An Uncertainty Analysis for Two Freeway Sites
Submission Date: August 1, 2002
Word Count: 7,246
Nathan Higgins
Department of Civil and Environmental Engineering
Rensselaer Polytechnic Institute
110 8th Street, Troy, NY 12180
Phone:(518) 276-8306
Fax:(518) 276-4833
Email: higgin@alum.rpi.edu
Corresponding author: George List
Department of Civil and Environmental Engineering
Rensselaer Polytechnic Institute
110 8th Street, Troy, NY 12180
Phone: (518) 276-6362
Fax: (518) 276-4833
Email: listg@rpi.edu
Stacy Eisenman
Department of Civil and Environmental Engineering
Rensselaer Polytechnic Institute
110 8th Street, Troy, NY 12180
Phone: (518) 276-8306
Fax: (518) 276-4833
Email: eisens@rpi.edu
Uncertainty in LOS for Basic Freeway Sections
TRB 2003 Annual Meeting CD-ROM
Original paper submittal – not revised by author.
Abstract
Uncertainty is a major issue in many phenomena. The operation of highways is no exception.
Yet, one of the most common methodologies for assessing the performance of intersections,
freeways, etc. lacks the ability to reflect uncertainty in the inputs or the outputs. Methods that
allow explicit examination of the variations that arise in the performance of given facilities or
systems are necessary. This paper examines the issue of uncertainty for two freeways that are
located in upstate New York and provides a commentary on the perspectives that might be
adopted to portray the stochastic performance of such facilities to analysts.
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1 INTRODUCTION
Traffic engineers and planners often model highway facilities using fixed values for the inputs.
Implicitly, one surmises, they are assuming that the data represent average conditions or average
values for a specific condition, such as a peak hour. In reality, it is likely that the analysts know
the inputs are stochastic. For example, they realize that the volumes vary by time of day, day of
the week, etc. They know that driver behavior also varies, as does vehicle mix, the ambient light
conditions, etc.
One of the reasons the analysts are accustomed to fixed inputs is that the procedures in the
Highway Capacity Manual (HCM) (1) are deterministic. Single valued inputs yield single valued
outputs. The HCM gives no indication of the variability in the answer. However, efforts are
underway to develop that capability.
The importance of conveying this uncertainty is highlighted when one considers the impetus
behind performing the analysis in the first place. That is, the analyst wants to know what quality
of service should be expected for a given facility. In addition, they need to know what
incremental improvements result from incremental capacity additions (e.g., extra lanes).
This paper examines the issue of performance characterization for two freeway sites in upstate
New York. One is I-87 between exits 18 and 19 and the other is I-84 between exits 15 and 16.
The paper provides a commentary on how uncertainty might be treated for such facilities and
how the results of that analysis could be interpreted.
2 EMERGING RESULTS FROM RESEARCH
The treatment of uncertainty within highway capacity is just beginning. Tarko et al. (2)
proposed a framework from which uncertainty can be viewed. That framework builds on earlier
work by Tarko (3), Tarko and Tracz (4), Kyte and Khatib (5), Roess and Prassas (6) and,
Luttinen (7).
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Tarko et al. (2) provided a framework for categorizing the uncertainties that can arise. One
category is uncertainty in the demands, another is uncertainty in the model parameters, and a
third is uncertainty in the model itself (i.e., limitations in the model’s ability to completely and
accurately capture all of the relevant cause and effect relationships). If the uncertainty can be
portrayed in a well-defined manner, then the results of the analysis can be interpreted reliably
and succinctly. Additionally, the results will be reliable in terms of the cause-and-effect
relationships between the variance in the inputs and the expected variance in the predicted
performance. (In principle, it is possible to directly relate the variance in the causal variables and
the variance in the outcome variable, but no such models have been proposed to date. For
example, if z = x + y, then the variance of z is the sum of the variances in x and y.)
