RECOMMENDED CHANGES TO THE PROGRESSION FACTOR
IN THE 2000 EDITION OF THE HIGHWAY CAPACITY MANUAL
by
Daniel John Sailer
B.S. University of Maryland, 1994
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Civil Engineering
2004
This thesis for the Master of Science
degree by
Daniel John Sailer
has been approved
by
Bruce Janson
Date
Sailer, Daniel John (M.S., Civil Engineering)
Recommended Changes to the Progression Factor in the 2000 Edition of the
Highway Capacity Manual
Thesis directed by Professor Bruce Janson
ABSTRACT
This study investigates the validity of the supplemental adjustment factors for
the effects of platoon arrivals (fpa) on traffic delay at a signalized intersection.
The objective is to look at potential problems with the default values listed in
the Highway Capacity Manual (2000 edition) for the six different arrival types
(AT) and to recommend changes to these values if found to be warranted. It
is found that the recommended changes to the supplemental adjustment
factors for platoons arriving during green (fpa) will bring the calculated control
delay (d) more in line with the control delay measured in the field.
This abstract accurately represents the content of the candidates thesis. I
recommend its publication
Signed
Bruce'Janson
hi
DEDICATION
I dedicate this thesis to my wife, Barbara, for her support and patience while I
was completing this degree.
ACKNOWLEDGEMENT
Special thanks to my advisor, Bruce Janson, for his guidance during this
thesis process. Thanks also to Sarosh Khan and Bruce for their exceptional
instruction over the past two years.
CONTENTS
Figures..........................................................vii
Tables..........................................................viii
Chapter
1. Introduction and Overview.......................................1
1.1 Estimation of Calculated vs. Measured Delay....................2
1.2 Understanding the Signal Network...............................5
2. Analysis of fpa.................................................8
3. Recommendations................................................18
3.1 Summary.......................................................19
Appendix
A. AT1 Through AT6 Field Data and Calculations (on Disc).20
References........................................................21
VI
FIGURES
Figure
1.1 SimTraffic Signal Network.........................................7
2.1 Normal Probability Density Function for AT1..................10
2.2 Linear Regression Applied to AT 1 Flow Rate and fpa.............11
2.3 Normal Distribution and Flow vs fpa Graphs for AT2 Through
AT5...........................................................1316
vii
TABLES
Table
1.1 Results Summary Table for AT1 .............................5
2.1 AT1 Table of Sample #, Flow Rate, and fpa Values Sorted by
Increasing fpa Value.......................................9
2.2 Summary of fpa Population Mean Analysis of AT1 Through AT5.12
2.3 Summary of Correlation Analysis Between Flow Rate and fpa for
AT1 Through AT5...........................................12
2.4 fpa Values for Trials in AT6..............................17
2.5 Summary of AT6 Results for fpa.............................17
3.1 Recommended HCM Exhibit 1612.............................18
3.2 Recommended New Figure....................................19
viii
1. Introduction and Overview
Control delay (d) is the primary measure of performance used when
assessing the operational performance of a signalized intersection. The
2000 edition of the Highway Capacity Manual (HCM 2000) provides and
explains the formulas it prescribes to estimate control delay. The primary
formula for determining control delay per lane group is given by equation
(1.1):
d = di(PF) + d2 + d3
(1.1)
where,
d = control delay per vehicle (seconds/vehicle)
di = uniform control delay based on uniform arrivals (seconds/vehicle)
PF = uniform delay progression adjustment factor used to account for
coordinated timing plans
d2 = incremental delay used to account for random arrivals (secs/veh).
d3 = initial queue delay used to account for delay caused by leftover
vehicles in the queue from one analysis period to the next.
Each variable listed in equation (1.1) has estimation formulas of their own
which will be discussed later in this report. In some cases, these variables
can be measured in the field.
Appendix A of Chapter 16 in the HCM explains a procedure by which
to measure control delay (d) in the field. The premise of this report is that
the value of d as measured in the field using the procedures in Appendix A
ought to closely approximate the value of d as estimated by equation (1.1)
when computed with measured values for all adjustment factors that can be
measured in the field.
