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OCR for page 55
55
Chapter Five
CRITICAL GAP AND FOLLOW-UP TIME
This chapter documents the analysis of factors that
may affect the critical gap and follow-up time
measurement results. The relationship between the
critical gap and follow-up time is then investigated.
Finally, the recommended values of critical gap and
follow-up time are given based on the analysis
results and eng~neenug judgment.
ANALYSIS OF FACTORS AFFECTING
CRITICAL GAP AND FOLLOW-UP TIME
Critical gap and follow-up time measurements for
each site were based on its unique intersection
geometry and traffic flow conditions. It was
important to identify those factors that could affect
the cntical -A and follow-up time values, so Hat
appropriate values could be recommended for
general use. The mean values of critical gap and
follow-up time were calculated according to
categories such as movement type, intersection
geometry, major street spool, and geographic sector;
however, this approach only provided a general
sense of what the values of the critical gap and
foDow-up bme under various U.S. conditions were.
Other investigation methods were used to identify
some of the factors which may cause larger
variations In the cntical gap and follow-up time.
Regression analysis was used to iden~ these
general factors. More specific analyses were then
conducted to investigate the effects of delay, major
street movement type, and vehicle type on the
critical gap and follow-up time.
Regression Analysis
A database was established for conducting the
regression analysis. To increase the sample size, the
entire intersection data set was divided Into several
sub-sections if the number of observations was
more in 200 for the cntical gap measurement. For
example, if more than 200 minor street vehicles
were observed In a 2-hour penod, the cntical gap
estimation was conducted for both the entire penod
end for each 15-minute period. Some measured site
attributes that could potentially affect cntical gap
were then added into the database. These attributes
included geographic sector, number of legs, multi-
lane or single lane on major street, number of lanes
a minor movement crosses, major street volumes,
major street right turn percentage, percentage of
traffic on the conflicting lane for multi-lane sites,
minor movement with shared lane or exclusive lane
condition, Once oftwo-way left-turn lane on the
major street, distance to upstream signals, grade of
minor approach, and average vehicle delay. The
database used for the regression analysis was
provided as Appendix m of Working Paper 16
(NCHRP 3-46, 1995~. Step-ur~se linear regression
was then conducted. Through the step-wise
regression process, those factors that did not show
a significant effect on the critical gap were
eliminated Regression models were then developed
to estimate the cntical gap given certain traffic flow
and intersection geometry conditions. Initially, the
regression analysis factors were selected based
purely on statistical significance. Based on these
preliminary results, He factors were judged and
included only if they were applicable to "real-worId"
conditions.
The independent variables considered are described
below.
Client is the minor movement for which the
critical gap was measured. MajET - major
sheet left tom, MinorLT - minor street left
turn, MinorRT - minor street left turn, and
MinTH - minor street through.
to is the mean critical gap measured for Mat
site estimated using the maximum
likelihood method.
S. D. is the standard deviation of He critical
gap for that site.
Obs is the member of observations of the
critical gap measurement, which is also the
number of subject vehicles observed.
The geographic sector information was described
OCR for page 56
56
using dummy variables.
.
.
.
Over
as combo when there are no exclusive
left turn lanes on the major street.
MajVolL Is the tragic volume from the left
side on the major street. For the major
street left turn movement, this volume
represents the total conflicting volume.
MajVolR is the traffic volume Mom the
right side on the major street.
%RT-Left is the percentage of right turn
movement on the major street from the leR.
%RT-Right is the percentage of right turn
movement on the major street from the
right.
%ConfLnVol is the percentage of traffic
volume on the conflicting lane at a multi-
lane site (calculated only for the through
movement). If traffic on the non-
conflichug lane also affects the minor street
driver, the higher the percentage on the
non-conflicting lane, the larger the critical
gap.
MajSp~ is the posted speed limit on the
major street, mph.