Kyte and Khatib (5) consider the treatment of uncertainty and how it propagates through the
analysis of signalized and unsignalized intersections. They discuss the effect of uncertainty in the
input parameters and the effect of those uncertainties on the performance predictions. They look
specifically at three categories of uncertainty: “uncertainty in the volume forecast, uncertainty in
driver behavior, and uncertainty in the nature of quality of forecasting model itself.” The
motivation for this examination is a search for a higher degree of confidence in decision making
during planning and design.
In examining unsignalized intersections, Kyte and Khatib (5) consider three cases that have
different volumes and levels of service (LOS). They vary the volume and the critical gap ± 10%
and study the outcome. They find that as the LOS degrades, a greater variation in delay occurs
when either the volume or the critical gap is varied. For example, when the initial LOS is B and
a 10% increase in volume is considered, the delay remains nearly unchanged while when the
base LOS is E, a 10% increase produces a significant change in delay.
In the case of signalized intersections, Kyte and Khatib (5) consider the effect of variations in
volume and four other parameters (lane width, heavy vehicle percentage, grade, and driver type)
on delay and the volume-to-saturation-flow-rate ratio (v/s). Changes in volume of ± 10% and ±
20% are considered across five volumetric conditions. They find that when the lane width is
changed by 1 foot, the delay changes noticeably while the v/s ratio changes little. However,
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when the heavy vehicle percentage is changed, it has a significant effect on both delay and the
v/s ratio. Additionally, if the grade is changed it has a minimal affect while if driver behavior is
selected it has a significant effect.
This paper approaches the issue of uncertainty in a manner similar to Kyte and Khatib (5) and
Tarko (3). However, variations in traffic flow are not hypothesized but based on real
observations from the field. Questions are addressed such as: what is the probability density
function for vehicle density (i.e., pc/km/lane, the LOS metric) during a given timeframe? If the
facility’s peak-hour performance is studied, what does that tell us? If a different perspective is
employed, what else do we learn? Can ways be devised to synthesize results for a given
operating condition and anticipate the variation in LOS performance that would result?
3 CASE STUDY SITES
Real-world facilities always provide a proving ground for methodological developments. This
paper uses two as sites for that purpose. The first is I-87 between exits 18 and 19 near Glens
Falls, New York. The second is I-84 between exits 15 and 16 in Dutchess County. Both of these
are continuous count stations monitored by NYSDOT. Because the stations gather data
automatically, a variety of count-related data can be collected. We obtained two datasets for each
site. The first contains volume counts by hour and lane from April 2001 to December 2001. The
second has 15-minute counts for up to four days from Friday January 4, 2002 until Monday
January 7. We obtained trends in volume by day, week, etc. from the former and, from the latter
we examined trends in the peak hour factor, etc.
3.1 I-87 between Exits 18 and 19
This site lies along I-87 between Albany and Montreal near Glens Falls, NY. Glens Falls is at the
foot of the Adirondacks. The traffic is going to and from work, Canada, winter ski resorts, and
summer vacation spots. The facility has three lanes in each direction and an interchange density
of 0.3 interchanges/km.
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Figure 1 shows the distribution of volumes by direction. The most common condition is a
volume of 0-200 vph. Of the 6576 hours in the dataset, volumes in that range occur for 1072
hours northbound and 1390 hours southbound. A second common condition is a volume of 10001200 vph. It occurs for 806 hours northbound and 841 hours southbound.
Figure 2 provides a plot of the average two-way traffic volumes for each day of the week. The
heaviest day is Friday; the lightest is Tuesday. The heaviest hour of use is in the afternoon
regardless of which day is studied. The maximum of the maximums is 4-5pm on Friday. The
volume then is 4045 vph (2275 northbound + 1770 southbound or an average of 760 and 590
vehicles per lane per hour respectively).