Because the formula for uniform delay (di) is widely accepted as an
accurate depiction of delay for the case of uniform arrivals1, the progression
1
factor PF (specifically fpa) was chosen to be evaluated due to the fact that the
HCM 2000 only provides one default value for fpa to be used for each arrival
type. The default values for fpa as listed in the HCM are as follows:
Arrival Type (AT): 1 2 3 4 5 6
Default fpa value: 1.00 0.93 1.00 1.15 1.00 1.00
The objective of this report is to assess the default fpa values as provided by
the HCM and to recommend changes to these values if warranted.
1.1 Estimation of Calculated versus Measured Delay
The two methods of estimating control delay d mentioned earlier will
be referred to as calculated delay versus measured delay. Calculated delay
involves the following formulas. If a given factor can be field measured, it is
listed with an asterisk (*) in front of it. The details of how to measure these
variables will not be discussed, but are outlined in the HCM:
PF = [ (1 P)fpa ] / [ 1 (g/C) ] (1.2)
Rp = P / (g/C) (1.3)
di = [ 0.5C(1 g/C)2 ] / [ 1 min(1 ,X)g/C ] (1.4)
c = s(g/C) (1.5)
d2 = 900T[ (X1) + V[ (X1)2 + 8klX/cT] ] (1.6)
where,
PF = progression adjustment factor
fpa = supplemental adjustment factor for platoon arriving during green
*P = proportion of vehicles arriving on green
g = effective green time (seconds). Under pretimed signal
operations, g is taken as the actual controller green time.
C = Signal cycle length (seconds)
Rp = Platoon ratio
di = uniform control delay (seconds/vehicle)
X = voiumetocapacity ratio (v/c) or degree of saturation of the lane
group
c = capacity of lane group (vehicles/hour)
*s = saturation flow rate of the lane group (vehicles/hour)
d2 = incremental control delay (seconds/vehicle)
T = duration of analysis period (hours)
k = incremental delay factor that is dependent on the controller
settings. It is equal to 0.5 for a pretimed signal operation.
I = upstream metering adjustment factor
Measured delay involves the following formulas, again with factors that can
be field measured proceeded by an asterisk (*):
dvq = ls(XViq/Vtot)0.9
(1.7)
# vehicles stopping per lane per cycle = Vstop/(Nc x N) (1.8)
FVS = Vstop / Vtot
(1.9)
dad = FVS(CF)
(1.10)
d = dvq + dad
(1.11)
where,
Is = survey count interval (seconds)
*
Viq = # vehicles in queue after each Is
dvq = time in queue per vehicle (seconds)
3
*Vtot = total vehicles arriving during survey period (vehicles)
*Vstop = # vehicles that stop during each Is (vehicles)
Nc = # of cycles surveyed
N = # of lanes in lane group
FVS = Fraction of vehicles stopping
dad = accel/decel correction delay (seconds)
CF = accel/decel correction factor
Because measured delay does not account for residual queue delay
d3 as estimated by equation (1.1), the observations made to obtain
measured delay must not contain any vehicles left in queue from a previous
signal phase. This reduces formula (1.1) to the following:
d = di(PF) + d2 (1.12)
Appendix A in Chapter 16 of the HCM explains how to field measure
delay at signalized intersections. Due to the complexity of measuring field
values for numerous parameters, only a simple signal network was studied
for this report. The procedure used consisted of collecting all fieid values for
those factors that could be measured. This was performed at a downstream
signalized intersection of a simple coordinated signal network. Once all
obtainable factors were measured in the field, these factors were used in the
formulas for both the calculated and measured delays as needed.
The next step was to plug the field measured delay (d) given by
equation (1.11) into the equation (1.12) for calculated delay. The purpose of
this is to reverse the calculated process with the measured (d) for purposes
of computing new PF values (in particular, fpa values). The values obtained
for di and d2 in the calculated process were substituted into this equation
and a new PF value computed. This value for PF was then used in equation
(1.2) to calculate a new fpa value. These steps were completed over several
trials for each arrival type in order to allow for statistical comparison to be
made of the fpa values.
4
A spread sheet was set up to process all the formulas. Table 1.1 is a
screen capture of the summary results of this process for AT 1.