Excl is a dummy vanable that has be
value ~ if the minor approach has an
exclusive lane and O if the minor approach
has a shared lane.
1~11 N is a dummy variable that has be
value ~ if there is a two-way left turn lane
on the major street. Otherwise TVVETLN
has the value of 0. The existence of a
TWLTLN makes it difficult to accurately
define the gap event, because some of be
drivers simply merge into the TWLTEN
and seek another gap among the traffic
from the right side.
MajDir is a dummy variable that has the
value of ~ for one way and 2 for two way
on the major street.
UpSigL is the distance to the nearest
Upstream signalized intersection on be left
side, miles.
UpSigR is the distance to the nearest
upstream signalized intersection on be
right side, miles.
Gradle% is the approach grade at be
location of the stop line.
TurnAngle is another indicator of be
Sector NE is a dumm r variable which has
the value ~ if the site is In be northeast
sector and 0 otherwise.
Sector CE is a dummy variable which has
the value ~ if the site is In the central sector
and 0 otherwise.
Sector NWis a dummy variable which has
the value ~ if the site is in the Northwest
sector and 0 otherwise.
Sector SE is a dummy variable which has
the value ~ if the site is In the Southeast
sector and 0 otherwise.
If a site is In the SW sector, all other
variables (Sector NE, Sector CE, Sector
NW, Sector SE) are zero.
Independent variables include:
NoLeg is the number of legs of the
Intersection.
Mmaj is a dummy variable that has the
value of ~ if the site is Multi-lane. It has the
value of zero if it is a single-lane site. A
multi-lane site usually has more than two
through lanes on the major street for both
directions. For one way on the major street
with two lanes, Me site is defined as multi-
lane site for minor sheet left and nght turn
measurements, but as a single-lane site for
minor sheet through movement. A multi-
lane site usually requires that traffic of a
particular movement travel on more than
one lane, and only the vehicles on the
conflicting lane are considered when
measuring the critical gap.
InCrs is the number of lanes a minor
movement needs to cross. This is perhaps
a better indicator of the difficulty of a
movement maneuver compared to using
just a multi-lane or single lane. For
example, be existence of exclusive left turn
lanes on the major street requires Me minor
streetleR turn movement to cross atleast a
two-lane width and the minor street through
movement to cross at least a four-lane
width, which may prove to be more difficult
.
.
.
.
.
.
.
OCR for page 57
57
difficulty of a turning movement maneuvre.
This variable has the value of ~ for a small
angle and O for a perpendicular angle. It is
considered easier to make a turn at sites
with a small angle approach. Sites
CET305 and CET314 have small turn
angles. The nght turn movement at these
intersections are similar to a freeway on-
ramp and the critical gap is smaller.
Delay is the average delay for those minor
stream vehicles dming the period of critical
gap measurement, sec/veh.
Other factors thalmay affect the critical gap include
the city size and intersection location (rural, urban).
Because data regarding Me city size and location are
limited, deer have not been listed; however, it should
be noted that most of the intersections In the SE
sector are located In aural areas.
.
The following set of equations was initially obtained
based on the regression analysis:
Major Street Left Turn:
tc = 4~3 ~0.31*MMaj `~0~'
Minor StreetRight Turn:
tc = 6.27+0.82*A~aj-0.0345*°/R~e.~- 109
0.38*3Leg+0.123*G=de-2.43*Tun2~gle
Minor Street Left Turn:
tc = 6.79 -0.72 *3Leg-0.0216 *°/oRTLeJi+
0.22*G~de+0.39 *LnCrs- 1.03 *1WLTLN
Minor Street Through Movement:
it = 4.91+0.0611 *°/oRTLe.,~+0.104*(MaySpd-30) `1 1-v
The results of variance analysis of the above
regression equations are given in Table 23.
Through We regression analysis, Me following
conclusions were reached:
.