Figure 3 shows a plot of the variations in the peak hour factor (PHF) for this site in the
northbound direction. Rather than just computing the PHF each hour, every 15 minutes it has
been recomputed. PHFi , where i is a time period, is based on 15-minute periods i, i+1, i+2, and
i+3. As can be seen, the PHF ranges from 0.53 to 0.99. It stays between 0.9 and 1.0 during the
daytime hours and is lower otherwise. This is consistent with what would be expected: it should
be higher and more consistent when the volumes are larger, and lower and less consistent when
the volumes are smaller. In the southbound direction, the PHF ranges from 0.51 to 0.98. It is
also highest and most consistent between 11am and 5pm. (The trend is very similar to Figure 3.)
3.2 I-84 between exits 15 and 16
The second site is a segment of I-84 between exits 15 and 16 in Dutchess County. This site is
east of the New York State Thruway (I-87) and west of the Taconic Parkway. The traffic is a mix
of suburban trips (around New York City) and intercity trips going between New England and
points south and west. The facility has two lanes in each direction and the interchange density is
0.3 interchanges/km.
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Figure 4 shows the distribution of traffic volumes in the east and westbound directions. The most
common condition is a volume between 200-400 vph, which occurs for 1183 hours eastbound
and 782 westbound out of the 6576 hours in the dataset. The next most common volume is 12001400 vph. That occurs for 837 hours eastbound and 767 hours westbound.
Figure 5 shows the temporal trends in the average daily two-way traffic volumes. Friday has the
most traffic while Sunday has the least. On the weekdays, the heaviest volumes occur late in the
afternoon while on the weekend the heaviest volumes are closer to midday. The hour with the
most traffic is Friday from 5-6pm, when the average is 4717 vehicles in both directions (2853
westbound, 1865 eastbound, or an average of 1427 and 934 vph per lane respectively).
The PHF varies similar to the way it did for the I-87 site. The trends for the eastbound direction
show that during the daytime, the PHF ranges from 0.9 to 1.0 while at other times it is lower.
The trends in the westbound direction are similar. The PHF ranges from 0.65 to 0.99. It is
highest and steadiest between 11am and 5pm. During the other hours, it is lower and more
variable.
4 UNCERTAINTY ANALYSIS
The basic freeway section methodology in the HCM uses five demand (volume) related inputs
(volume, PHF, truck/bus percentage, RV percentage, and driver population adjustment) and two
facility-related inputs (number of lanes and terrain). In addition, if a free-flow speed has to be
estimated (i.e., none has been measured), three more facility-related inputs are needed (lane
width, right-shoulder lateral clearance, and interchange density). Some of these are arguably
invariant for a given site (number of lanes, right-shoulder clearance, interchange density) while
others are not.
4.1 Preparing for the Analysis
A comprehensive study of uncertainty in performance would need field data for all of the
variable inputs (volume, peak-hour factor, truck/bus percentage, RV percentage, driver
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population). However, today’s sensor and instrumentation technology cannot provide data
concerning the truck/bus percentage, the RV percentage, or the driver population factor. Still,
volume and peak hour factor data can be collected with ease. We will focus on these two inputs.
Given the methodology, we also need to select at least one “case study hour” or “peak hour” as it
is often called. The HCM operational procedure for freeway sections assumes an hour has been
identified when the facility is at its “intended”, “design,” or “peak use.” The methodology
provides assessments of performance that are predicated on that hour.
From Figures 2 and 5 it seems that picking the “case study hour” may not be as easy as one
would like. In the case of Figure 2, for example, 4-5pm is the peak hour during the week, but the
peak hour on Sunday is 12-1pm and on Saturdays, it is 11am-noon. Moreover, the Saturday peak
is about 10% larger than Monday through Thursday. (In the northbound direction, it is 33%
larger.) So is the right hour to select Friday 4-5pm, the average weekday, Saturday 11am-noon,
or some combination? What is the “correct” characterization of the performance of this facility?
