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29
Chapler Three
CANDIDATE MODELS
The previous chapter provided a summary of the main
theoretical models available for estimating critical gap,
capacity, and delay at TWSC intersections. The purpose of
As chapter is to summarize the methods and models that
were selected as candidates for evaluation against field
data.
CRITICAL GAP ESTIMATION PROCEDURES
Some ofthe well known estimation procedures for critical
gaps have been checked for consistency in MiDer (1972)
and Troutbeck (1975~. The effectiveness of the maximum
likelihood memos has been evaluated in further studies by
Briton and Troutbeck. In a recent report(Brilon, 1995),
Briton concluded that the estimation process for critical
gap distribution should be consistent. There was a strong
suspicion by Me research team that a great deal (if not Me
majority) of all inconsistencies found in the literature
regarding the relationship of critical gaps to other
parameters (such as traffic volumes, delays, delays at the
stop line as service times of the imbedded queuing system,
and geometric characteristics of intersections that are
normally studied at different sites under different traffic
volumes) might not actually exist but may instead result
Dom inconsistencies in the estimation procedures.
The estimation of critical gaps is not an end In itself.
Critical gaps are used in models to estimate the capacity of
unsignatized intersections. Estimating cntical gaps using
different procedures could influence the model's capacity
output. It should be guaranteed to the user community that
the estimated cntical gap, In conjunction with the follow-
up brne, and their respective estimation procedures give a
reliable and realistic estimate of intersection capacity, i.e.
an estimation procedure that is independent of the external
parameters mentioned above, especially the major street
volume. Further, the estimation procedure for cntical gaps,
the capacity model, and the delay mode! must form one
integrated system using consistent definitions and
assumptions.
Therefore, Briton (1995) conducted a simulation study to
investigate the independence to traffic parameters of
different critical gap estimation methodologies. The
simulation study for testing different procedures used two
traffic streams: one on the major street (volume vp) and
one on the minor street (volume vn; see Figure 3~.
S~mulation runs with constant traffic volume vp and vn
were conducted over 10 hours of simulation. This was
repeated for 46 different combinations of vp and vn (where
vn < capacity3. The arrival headways were generated
according to a hyper-eriang-distr~bution. The critical gaps
tc and the follow-up times if were generated according to
a shifted Eriang-distribution. These distnbutions reflect
stochastic yet realistic traffic situations.
The critical gaps were estimated from the simulated flow
patterns using the following methods:
.
.
.
.
.
maximum likelihood procedure (e.g., Troutbeck,
1992)
Meshwork (1970)
Raff (1950)
Harders (1976)
Hewitt (1992)
logy mode]
Siegioch (1973)
In addition, for the maximum likelihood method, an
estimate for the standard deviation of the critical gap was
obtained from the procedure. The SiegIoch estimation
procedure was also tested with constant queues in the
minor street approach. For each procedure, two different
combinations of the critical gap tc and the follow-up time
If were used:
for 50 km/in: to = 5.S see, If = 2.6 see
for 70 tenth: tc = 7.2 see, If = 3.6 see
(The annotation of the cases corresponds to Harders'
(1976) findings regarding the speed dependency of tc and
if in Germany, a finding which has not been confirmed by
this study). The stochastic distributions of critical gap and
follow-up time assumed ~at:
witch = 0.308 x tc
city = 0.387 x if
mints = Max(0.3Ixtc, 2.0)
max to = Min ~ 2.16 x to, 20.0)
In all simulation runs, the drivers behave consistently:

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30
although tc and tf for each Diver are sampled from an
assented distribution, each individual driver uses the same
tC and tf value in any situation (e.g. regardless of time spent
in queue). In subsequent studies it could also be shown
Mat the significant conclusions, drawn from this study, are
also valid if inconsistent drivers - i.e. drivers with varying
critical gaps - are represented In We simulation programs.
Figure 17 illustrates the simulation results using the
maximum likelihood method. Figure IS shows the
comparison between the regression functions of different
mesons for Me case oftc = S.S see and tf= 2.6 sec. Note
that the scales of the y-ax~s on Figure 17 and IS are
different An unbiased estimation method should yield the
same average critical gap estimate over multiple
simulations, independent of the major street volume (or
over traffic parameters).