The following factors that have significant
effects on die critical gap were identified:
Anal - whether the intersection has multi-
lane or single lane approaches on the major
street
SORT - percentage right turn movement on
the major street
NoLeg - whether Me intersection has 3 legs
or 4 legs
Grade - the grade of the minor steam
approach
Tu~nAngle - whether the minor stream
movement has a small turn angle
The regression analysis included some sites
Ninth unusual geometric charactenstics,
such as a one-way major street or a two-
way left-turn lane. These may have caused
some bias in the regression equations;
therefore, judgement needs to be used in
recommending final cntical gap values.
Some factors were studied at a detailed level to
determine Weir effect on critical gap according to
each gap characteristic, including the accepted and
rejected gaps. The following factors were reviewed:
(110) .
the delay effect;
the effect of heavy vehicles;
the effect of major street traffic from Me
left side and Dom the right side of Me
minor street approach.
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58
Table 23. Summary of Variance Analysis of Different Regression Equations
, -; ... .. . ; . ;;;; -; ; ; .; .;; ................ .. ; ;; .;;;;; . -;;; - -;; -
~ . ~f,~ ............ ............... ~ . . ~= . ~
MaJor Left Turn
Regression 1 1.99 1.99 3.37
Residual 8S 45.37 0.53
Total 86 47.36
if 1 0.042 1[
Minor Right Turn
Regression S 140.73 28.1S
Residual 127 97.69 0.77
Total 132 238.42
R2 0.4S9
Minor Left Turn
Recession 1 5 78.96 1 15.79 16.78 1[
Residual 12 1 113.88 0.94
otal 1 126 1 192.84
R2 0.409
. ~
Minor Through
Regression 1 2 1 25.30 1 12.65 13.49
Residual 23 21.S8 0.94
Total 2S 46.88
R2 1 0.409
The Effect of Delay on Critical Gap
Investigation of any of single factor should ensure
that other factors Hat may affect Be critical gap
have held constant. For this reason, it is better to
analyze groups of sites with similar intersection
characteristics. Two "groups" of sites were selected
for analyzing the delay effect on the critical gap.
These sites were actually videotaped from two
intersections during different periods with a high
number of observations. The entire period was
divided into 15 minute sub-sections and the critical
gap was estimated for each sub-section. Figure 27
illustrates the critical gap measured for each period
and the associate average vehicle delay during that
period A regression line for aD of the data points is
shown In Figure 27. It can be clearly observed that,
vv~th~e increase of delay, the critical gap decreases.
This implies that an iterative process would be
required if this delay effect is to be taken into
account since delay is not known until an Initial
capacity estimate has been obtained. Alternatively,
36.ss
the cntical gaps could be related to the major street
flow so that the iteration could be avoided.
However, since the cntical gap is also related to
over factors, the relationship between major street
flow rate and cntical gap is always difficult to
establish.
Figure 27 reveals a concern regarding estimates of
critical gap and follow-up time obtained Tom
obsenabons curing understated conditions. Such
estimates are valid for estimating capacity and then
delay for given traffic conditions. However, they
may underestimate capacity and overestimate delay
at the same site when a site has higher traffic
volumes (near capacity) when important decisions
such as signal warrants or mitigation through lane
improvements may be necessary. Therefore, it is
recommended that site critical gap and follow-up
time be obtained (rather than using national
OCR for page 59
59
8.0
87.0
6.5
6.0
85.5
:~
~ 5.0
o
20 40 60 80
Average Delay, Seth
o Group1 o Group2 |
100 120
Figure 27. Analysis of He Delay Effect on Critical Gap
averages) when it appears that the critical approach
is near capacity, say v/c > 0.90. Another solution
would be simulation models.
The Effect of Vehicle Type on Critical Gap and
Follow-up Tune
Because of the limited number of observations, it
was impossible to estimate the critical gap for heavy
vehicles at each site. To conduct this analysis, the
gap events related to heavy vehicles were extracted
from each site and then aggregated based on
intersection geometry and movement type. The
maximum likelihood method was then applied to
calibrate the cntical gap. Table 24 lists the results.