Table 1.1 Results Summary Table for AT 1
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19
i 20: Tnal#1 Tnal #2 Tnaf #3 Trial *4 Tnal #5 Trial #6 Tnal #7 Trial #8 Tnal *9 Tnal FI 0 Tnal #11 Trial #12 Tnal #13 Tnal #14 Trial #15
Proportai ot
Vehctes Amvng on
GreenP
RowRateV.
fveh#f)
De^ee ot SAurtton
X
FWoor Rate Rd 
Profession Act
Facto PF 
(catcutfedj
Uniform Delay di
(sec)
hcreroertal Delay
.(ft(sec)
:Cortrol Delav d 
ifsec)
:Meesured Conroi
:OeMy d (sec)
ip* subtongr
^Measured Control
: Delay
0 06024 0 11364 0 09639 0.08642 8.140B5 CU32341 002617 0.02893 0.03077 019643 0.0339 006383 OOf98 8.04302 0.84762
580 600 S3333 540 471333 453,333 473333' * 46Q 433333 373333 333333 306667 673333 " 630 700
,*  ' S,  ! ffv *
<058784 &61t 056081 65473 047973 045946 047673 046527 3 4391S C 32698.0.39865 63t031 088243"0.30919 G.70946
0.11295 0.2130? 0.18072 016204 026408 005515 005262 0 05435 005769 0.3683 006356 0.11968 0.03713 0.09l9t 0.08329
201377.189936 193532 1.9576? 184105 2.07383 2.08249 206075 207692 1721942.07022 200608 2.10042 203782 204082
14.356 14:9341 13.962 13.8397.4316 129603 13.1696 130432 07978 12.2777:124462 11.7473 15.4679 154957 1S.fflO
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26.8966 29222 30.6747 332346 29:6056 266785 3D.GB4S 31.B812 2S07S9 77.3929 2508*7 31.6739 32.7821 32S76 33466?
32
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1.2 Understanding the Signal Network
A big question to answer was how to collect the field data. The
intersection had to be part of a coordinated signal network. Additionally,
finding coordinated signal networks that experienced all six arrival types was
anticipated to be too difficult a challenge. For these reasons, a computer
simulated signal network was chosen to create the six arrival type scenarios.
Using computer simulation allows for the signal network environment to be
controlled. The sole purpose of the simulation is to create virtual vehicles.
The computer software programs Syncrho and SimTraffic were
used to create the signal network and traffic conditions. Once established,
5
field measurements were undertaken for those variables that could be
measured using the same techniques outlined in the HCM just as they would
be done at an actual intersection. No calculated values that are produced in
the software were used for any part of the two procedures.
The signal network is comprised of two signalized intersections. Both
intersections were made up of single lane approaches. This allowed for
single through lane groups at both the upstream and downstream signals.
Turning movement volumes were set up for each trial such that no left or
right turns were made. Volumes consisted of equal through movements at
both intersections.
The signal timing was set up so that both intersections operated on a
pretimed cycle length. The timing plans shared identical signal phases and
phase times so that the volumetocapacity ratios (also called the degree of
saturation X) were equal for the through lane groups of these intersections.
This decreased the amount of field data needed by only having to examine
the downstream intersection. If the degree of saturation for the upstream
signal approaches did not equal the downstream lane group under study, the
upstream lane groups that contribute volume to the downstream lane group
would need to be obtained.
The cycle length was set to 90 seconds for both signals. According to
the procedures for measuring the control delay in Appendix A of Chapter 16
of the HCM, the interval time for measuring queued vehicles should not be
an integral divisor of the cycle length and should be between 10 and 20
seconds. For all trials, 20 seconds was used as the survey interval time for
measuring queued vehicles, 90 seconds was used for both cycle lengths,
and 9 minutes was used for the analysis period.
The six arrival types were created by adjusting the signal offset of the
down stream intersection. Changing the offset changed the number of
vehicles that arrived during green and red times. This changed the Rp value,
which defines the six arrival types. All field measurements were collected at
the downstream intersection on the through lane group. Figure 1.1 shows a
screen capture of the network from SimTraffic.
6
Figure 1.1 SimTraffic Signal Network
StmTratflc 5: C:\SctooHTtofcUMeHI0jj* /
Â£te Arwnace $ap*cs Options tlalp
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7:15A 7:17:Q9A 7.33A ; 
7
2. Analysis of fpa
A sufficient number of trial samples had to be completed for each
arrival type to allow for statistically significant results to be achieved. The
strategy was to use inference procedures to investigate the total population
mean value of fpafor each arrival type. The two inference procedures used
were the confidence interval comparisons and hypothesis testing, commonly
called tintervals and ttests2.