Based on the simulation Study, Me following conclusions
were reached:
.
The Siegioch method' gives values close to Me
hue values; however, Me variance is large, and Me
results depend on Me major street flow vp.
Overall, Me SiegIoch method is reasonable.
However, it is not the least biased compared to
over melons.
~ t-7 te~ ~ I · ~ ~
· lne maximum likelihood method had an
extremely 0th correlation between the true tc
value and the estimated values. For the 10-hour
simulation period, the maximum deviation
between estimation and true value is below 0.15
seconds. The regression function (Figure 17) of
all Me values exactly coincides with the Rue value
and the result does not depend on the priority
street volume. The maximum likelihood
estimation procedure is also not influenced by
traffic volumes In the nompnority stream.
· The probit estimation procedure described by
Hewitt has an extremely good correlation between
the hue value and He estimated values. The
max~mumdev~ations are below O.2seconds,and
He regression line Is horizontal and coincides with
He Sue value (see Figure IS). Therefore Hewitt's
estimation procedure performs extremely well.
Hew~tt's procedure is also independent of He
minor sheet volume. There is, however, some
indication that He results show sightly more
variance Man He maximum likelihood method
with very small traffic volumes.
.
In conclusion:
.
.
.
A logit model using bow lags and gaps does not
include He hue value and has a strong
dependency on the major street volume.
Therefore, a logit model that uses lags and gaps
see~nsto produce biased results.Ashwor~1968)
indicated that this bias can be predicted under
some circumstances.
The estimation method for critical gaps must be
selected with great care.
Among the tested methods, only the maximum
likelihood method and the Hewitt method are
consistent with respect to major street and minor
street flows.
The recommended method used for this project is
the maximum likelihood method. The theory for
implementation of this method was presented in
chapter two. Inputs to the procedure are the
observed accepted gap and maximum rejected gap
for each driver in a sample. The distribution of
critical gaps is assumed to be log-normal. An
iterative procedure provides an estimated critical
gap, as well as the estimated variance of He
critical gap.
6.0
5.9 - x x ,,,: x
~ 5.9 x x x x
In x X x
58¢ x ~ y x x
5.7 x x xx
57 ~1 ~1 ~ 1 ~ 1 ~ 1 ~ 1 ~1 ~1 ~
0 100 200 300 400 500 600 700 800 900
Major Street Volume, veh/hr ~ = S.8 sac
x Estimated Values
Figure 17. Estimating the Critical Gap by the MDum Likelihood
Method

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31
7.0
6.5
~ 6.0
g 5.5
:~
5.0
4.5
Figure 18. Estimating the Cntical Gap by Various Methods
CAPACITY MODELS
l ~ I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ | Hewitt
_~
0 100 200 300 400 500 600 700 800 Lo' x
Major Street Volume, vehlhr
Is.
Two general categories of models for capacity analysis of
TWSC intersections have been addressed In chapter two:
gap acceptance models and empirical models. The
candidate capacity models considered for testing as part of
NCHRP 3-46 are summanzed in this section.
Mode! I.1
The 1985 HCM memos Is based on gap acceptance theory
developed by Harders. It requires an understanding of the
availability of major stream gaps, He usefulness of each
gap, and the relative hierarchy of the traffic streams at the
intersection. It assumes random arrivals on the major
street and minor stream driver behavior that is both
consistent and homogeneous In accepting or rejecting
major stream gaps.
v e ~(~P ~) ~ ic - try
c = ~ P (8S)
ewpirl3600_ 1
where en is the capacity of He non-pnority stream in
veer, vp is the priority flow rate In Their, tc is the critical
gap In seconds, and If is the follow-up time In seconds.
Mode] 1.2
Another gap acceptance capacity model, developed by
Siegloch and now included In He 1994 HCM Update, is
based on a somewhat different formulation of the critical
gap.
3600 -v ~ / moo
en = e p o (86)
If
where to is the zero gap and the other parameters are
defined as before. Both the Harders model and the
Siegioch model yield nearly identical estimates of capacity
despite somewhat different formulations.