As indicated, critical gaps for heavy vehicles are
significantly higher than those for passenger cars;
large variations in the values also exist for heavy
vehicles.
Table 24. Critical Gap Measured for Heavy Vehicles by Geometry and Movement Type
.. ... . . . .. . . .
art::::::: :-:: .~ ~
| 3-leg Single | LT r 7.2 | 2.9 T 166 | 6 l
RT 6 2.3 65 5.2
Leg Single LT 7.6 2.1 58 7.1
TH 6.3 2.9 24 6.4
RT 6.7 3 37 5.9
3-leg I LT I 1 2.1 1 9 1 7.2 |
Multi-lane RT 9.4 3 .8 104 6.9
Leg LT 9 4.5 13 7.4
Multi-lane TH 9 .5 6 .3 17 7.6
RT 8.2 2.7 25 6.8
Mulli-lane 1 LT | 9 | 3.3 1 22 1 7.3 l
TH 9.5 6.3 17 7.6
RT 8.8 ~3.3 90 6.8
~
Singl~lane LT 7.4 2.5 224 6 .4
TH 6.3 2.9 24 6.4
RT 6.4 2.7 102 5.5
I MajLT 5.5 1.7 202 4.1
The foHow-up time for heavy vehicles was also
calculated Whenever a heavy vehicle is involved in
a foDow-up time event, i.e., either Me first vehicle or
the second or both In a follo~up time event are
heavy vehicles, the foRow-up time was classified as
a heavy vehicle related follow-up time. Table 25
lists the results measured for heavy vehicles, for
passenger cars, and for all vehicles grouped
together. The measurements are grouped based on
intersection geometry and vehicle movement type.
Figure 2g illustrates the follow-up time
measurement results averaged for passenger cars
OCR for page 60
60
and heavy vehicles across all sites. It can be
observed ~at ~e foDow-up times for heavy vehicles
Table 25. Follow Up Time Measuremen~ Based on Vehicle Type and Geomeby
MajLT
are about ~ second higher than those for passenger
cars.
Mcan
S.D
Obe.
~......... ~::::: ::: <( :::3~::: :: :::: : :: :: ::::~::: :: ::: ::::: :~ :::: :: ::::::: :
: : :--: :-:-:::::::::::: :: :-:-:-:-:-: ~:::- : :::::::::: :::-::-::::: :-:::::::-::::: :::::::::::: .::::: - - ·:-:
.. - . . ~ . . . ~ . ~: . .- .. - . . ..........
22 3.1 2.3 l.9 ~
0.7 0.9 0.8 05 *
388 27 41S 2S *
33 4.0 3.4 3.4 43
12 12 12 2.4 1.1
7S@v 26 784 71 S.0
* 3.? 4.8
* * * 1.4 0.0
* * * 60 1.0
3.4 4.0 3.4 33 4.4
12 12 12 15 1.1
90g 27 83S 331 17
1.9
05
2S
35
2.4
76
3.7
1.4
61
3.4
15
~Ao
22
0.8
72
2.8
03
3.0
22
0.8
7S
2S
0.8
66
4.0
1.8
181
4.1
1.4
2S9
4.0
1S
SS1
3.2
0.1
_3.Q
2S
0.?
6'*~
2.2
0.8
SS1
3.1
0.8
33
MinRT
Meen
S.D
Obe.
3.1
1A
S49
3.1
1.4
13
3.1
1.4
S62
S.1
22
11
4.0
1.8
192
3.3
1.4
1S59
4.2
1.6
SS
MinTH
Mean
S.D
Obe.
*
* _
_
*
*
*
*
4.7
0.8
S
4.1
1.4
264
4.0
1.4
319
4.7
0.7
6
MinLT
Mean
S.D
Obe.