The confidence interval for fpa is an interval that contains plausible
values of the mean fpa value for the entire population. The ttest allows for
determining whether a specific value for the population mean fpa is a
plausible value.
AT 1 was used as the test case to determine the number of trials
needed to produce significant results. A sample size of 15 was chosen as a
starting point. The two processes described earlier were followed for 15
trials to calculate the corrected fpa value for each. The summary table shown
in Table 1.1 shows these values for all 15 trials for AT1. The analysis results
reported in the following paragraphs are for AT 1.
The next step involved determining whether the fpa values were
normally distributed. The importance of being normally distributed has to do
with the smaller sample size. For sample sizes that are less than 30, the
validity of the two inference procedures occurs only if the data are normally
distributed2. The sample number, flow rate, and fpa values were listed into a
table and sorted by fpa value in increasing order. The mean, the standard
deviation, and the variance of the fpa values were calculated as well as the
normal probability density function value for each fpaas shown in Table 2.1.
8
Table 2.1 AT1 Table of Sample #, Flow Rate, and fpa Values Sorted by
Increasing fpa Value.
Sample # Flow Rate fP value Norm Prob. Density Funct
1 580 0.8672 0.894215048
13 673 0.9367 2.182919168
15 700 0.9533 2.182919168
14 680 0.9524 2.16827973
2 600 0.9833 2.647270272
6 453 1.0133 3.020315998
7 473 1.0449 3.248592872
9 433 1.0448 3.248190843
3 553 1.0678 3.282825387
11 393 1.0858 3.228382575
8 460 1.1174 2.972720371
4 540 1.1579 2.422318715
5 473 1.1621 2.356378398
10 373 1.2476 1.035861044
12 307 1.312 0.402058199
Mean: 1.0631 StdDev: 0.121433 Variance: 0.014746
The normal probability density function is graphed to show that the
function does have a bell shaped curve, with the random variable fpa being
symmetrically distributed about the mean fpa value of the15 trials. These are
the characteristics of a normally distributed random variable2. Figure 2.1
shows the graph of this normal probability density function.
9
Figure 2.1 Normal Probability Density Function for AT 1
Normal Probability Density Function
Normal Probability
Density Function
Using a 99% confidence level, the confidence length of this interval for
this sample size of 15 is 0.162. This means that with 99% confidence, the
population mean of fpa is between 0.982 and 1.14. By comparison, to obtain
a smaller confidence length of 0.1 with a 99% confidence level using the
same sample standard deviation, the number of trials that would be needed
is 57 or more. The confidence interval obtained with the sample size of 15 is
deemed acceptable and this sample size was used for all of the remaining
arrival types.
The ttest was carried out next to evaluate whether the default fpa
value of 1.0 as listed in the HCM is plausible. The hypothesis statement for
this test is as follows:
Ho: fpa = 1.0 vs Ha: fpa ? 1.0
This test is measured with a pvalue, which is a probability value between 0
and 1. The hypothesis is accepted, and the value considered to be true if the
pvalue is larger than 0.10. For this specific test, the pvalue is found to be
0.071. Based on this, the hypothesis is rejected and the value of 1.0 for fpa
as stated in the HCM is not considered to be plausible.
10
A new ttest was carried out with the following hypothesis:
Ho: fPa = 1.06 vs Ha: fPa 1.06
The pvalue obtained for this test is 0.99. Based on this, the hypothesis is
accepted and the population mean value of 1.06 for fpa is considered to be
plausible.
Based on the changes to the value of fpa collected in the 15 AT1 trials
with the changes in flow rate, a test for correlation was completed. The
correlation coefficient between the flow rate values and the fpa values for AT 1
is 0.79. This indicates that the two data sets do share a significant inverse
relationship. Linear regression was used to determine what this relationship
is and the R2 value calculated to determine whether the equation is a
significant fit. Figure 2.2 shows the plot of Flow Rate vs. fpa along with the
linear regression formula and the R2 value.