If Monday through Thursday 4-5pm is selected as the “peak hour”, on two days each week the
peak hour LOS will be worse than the modeled condition. If Friday 4-5pm is selected, the
“average” peak hour LOS will be better than reported. This may mean that the idea of selecting
one or more “study hours” should be revisited.
Next, a stochastic dataset that captures the variation in operating conditions for the peak hour is
needed. The dataset has to contain a large number of “representative” combinations of volume
and PHF. Empirical data would be best with an assumption that the other parameters are
constant. In lieu of that, a synthesized dataset must be created.
In this case, no large dataset of field observations containing volume and peak hour factor values
was available. In the datasets from NYSDOT, the first contained hourly data for nine months and
the second had 15-minute counts for a few of days.
Consequently, the “observations” for the two sites had to be synthesized. Combinations of
volume and peak hour factor were generated from assumed density functions. Those values were
then combined to form a large synthesized dataset.
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In this instance, the synthesized observations were created in this way. First, the mean µv and
standard deviation σv for the peak hour volumes were computed. The hourly data (for the nine
months) were tallied by day-of-the-week and hour of the day. The “peak hour” was identified by
finding the weekday hour with the largest total volume. Based on the data for that hour, µv and σv
were computed. Fifty volume observations were then synthesized assuming the underlying
density function was normal (truncated). That is, each volume observation, vi was computed
using:
vi = NormInv(ω i | µ v , σ v ) ∀ i
(1)
where NormInv is the inverse normal function, ωi is the random variable (from a uniform density
function between 0 and 1) associated with the ith volume observation, and µv and σv are the
volume-based mean and standard deviation values respectively.
The following steps were taken to compute the mean µβ and standard deviation σβ for the peakhour PHF. First, µv and σv were used to derive upper and lower bounds for the peak hour volume.
The minimum was V- = µv - 2.5 σv and the maximum was V+ = µv + 2.5 σv. This encompasses
about 99% of the volumes that might be observed. The next step was to use V- and V+ to select
plausible “peak-hour” PHF observations from the PHF dataset. This was done by finding those
(volume, PHF) combinations for which the volume was between V- and V+. The resulting PHF
observations were then used to compute µβ and σβ. Fifty PHF observations were synthesized
assuming the underlying density function was normal (truncated). Each PHF observation, βj was
computed using:
β j = NormInv(ω j | µ β , σ β ) ∀ j
(2)
where ωj is the random variable (from a uniform density function between 0 and 1) associated
with the jth PHF observation, and µβ and σβ are the PHF-based mean and standard deviation
values respectively.
The 50 values of vi were then combined with the fifty values of βj to generate 2500 pairwise
combinations of volume and PHF (vi, βj) (i.e., for all (i,j) combinations). Additionally,
assumptions were made about the other parameters needed by the HCM procedure: the terrain
was level; the lane width was 3.6 meters; the right shoulder clearance was 1.8 meters; the
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truck/bus percentage was 13.5%; the RV percentage was 1.5%; the basic free flow speed was
120 km/h; and the driver population factor was 0.925.
4.2 I-87 between Exits 18 and 19
For the I-87 site, µv is 1606 vph and σv is 485 vph in the northbound direction. Thus, V- is 394
vph and V+ is 2817 vph. Based on the PHF observations with volumes within these limits, µβ is
0.907 and σβ is 0.051 (220 samples are in the PHF dataset).
The 2500 synthesized combinations of peak-hour volume and PHF yield the I-87 northbound
CDF for freeway density (pc/km/lane) shown in Figure 6. The minimum density is 1.83 (LOS
A), the maximum is 10.34 (LOS B), the mean value for the density is 5.89, and the standard
deviation is 1.61. Since the boundary for LOS A/B is 7 and for LOS B/C it is 11, approximately
74% of the time this facility should be at LOS A during the peak hour in the northbound
direction and 26% of the time it should be at LOS B.
