’ isx∆ c from the assumption.
C

Therefore, the correction of offset and loss time of one stream per cycle is shown as following
equation, using ∆ ′φ = ( ∆ ′s , ∆ ′o ) .
∇F ′ ⋅ ∆′φ = A∇F∆ φ ∆−φ1∆′φ = ∇F∆φ ⋅
(C + ∆ c )∆ c  GL∆ s+ x∆ o 
22 
2C (∆ s + ∆ o )  (C − L)(Eq. 6)

Finally, the correction of offset and loss time of whole the sub-area per unit time in the caseof increase of the cycle time is obtained.
  ∂F ∂F  ∆  ∆c GL∆ s+
Coro _ l = ∑∑   + , +  s   ⋅ 2+ x∆ o ( Eq. 7)22 ∂o  ∆ o   j C (∆ s + ∆ o )  (C − L)ij   ∂sWhere i and j show the j th stream at i th intersection.
Corrected curveconsidering loss timeCum. # of
 C + ∆c − L vehicles L + R ×  C − L 

Enlargedsimilarly
 C + ∆c − L G× C−L time
 C + ∆c 
( L + R) ×  C Figure 22 Correction of loss time

 C + ∆c 
G ×

 C 

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