Figure 2.2 Linear Regression Applied to AT 1 Flow Rate and fpa
y = 0.0008x+ 1.4781
R2 = 0.6196
Flow vs fp
Linear (Flow vs fp)
Flow Rate vs fp
The R2 value of 0.62 indicates that the linear equation ipa = 0.0008 *
(Flow Rate) + 1.48 is a good fit. Based on the strong correlation between the
two variables and the reasonably good R2 value of the linear regression, it is
recommended that fpa values for AT1 be calculated using the formula above
with measured flow rates for each coordinated lane group.
11
The same analysis procedure was performed for arrival types 25.
Table 2.2 is a summary of the population mean fpa analysis for AT 1 through
AT5. Table 2.3 is a summary of the correlation analysis between the flow
rates and fpa values for AT 1 through AT5. It is interesting to note that the first
three arrival types share a linear relationship with the flow rate. AT4 and
AT5 do not, however. The linear relationship between the two gets weaker
as the arrival types change and the progression gets better. Intuitively, this
seems to make sense considering that the progression factors influence on
control delay decreases with increased progression proficiency. Figure 2.3
shows the distributions of flow rates versus fpa values for AT2 through AT5.
Table 2.2 Summary of fpa Population Mean Analysis of AT1 Through AT5
Arrival Type Default fpa value Average fpa value calculated Standard Deviation Normally Distributed? 99% confidence Interval Accepted Mean Population fpa value
1 1.00 1.06 0.12 yes 0.981.14 1.06
2 0.93 1.00 0.07 yes 0.95 1.05 1.00
3 1.00 0.87 0.12 yes 0.79 0.95 0.87
4 1.15 1.16 0.09 yes 1.101.22 1.15
5 1.00 0.62 0.24 yes 0.46 0.77 0.62
Table 2.3 Summary of Correlation Analysis Between Flow Rate and fpa for
AT 1 Through AT5
Arrival Type Flow Rate and fpa Correlated? Correlation Coefficient Regression Formula R2 Significant Fit?
1 Yes 0.79 fpa = 0.0008(Flow) + 1.48 0.62 good
2 Yes 0.72 fpa = 0.0005(Flow) + 0.72 0.52 o.k.
3 Yes 0.70 fpa = 0.0007(Flow) + 0.50 0.48 o.k.
4 No 0.09 na na na
5 No 0.07 na na na
12
Figure 2.3 Normal Distribution and Flow vs fpa Graphs for AT2 Through AT5
Probability Density Function
AT2
Flow Rate vs fp
Flow Rate (veh/hr)
y = 0.0005x + 0.7183
R2 = 0.5188
Flow Rate vs fp
Linear (Flow Rate vs
fp)
AT2
13
Figure 2.3 (Cont.)
AT3
Flow Rate vs fp
y = 0.0007x +0.4984
R2 = 0.4781
Flow Rate vs fp
Linear (Flow Rate vs
fp)
AT3
14
Figure 2.3 (Cont.)
fp vs Probability Density Function
AT4
fp vs Flow (veh/hr)
Flow (veh/hr)
y = 5E05x + 1.183
R2 = 0.0077
Series 1
Linear (Series 1)
AT4
15
Figure 2.3 (Cont.)
AT5
fp vs Flow (veh/hr)
y = 0.0002X + 0.5349
R2 = 0.0045
Seriesl
Linear (Seriesl)
AT5
16
Arrival type 6 (AT6) is an arrival type that is very elusive to actually
achieve in field conditions. At the higher g/C ratios (> 0.5) the proportion of
vehicles that arrive on green must be extremely close or equal to 1.0 in order
to achieve a platoon ratio value (Rp) > 2.0 which characterizes AT6. When
this occurs, P = 1 and because of the (1P) term in the numerator of equation
(1.2) for calculating PF, PF = 0 and the uniform delay value is canceled out.
Control delay thus equals d2, which is reflected in exhibit 1612 of HCM 2000.
After applying the same process to AT6, as was done for the first five
arrival types, for a green ratio (g/C) value less than 0.40, the corrected
values for fpa fell below zero. These fpa values for the AT6 trials completed
are shown in Table 2.4.