The southbound direction is similar: µv is 6.02 and σv is 1.27. Approximately 82% of the time the
facility should be at LOS A during the peak hour in the southbound direction and 18% of the
time it should be at LOS B.
4.3 I-84 between Exits 15 and 16
For the I-84 site, µv is 1822 vph and σv is 286 vph for the eastbound direction. Thus, V- is 1108
vph and V+ is 2536 vph. Based on the PHF observations with volumes within these limits, µβ is
0.941 and σβ is 0.028 (82 samples are in the PHF dataset).
The minimum density among all 2500 synthesized conditions is 6.12 pc/km/lane (LOS A) and
the maximum is 13.74 (LOS C). The mean value for density is 9.84 pc/km/lane and the standard
deviation is 1.34. Approximately 2% of the time the eastbound direction should be at LOS A
during the peak hour, 82% of the time it should be at LOS B, and 10% of the time it should be at
LOS C.
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In the westbound direction, µv is 2512 vph and σv is 388 vph for the peak hour. This means V- is
1542 vph and V+ is 3482 vph. Based on the PHF observations with volumes within these limits,
µβ is 0.894 and σβ is 0.043 (19 samples are in the PHF dataset).
Figure 7 shows the CDF for density in the westbound direction. The minimum is 7.98 (LOS B)
and the maximum is 26.07 (LOS D). The mean is 14.68 and the standard deviation is 2.61.
There are no times during the peak hour for the westbound direction when LOS A should
pertain. Approximately 5% of the time, the LOS should be B, another 70% of the time it should
be C, 22% of the time it should be D, and 3% of the time it should be E.
5 AN ALTERNATIVE PERSPECTIVE
Selecting a single peak hour to analyze presents an interesting dilemma. In the case of the I-87
situation, for example, it limits one’s ability to depict the actual range of operating conditions
that pertain to the facility’s performance. In this light, the 1965 Capacity Manual’s notion of an
Nth highest hour seems like a good idea to revisit (8).
Brilon (10) describes a way to view capacity analysis that is predicated on marginal economics.
His idea is that as capacity increases the total user cost falls while the total facility cost rises. At
some point, the total cost reaches a minimum. Capacity investments should be made up to that
point, but not further.
Brilon’s (10) idea provides a thought about how a comprehensive uncertainty analysis should be
conducted for a given facility. Maybe it should be predicated on a complete year’s use of the
facility (that is, one complete use cycle). It would be even better to consider its performance
across a series of years. The use cycle (an entire year) analysis would give a comprehensive
picture of how well the facility could be expected to perform.
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5.1 Doing a Use Cycle (Whole-Year) Analysis
Brilon (10) argues that one should examine a facility’s performance across an entire use cycle. In
most instances, that cycle is one year. He says that focusing on this timeframe provides a more
accurate sense of the operational conditions that pertain for the facility.
An approximation to a use cycle analysis was conducted for the two case study sites as follows.
The 6576 volume observations were treated as a use cycle. A PHF was computed for each (see
text below). Based on the volume and PHF combinations, a density (pc/km/lane) was computed.
These were used to identify the level of service and a histogram was prepared showing the
percentage of time the facility spent in each LOS.
To develop PHF values corresponding to the 6576 volume observations, the following procedure
was used. (Remember that the 6576 volume observations were whole hour observations, not 15minute observations, so PHF values had to be synthesized.)
As Figure 8 shows, there is a relationship between hourly volume and PHF for I-87 southbound.
This relationship cannot be overlooked if credible density (pc/km/lane) estimates are to be
developed. The PHF increases as the volume increases. It also becomes more consistent. Higher
traffic levels produce more consistency in the 15-minute counts and less variation in the PHF.
Log-log regression applied to the PHF data for all the 15-minute data yielded the following
relationship:
PHF = e 6.44 * (V/n)0.0621
(3)
Where PHF is the estimated peak hour factor, V is the hourly volume, and n is the number of
lanes. If the sites are examined individually, the values of the exponents are slightly different, but
the differences are not substantial.