3S
13
2760
4.4
1.6
82
3.6
13
2842
2.2
2.1
18
4.1
1.6
S69
3.S
13
44SO
4.4
1.6
144
Notes: PC is passenger care; li~y i8 heavy vehicles. Categories in top row are geometric types.
-
3
o
5 _ 1
H^T ~T - ~ ~T
~-o~ 1
Figure 28. Weighted Average Values of Follow-Up Times
Measured for Passenger Cars and Trucks
The Effect of Direction of the Major Street Flow
In the max~m~ ~e~ihood procedure, ~e maximum
rejected gap of each vehicle dete~m~nes the lower
limit of the critical gap value. To investigate ~e
effect of different major street movements on the
minor street driver, the maximum rejected gaps were
extracted and classified by different ending gap
movement type. The data were based on the minor
street left tum movement at 3-leg intersections.
Table 26 shows ~e results. It can be obsened ~at
the major street left turn always yields ~e highest
value. This may be explained by the "slow down
effect" of the left turn movement. However, it is
clear that the through movement from the right side
always y~elds a higher value than that from the leR
side. This means that vehicles from ~e right side
usually put more pressme on~e minor street driver,
because the minor slleet driver needs to accelerate to
the desired speed if he or she decides to enter the
intersection.
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61
Table 26. Maximum Rejected Gaps by Different End Gap Movements For Minor Street LeD Tum
~-~1 it,
~ 1~1~ ~ 1~.~:-:$~:
1.7 1S7 * * * * * * 4.0 1.7
1.S 192 * * * * * * 4.0 1.S
1.6 1S3 * * * * * * 4.0 1.7
2.1 27 * * * * * * 4.8 2.1
1.9 109 4.0 2.3 123 4.8 2.7 18 4.6 2.1
2.0 108 4.0 2.7 117 S.9 2.3 49 3.9 2.0
2.1 111 3.7 2.0 183 S.4 2.1 18 4.4 1.7
2.1 44 3.9 2.2 1 4.3 1.3 4 4.1 1.8
1.9 901 3.9 2.3 424 S.1 2.1 89 4.2 1.8
, _
..... -.- -
:::::::::::::::::::- ::::::- ::-. 1
.................. 1
.................
SWT008
SWT017
SWTOlg
CET306
NWT402
NWT40S
SW1010
NET209
Average
3.9
3.7
3.6
4.3
3.4
3.7
3.7
3.3
3.7
240
196
303
176
27
73
9S
74
1184
A RELATIONSHIP BETWEEN CRITICAL
GAP AND FOLLOW-UP TIME
In the process of studying the critical gap and
foDow~up time measurements, a consistent ratio was
found between the critical gaps and the follow-up
times. Table 27 lists He ratio of the follow-up time
and the cntical gap measurements for each
movement. Figure 29 shows the relationship
between cntical gaps and the follow-up times
measured for all He sites. Figure 30 shows the
Table 27. Ratio of Critical Gaps and Follow-Up Times by Movement
distribution of the critical gap/follow-up ratio. It can
be observed that the ratios of He follow-up time and
the critical gap range between 0.4 and 0.9. The
majoribrofallobsewabons are around 0.6, which is
the value used in the old version of the HCM.
Further investigation of this relationship may be
warranted As mentioned earlier, the follow-up time
can be mms~1 directly in the field. Thus it may be
feasible for the practitioner to measure the follow-
up time and compute He critical gap from this
measured value.
.Mo ~tM ~ ~d ~ 06~Ob~o"
MajLT 0.55 0.08
MinRT 0.62 0.11
MinTH 0.58 0.07
MinLT . 0.58 0.12
All 0.59 0.11 .