Troutbeck has extended both of these models to account
for non-random or platooned major stream flow patterns
and for multiple lanes on the major sheet. He modified
both Harders and SiegIoch models using a dichotomized
headway distribution.
Mode! 1.3
The modified Harders mode! is given in Equation 87.
a vp e -A ~ i~ - tr - Jay
eat! _] (87)
a v
3600 ~ tm vp (88)
where ~ is He proportion of free-vehicles, tm is the
minimum gap in the major traffic stream, and the over
parameters are defined as before.
Model 1.4
The modified SiegIoch model is given in Equation 89.
av e-A(io~im)
en - (89)
A If
a v
A = P (90)
Model 1.5
An alternative formulation of capacity of TWSC
intersections, developed by Kimber, is based on
emp~ncally derived relationships between capacity, flow
rates, and intersection geometry.

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32
en = A - EBf Vp f - If ~ Vp i Vp ~ (9 l)
where v is the volume on the various opposing and
conflicting approaches, A, B. and C are regression
coefficients, and the other parameters are defined as
before.
Mode! 1.6
Kyte also developed an emp~ncal mode} based on a
somewhat different method of measuring capacities. Both
the Kimber mode} and He Kyte mode! relate the minor
stream capacity directly to the flow rates of the conflicting
traffic streams.
C = Deaf Vi (92)
where Ai's are regression coefficients, and vi is traffic
volume of movement i.
DELAY MODELS
There are four general categories of models for estimating
average vehicle delay at TWSC intersections: gap
acceptance models, queueing theory models, empirical
models, and hybrid models based on t~me-dependent flows.
Models for estimating average vehicle delay based on gap
acceptance theory were developed by Harders and
SiegIoch. Both models require simplifications ~ havoc
flow assumptions to allow for practical application.
Mode! 2.l
Harders developed the delay mode} based on gap
acceptance theory, which is also used In the 1985 HCM.
The mode! is given In Equation 93.
d = 3600
( 1 - e ( P c ., ) (93'
C - V
where d is He average delay of He subject approach and
the other parameters are defined as before.
Mode] 2.2
Siegioch also proposed a delay mode} using the gap
acceptance theory, and the model is given in Equation 94.
3600 ~ e-VS J./ 3600
d= ~_
v5 ~1 - X 1) t94'
where x is the degree of saturation and He other
parameters are defined as before.
Mode! 2.3
Troutbeck used the PolIarczek-Klintchine formula Tom
queueing theory for delay estimation.
d = ~ [1 + XC 1
on ~1 ~ xJ
where cnis the capacibrofthe subject approach and C is as
defined following Equation 50; other parameters are
defined as before.
Mode! 2.4
Kimber and Hollis developed He following delay
formulation based on time-dependent flows. A
simplification of this mode! assumes steady state
conditions for undersaturated flows and a coordinate
transformation to account for oversaturated flows.
d=~i +E+ 3600 too
dim = ~ [~(F2+G)-F] t97'
_ 2vo
E= V (98)
F = V ~ tc ~ fin) y + 2 by --~ ~ + E t99'
G = 2Ty [_ _ TIC - V ~ E ] (100)
h = c
Co + Vo (101)

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33
Y vn
Mode] 2.5
Troutbeck and Akcelic developed a simplified version of
He Kimber-Hollis model Hat depends on He length of the
peak period, T.
d = 3600 + 900 T|X - 1 + ,|(x _1)2 +(C/36oo~xl(lo
This is He equation used In the 1994 HEM Update.
Mode! 2.6
Kyte proposed an empirical approach directly relating
delay to reserve capacity (cn - vn).
d=`5le-P2(c~ ~
where ,B's are regression coefficients.
MODEL EVALUATION AND SELECTION
(104)
A primary objective of NCHRP 3-46 is to develop new
methodologies for computing capacity and level of service
for stop-controlled intersections based on data that are
representative of U.S. conditions. At the heart of these
methodologies must be models that produce reasonably
accurate forecasts of capacitor and delay (assuming that
delay is the basis for determining level of serviced using
input data normally available to practicing tragic
engineers.
What procedure is used to determine if a model is able to
meet this objective? Consider the following statements:
· Specification. The model must be specified using
standard traffic engineering parameters.
ti02' ~
.