A disappointing result from the regression analysis is that the scatter plot that compares the
observed PHF values with the estimated ones does not produce a 1:1 correspondence. (Most
likely, the data points for high PHF values mask the ability of the low PHF observations to
influence the regression line.) The PHF estimates are too high at low volumes (i.e., the PHF
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estimates are higher than those observed) and too low at high volumes (i.e., the PHF estimates
are lower than those observed).
A more visually convincing relationship is provided by the following equation. It was developed
by trial-and-error (which is heresy from an analytical perspective):
PHF = e 5.74 * (V/n)0.163
(4)
Figure 9 shows that this function predicts PHF values that have the same trend as the observed
PHF values. The observed values are plotted along the horizontal axis and the estimated values
along the vertical axis. A few PHF values (technically 13 out of 1156) exceed 1.0, but the
correspondence in general is good. (Future work in this area would be beneficial.)
Based on equation (4), we can synthesize PHF value for each volume (6576 values). Since
equation 4 does not guarantee an upper bound of 1.0 that limit should be applied. The
corresponding density (pc/km/lane) and LOS can then be computed.
5.2 I-87 between Exit 18 and 19
The use cycle analysis (in this instance, the 6576 hourly observations that were part of the ninemonth sample provided by NYSDOT) of the southbound direction for the I-87 site produces the
cumulative density function (CDF) for density (pc/km/lane) shown in Figure 10. We see that
93% of the use cycle, the facility should be at LOS A, 6% of the time it should be at LOS B, and
1% of the time, it should be at LOS C.
The results for the northbound direction are similar. During 93% of the use cycle, the facility
should be at LOS A and 7% of the time it should be at LOS B.
5.3 I-84 between Exits 15 and 16
Figure 11 shows the corresponding CDFs for density (pc/km/lane) in the westbound direction for
the I-84 site. These results are predicated on the directional hourly volume distributions across
the nine-month timeframe presented in Figure 3. In the westbound direction, the results are 55%
at LOS A, 37% at LOS B, 8% at LOS C and less than 1% at LOS D. Similarly, in the eastbound
direction, the results are 55% of the use cycle in LOS A, 39% in LOS B, and 6% in LOS C.
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6 CONCLUSION
Two very different perspectives have been presented concerning the analysis of uncertainty in
basic freeway sections. Consistent with the current HCM, the one considers a “study hour” and
looks at uncertainty in terms of quantifying the variations in LOS that might occur during that
hour. This yields statistically defensible predictions of how the facility will perform given those
“peak hour” conditions.
The other perspective considers the performance of the facility across an entire use cycle (one
year). From this one learns about the percentages of time that the facility will function at various
levels of service. With this information, the analyst can decide whether the percentile point for a
given LOS is high enough or if design changes are needed.
In either case, such analyses should be helpful to those who need more than a point estimate of
how well a facility is performing. In addition, these ideas can be carried into other analysis
situations and provide a richer and more comprehensive picture of the performance of a given
facility or system.
Future considerations with regard to this research include:
•
Finding a more defensible relationship between volume and peak hour factor
•
Analyzing the uncertainty of a dataset produced by deriving PHF by volume for an entire
years worth of data
•
Considering the nature of uncertainty with regards to the optimization of economic
investment and quality of service
•
Looking into the nature of the relative proportionality of volume and density
Acknowledgment
The authors are deeply indebted to Bernard Schatz, Michael Shamma, Todd Westhuis, and
Michael Fay from NYSDOT who made the continuous count station data available.
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References
1. Transportation Research Board. Highway Capacity Manual, Special Report 209, National
Research Council, Washington, D.C., 2000.
2. Tarko, A., R. Benekohal, J. Bonneson, E. Elefteriadou, and J. Sacks. Uncertainty in HCM Fundamental Concepts and Issues, Working Paper, January meeting, Highway Capacity and
Quality of Service Committee, Transportation Research Board, Washington, DC, January, 2002.