14
25
32
79
OCR for page 62
62
10
9-
8.
on 7
~ 6-
F 5
~ 4-
_ 3
o
LO 2
O
o
Figure 29. Relationship Between Critical Gap and Follow-
Up Time
0.5
0.4
0.3
0.2
0.1
0
1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 11 ~ 1
1 1 1 1 1 ~51T1 11
1 1 1 1 1_111~: 1 1
. 1 1 1 1 11111 1 1 1 1
1 1 1 11115_1 1 1
. I I 1 1 111111111 1 1
. I 1~ 11111111 1 1
: I I I l An I
0 0.10.20.30.40.50.60.70.80.9 1
Or heal Gap/Follow-up Time Ratio
1
Figure 30. Distribution of Critical Gap/Pollow-Up Time
Radio
RECOMMENDED VALUES
Both the regression analysis and the more detailed
(microscopic levep analysis were conducted to
identify those factors that may affect He critical
gaps and follow-up times. Overall, Be regression
technique provides more insights regarding these
relationships. Generally, the following factors have
been found to have a significant effect on the cntical
gap and follow-up time:
.
.
major street volume or minor stream
vehicle delay
intersection geometry, including the
number of lanes on Be major street and the
number of legs of the intersection
right turn volume proportion on Be major
street
minor stream approach grade
movement turn angle
With the increase of major stream volume or minor
stream vehicle delay, He critical gap and follow-up
time tend to decrease. However, the critical gap
value cannot be reduced to the minimum threshold
determined by the follow-up tune value or the
minimum accepted gap value. With the increase in
the number of lanes on the major street or the
number of legs at the intersection, He critical gap
tends to increase because of the increased difficulty
of the movement maneuver. With the increase of the
right turn volume on the major street, the critical
gap tends to decrease because the right turn
movement on the major street causes less conflict to
the minor street vehicles compared to the through
movement. With He increase of the approach grade,
the critical gap tends to increase. With a small turn
angle, the movement maneuver is easier compared
to a perpendicular angle or a large angle, and the
critical gap tends to decrease.
Based on the regression analysis, the microscopic
analysis, and practical engineering judgement, the
recommended critical gap and follow-up time values
are listed in Tables 28 and 29. A factor addressing
the proportion of right turns is not included in this
set offactors. This factor is handled through the use
of weighUngLactors when calculating the conflicting
steam for a minor movement.
OCR for page 63
63
Table 28. Recommended Critical Gap Values
.... ; : : ,:.,:.,,,,., ,, ,,,,,,,,., , , ~, . , ,,, , ............................................................................................
., .................................................... ................................................ .. ...................................................................... ............................................
................................................. ....................... ...................... ...................... ............. ........... ,. ............ ..................... ......................
2~''""""'' ""'' 1''' '1 "'"I ~.t
CridchlGap
p~asse~ger Cars)
Adjustment Facto" _
Heavy Vehicle*
Grade%
Three-Leg
4.1
..... ,., ,,., ,,,,,,., ,, , ,,, , .....................................................................
,,., """""""' . ........................................... ...........
.................................. ......................... .. . . . .. .: . . . ..... .... .
.. 2 ~: . ~my'' , , . , ~....... ,
6.2 65 7.1 4.1 6.9 6.S
1.0 1.0 1.0 2.0 2.0 2.0
0.1 0.2 0.2 0.1 0.2
ft q
~v./
7.5
1.0
-
2.0
0.2
-0.7
Notes: *Combined critical gap is computed based on the proportion of passenger cars and heavy vehicles
Table 29. Recommended Follow-Up Time Values
,.. .. ......................................
Mowm ~ J~T ~
. .,
Follow-up Thne 2.2 3.3 4.0
(Passenger Caps)
Adjushnent Factors
Heavy Vehicle. 0.9/1.0 0.9/1.0 0.9/1.0
............ ;;;; ; .;;; ; ; .
3.S
0.9/1.0
Note: *O.9-second adjustment for single-lane sites and 1.0-second adjustment for multi-lane sites. Combined follow-up time is computed based on
the proportion of passenger cars and heavy vehicles
OCR for page 64
64
Representative terms from entire chapter:
major street