.
.
.
Theory. The model specification must represent a
sound underlying theory of traffic flow.
Calibration. The model parameter must be
unmated using the data Hat have been collect
This process of determining the numerical values
of the model parameters is called mode!
calibration.
Range. The model must be able to account for a
wide range of traffic flow and geometric
conditions likely to be encountered by the
practicing traffic engineers.
Validation. The accuracy of the mode! forecasts
must be verified over a wide range of operating
conditions. When forecasts are verified using data
that was not used to calibrate the model, this
process is called model validation.
Qualily. The model must produce better forecasts
than other competing models.
If the standards inherent In these statements can be met, a
model can be put forward as the core of a new
methodology to forecast capacity or delay. These six
statements have been translated into five evaluation criteria
that are used to provide an initial assessment of the
candidate models.
The criteria used to assess each model is listed below.
Ideally, a model should be
.
.
.
.
theoretically sound
easily validated with field data
practical and easily applied by the practitioner
produce sufficient and appropriate measures of
effectiveness as output
relevant in terms of common situations
encountered by He practitioner.
Each of the candidate models described above were
evaluated using these criteria. The results of this
evaluation are given in Table 3. Based on this evaluation,
a few models were recommended for testing against field
data. If none of the recommended models were
subsequently found to adequately represent field data, then
another candidate model could be tested.

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34
Table 3. Model Evalaudon
1 ................................................................................................
................................................................................................................ .
TWSC CapacIb Models
Model 1.1
Model 1.2
Model 1.3
Model 1.4
Model 1.5
Model 1.6
TVVSC Delay Models
Model 2.1
Model 2.2
Model 2.3
Model 2.4
Model 2.5
Model 2.6
,..........
...........
...........
:-::::::::::
:,::: .y::::
::::~::::
P
P
Y
Y
Y
Y
. _ .
P .
Y .
Y .
Y l
Y l
.. . . . . . . . . . . . . . . . . . . . ..
. . . .. .. ...
..... ... , , ~
.......................... .. ..............
. ... ... . . ... . . . . . . . . . . .. . . . . . . .
::::::::::: :-:::: ::::::::::::::::: :::::::::::: :-:: :.: ::::-:-:::-:::-:-:-:-:-:::
.
:::::::- :~:::::::-:: :::::::::~::::: ::::-::::~::: :-: ::::::-:-:~-:-: :-:-:
........... ........ ............. ...................
Y N P Y
Y N P Y
Y N P Y
Y N P Y
P Y Y Y
P . Y Y Y
.
Y Y Y Y
Y Y Y Y
P Y Y Y
Y Y P Y
Y Y Y Y
P Y Y Y
Notes: Y = Yes, meets criterion; N = No, does not meet criterion; P =
Partially meets criterion
Recommen`ded TWSC Intersection Capacity Mo`dels
Mode! 1.4 is a theoretically sound gap acceptance mode}
that considers ~e effects of vehicle platoon~ng on the
major street. The Harders and SiegIoch models by
themselves do not o~er this advantage. Mode! ].5 is a
generali~d empirical approach to capacity estimation, and
is similar to the me~o~proposed by Kyte. The theoredcal
basis of the models based on gap acceptance theory lead to
the selection of the SiegIoch and Harders models, wid1 ~e
corrections proposed by Troutbeck to account for vehicle
platoon~g for test~ng agamst field data. The emp~ncal
models would be calibrated only if the gap acceptance
models were subsequently found to inadequately replicate
field data.
Recommended TWSC Intersection Delay Mo`dels
All six models listed in Table 3 are capable of estimating
delay for steady state conditions. However, o~ly Mode]
2.4 andMoa?eZ 2.5 account for ~e effe~ct oftime-vanation
in traffic flows. This effect is becoming more unportant as
analysts are more Dequently required to address more
highly congested tra~c facilities. The complexity of
Mode] 2.4 limits its practical application; thus Mode! 2.5
was selected for final test~ng. As discussed ~n chapter two,
the accuracy of the delay estimate using any delay models
is I~m~ted by the accuracy of ~e estimated capacit
parameter which is a prima~y input to all delay models.