3. Tarko, A. Uncertainty Issue in the Highway Capacity Manual, Presentation, Midyear Meeting,
Highway Capacity and Quality of Service Committee, Transportation Research Board, Lake
Tahoe, CA, July 24-28, 2001.
4. Tarko, A. P. and M. Tracz. Uncertainty in Saturation Flow Predictions, Fourth International
Symposium on Highway Capacity, Transportation Research Board, Maui, Hawaii, June 27 - July
1, 2000.
5. Kyte, M. and Z. Khatib. Uncertainty in Projecting the Level of Service of Signalized and
Unsignalized Intersections. Proceedings of the Transportation Research Board Annual Meeting,
Washington, DC, January 7-11, 2001.
6. Roess R. and E. Prassas. Accuracy and Precision in Uninterrupted Flow Analysis, Proceedings
of the Transportation Research Board Annual Meeting, Washington D.C, January 7-11, 2001.
7. Luttinen, T. Uncertainty in the Operational Analysis of Two-Lane Highways, Presentation,
Two-Lane Highway Subcommittee, Midyear Meeting, Highway Capacity and Quality of Service
Committee, Transportation Research Board, Lake Tahoe, CA, July 24-28, 2001.
8. Transportation Research Board. Highway Capacity Manual, Special Report 209, National
Research Council, Washington, D.C., 1965.
10. Brilon, W. Traffic Flow Analysis beyond Traditional Methods, Proceedings of the Fourth
International Symposium on Highway Capacity, Transportation Research Board, Maui, Hawaii,
June 27 - July 1, 2000.
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Tables and Figures
FIGURE 1 Distribution of Hourly Volumes for I-87
FIGURE 2 Daily Variations in Average Two-Way Volume for I-87
FIGURE 3 Variation in the PHF for I-87 Northbound
FIGURE 4 Hourly Volume Distributions for I-84
FIGURE 5 Daily Variations in Volume for I-84
FIGURE 6 "Peak Hour" Density (pc/km/lane) CDF for I-87 Northbound
FIGURE 7 CDF for "Peak Hour" Density (pc/km/lane), I-84 Westbound
FIGURE 8 Typical Relationship between PHF and Volume (I-87 Site, Southbound)
FIGURE 9 Estimated versus Observed PHF
FIGURE 10 CDF for Density (pc/km/lane) for One Use Cycle, I-87 Southbound
FIGURE 11 CDF for Density (pc/km/lane) for One Use Cycle, I-84 Westbound
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Distribution of Hourly Volume
1600
Number of Hours
1400
1200
1000
NB
800
SB
600
400
200
38
00
34
00
30
00
26
00
22
00
18
00
14
00
10
00
60
0
20
0
0
Volume (vph)
FIGURE 1 Distribution of Hourly Volumes for I-87
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Averag e Tw o-Way V olumes
4500
Av e rag e Tw o-W ay Volu me
4000
3500
Sun
3000
Mon
T ue
2500
W ed
2000
T hu
1500
F ri
1000
Sat
500
24
22
20
18
16
14
12
10
8
6
4
2
0
0
Ho u r
FIGURE 2 Daily Variations in Average Two-Way Volume for I-87
Uncertainty in LOS for Basic Freeway Sections
TRB 2003 Annual Meeting CD-ROM
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Original paper submittal – not revised by author.
PH F by H our of D ay
1
0.95
0.9
0.85
F ri
P HF
0.8
S at
0.75
S un
0.7
M on
0.65
0.6
0.55
0.5
0
4
8
12
16
20
24
Ho u r O f Da y
FIGURE 3 Variation in the PHF for I-87 Northbound
Uncertainty in LOS for Basic Freeway Sections
TRB 2003 Annual Meeting CD-ROM
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Original paper submittal – not revised by author.