Table 2.4 fpa Values for Trials in AT6
Trial #1 Trial #2 Trial #3 Trial #4 Trial #5 \ Trial #6 Trial #7 Trial #8
fpa 7.726 1.418 2.323 0.883 2.481 5.438 0.192 0.192
This is occurring based on the fact that the d2 term under AT6 seems
to overestimate the control delay compared to the control delay found using
the measured process. The negative fpa values are creating a negative PF
value in order to compensate for the larger d2 value. The corrected fpa
values for each trial are not as close in values from one to the next as was
the case in the first five arrival types. What is found is that fpa is highly
correlated with the proportion of vehicles arriving on green (P). The two
share a significant inverse relationship. The results of AT6 are summarized
below in Table 2.5.
Table 2.5 Summary of AT6 Results for fpa
Arrivai Type Default fpa value Average fpa value calculated Standard Deviation Normally Distributed? 99% confidence Interval Accepted Mean Population fpa value
6 1.00 2.533 2.73 yes (4.35) (0.72) 2.53
Arrival Type Flow Rate and fpa Correlated? Correlation Coefficient Regression Formula R2 Significant Fit?
6 P and f are 0.92 fpa = 125.8P + 118.1 0.85 yes
17
3. Recommendations
Exhibit 1612 in the HCM should be changed to Table 3.1 shown
below to reflect the population mean of each fpa values for each arrival type.
This exhibit is meant to provide the analyst with good approximations of PF
for each arrival type at various g/C ratios and should be described as default
PF values. The recommended changes are shown below in Table 3.1
Table 3.1 Recommended HCM Exhibit 1612
EXHIBIT 1612. DEFAULT PROGRESSION ADJUSTMENT FACTOR FOR UNIFORM DELAY CALCULATION
Arrival Type (AT)
Green Ratio (q/C) AT 1 AT 2 AT 3 AT 4 AT 5 AT 6
0.20 1.237 1.083 0.870 1.000 0.517 1.898
0.30 1.363 1.143 0.870 0.986 0.443 1.446
0.40 1.531 1.222 0.870 0.895 0.344 0.843
0.50 1.767 1.333 0.870 0.767 0.206 0.000
0.60 2.121 1.500 0.870 0.576 0.000 0.000
0.70 2.710 1.777 0.870 0.256 0.000 0.000
Mean fpa 1.06 1.00 0.87 1.15 0.62 2.53
Default, Rp 0.333 0.667 1.000 1.333 1.667 2.000
Notes: PF = (1 P )fpa/( 1 gC ). Tabulation is based on default values of fpa and Rp. P = Rp g/C (may not exceed 1.0) or proportion of vehicles arriving on green. PF may not exceed 1.0 for AT 3 through AT 6.
A new exhibit should be inserted in Chapter 16 of the HCM that
describes the formulas for fpa to be used when field measurements are
undertaken in order to calculate PF and control delay (d). The following
exhibit shown in Table 3.2 is recommended to be included:
18
Table 3.2 Recommended New Figure__________________
EXHIBIT 16(xx). fpa FACTOR FOR UNIFORM DELAY CALCULATION
Arrival Type (AT)
AT 1 AT 2 AT 3 AT 4 AT 5 AT 6
fpa 0.0008V + 1.48 0.0005V + 0.72 0.0007V + 0.50 1.15 0.62 125.8P + 118.1
Notes:
V = Hourly Flow Rate (veh/hr)
P = Rp g/C (may not exceed 1.0) or proportion of vehicles arriving during green.
3.1 Summary
The recommended changes to the supplemental adjustment factors
for platoons arriving during green (fpa) will bring the calculated control delay
(d) more in line with the control delay measured in the field. This finding
applies to lane groups that operate under coordinated signal timing plans.
Default values for the progression factor (PF) have been supplied, but its
recommended that PF be calculated with the recommended formulas for fpa .
along with field obtained values for the proportion (P) of vehicles arriving
during green in equation (1.2). Using either the default PF values in the
recommended HCM exhibit 1612 (shown above as Table 3.1) or using the
formulas for fpa (shown above as Table 3.2) should both provide the analyst
with better estimates of control delay at coordinated signals. Although this
study used simulation to generate field observations, similar results would
likely be found in actual field studies as recommended for future research.
19
APPENDIX A
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20
REFERENCES
1. Highway capacity manual. (2000). Washington DC: Transportation
Research Board.
2. Hayter, A.J. (1996). Probability and statistics for engineers and
scientists. Boston MA: PWS Publishing Company.
21