D istribution of H ourly V olumes
1400
Num be r of Hours
1200
1000
800
EB
600
WB
400
200
00
34
00
30
00
26
00
22
00
18
00
14
10
00
0
60
20
0
0
Volum e (vph)
FIGURE 4 Hourly Volume Distributions for I-84
Uncertainty in LOS for Basic Freeway Sections
TRB 2003 Annual Meeting CD-ROM
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Original paper submittal – not revised by author.
5000
4500
4000
S un
3500
M on
3000
Tue
2500
W ed
2000
Thu
1500
Fri
1000
S at
500
24
22
20
18
16
14
12
10
8
6
4
2
0
0
Ave ra ge Tw o-W a y V olum e (ve h)
Average Tw o-W ay H ourly V olumes
Hour
FIGURE 5 Daily Variations in Volume for I-84
Uncertainty in LOS for Basic Freeway Sections
TRB 2003 Annual Meeting CD-ROM
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Original paper submittal – not revised by author.
CDF for Density, I-87 Northbound
1
LOS B/C
Boundary at
100th percentile
0.9
0.8
0.7
LOS A/B
Boundary at
approximately
74th percentile
Percentile
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
11
Density (pc/km/lane)
FIGURE 6 "Peak Hour" Density (pc/km/lane) CDF for I-87 Northbound
Uncertainty in LOS for Basic Freeway Sections
TRB 2003 Annual Meeting CD-ROM
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Original paper submittal – not revised by author.
CDF for Density, I-84 Westbound
1
0.9
LOS D/E
Boundary at
approximately
99th
percentile
Cumulative Density Function Value
0.8
0.7
LOS C/D
Boundary at
approximately
75th
percentile
0.6
0.5
0.4
0.3
LOS B/C
Boundary at
approximately
5th percentile
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Density (pc/km/lane)
FIGURE 7 CDF for "Peak Hour" Density (pc/km/lane), I-84 Westbound
Uncertainty in LOS for Basic Freeway Sections
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Original paper submittal – not revised by author.
P eak H our Factor (P H F) versus V olume
1.2
Pe a k Hour Fa ctor
1
0.8
0.6
0.4
0.2
0
0
500
1000
1500
2000
Pe a k Hour V olum e (ve h)
FIGURE 8 Typical Relationship between PHF and Volume (I-87 Site, Southbound)
Uncertainty in LOS for Basic Freeway Sections
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Original paper submittal – not revised by author.
1100
1000
Estim a te d
900
800
700
600
500
500
600
700
800
900
1000
1100
Obse rve d
FIGURE 9 Estimated versus Observed PHF
Uncertainty in LOS for Basic Freeway Sections
TRB 2003 Annual Meeting CD-ROM
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Original paper submittal – not revised by author.
CDF for Density (pc/km/lane) for a Use Cycle
I-87 Southbound
1.1
1
0.9
LOS B/C
boundary at
approximately
99th percentile
0.8
LOS A/B
boundary at
approximately
93rd percentile
Percentile
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Density (pc/km/lane)
FIGURE 10 CDF for Density (pc/km/lane) for One Use Cycle, I-87 Southbound
Uncertainty in LOS for Basic Freeway Sections
TRB 2003 Annual Meeting CD-ROM
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Original paper submittal – not revised by author.
CDF for Density for a Use Cycle
I-84 Westbound
1.1
1
0.9
0.8
LOS B/C
boundary at
approximately
92nd percentile
Percentile
0.7
LOS C/D
boundary at
approximately
99.9th
percentile
0.6
0.5
LOS A/B
boundary at
approximately
55th
percentile
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Density (pc/km/lane)
FIGURE 11 CDF for Density (pc/km/lane) for One Use Cycle, I-84 Westbound
Uncertainty in LOS for Basic Freeway Sections
TRB 2003 Annual Meeting CD-ROM
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Original paper submittal – not revised by author.