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OCR for page 5
s
Chapter Two
AVAILABLE METHODOLOGIES
Capacity analysis at TWSC intersections depends upon a
clear description and understanding of the interaction of
drivers on the minor or STOP-controlled approach with
drivers or vehicles on the major street. Both gap
acceptance and emp~ncal models have been developed as
a means to describe this interaction. Critical gap and
follow-up time are the two major parameters used in
venous gap acceptance capacity models. Most delay
models were developed based on the degree of saturation,
or vol~ne/capacity ratio. This chapter documents the
theoretical background of critical gap and follow-up time
estimation, and various capacity and delay models Cat
have been developed for analyzing TWSC intersections.
CRITICAL GAP AND FOLLOW-UP TIME
Definitions
Gap acceptance theory relies upon three basic elements,
including the size and distribution (availability) of gaps In
the major traffic swam, the usefulness of these gaps to Me
minor stream Divers, and the relative Lions of the
various traffic sins atthe~ntersection. The first element
to consider is the proportion of gaps of a particular size in
He major traffic skeam offered to the driver entering Tom
the minor stream, as wed as the pattern of ~nter-a~Tival
times of vehicles. The second element to consider is the
extent to which drivers find the gaps of a particular size
useful when attempting to enter the intersection. It is
generally assumed In gap acceptance theory that drivers are
both consistent and homogeneous.
The critical gap Its) is defined as the minimum-length time
internal that allows intersection entry to one minor stream
vehicle. Thus, Me Diverts critical gap is the minimum gap
in the opposing traffic stream that would be found
acceptable. A particular Diver would therefore reject any
gaps less than this cntical gap and would accept any gaps
greater than or equal to this critical gap.
The follow-up time (t) is the time span between the
departure of one vehicle form the Tninor stream and Me
departure of the next, under a condition of continuous
queuing. Put another way, If is the headway that would
define the saturation flow rate for the lane if there were no
conflicting vehicles on movements that have the pnonty of
entering the intersection.
Another parameter used while measuring follow-up time
and field capacitor is called move-up time (to,,) Move-up
time is defined as the time span between the departure of
one vehicle Tom the minor steam to Me arrival at the stop
line of Me next vehicle under a condition of continuous
queuing. Move-up time is only a portion of the follow-up
time.
Critical gap and follow-up time are the two major
parameters for various gap acceptance capacity models.
The values ofthese two parameters significantly affect the
final capacity result; therefore, it is important to correctly
measure these two parameters based on certain traffic and
intersection conditions. The 1985 HCM adopts values
from German studies, however, some inconsistency has
been found when the values are applied to traffic
conditions in the United States. This study has produced a
comprehensive estimation of cntical gap and follow-up
time measurements based on U.S. conditions.
Several previous studies addressed critical gap and follow-
up time measurement procedures. However, the majority
of He studies only discussed the estimation methodology
from a theoretical point of view. Few studies have
addressed in detail how to measure these parameters in the
field. For example, most of the studies reviewed in the
literature search address the estimation process In relation
to available data regarding accepted and the rejected gaps,
however, none of the these studies has explained how to
define gap events when a minor steam driver faces
different conflicting streams. Without this discussion, the
procedure remains at a theoretical level and does not
provide enough guidance to be used in practical traffic
engineering applications.
Critical Gap Estimation Procedures
Inreality,thecridcal gap is not a constant value. Instead it
is a variable vnth different values for different drivers and
for each individual driver over time. Consequently, the
critical gaps Hat drivers use for Heir decision-making
process at unsignalized intersections have a stochastic
distribution, characterized by:
a minimum value as the lower threshold, which is
greater Man zero,
an expectation of average critical gap (or mean
OCR for page 6
6
critical gap), which is often denoted as "the
critical gap" In theoretical models,
a standard deviation, and
a skewness factor, which is expected to be
positive, that assumes and accounts for a longer
tad! on the right side of the distribution.
This distribution and its parameters cannot be directly
estimated because Me cntical gap cannot be observed.
Only the rejected and accepted gaps can be measured. To
estimate the critical gap, procedures must be used to
estimate the distribution or its parameters as closely as
possible using the accepted and rejected gap data.
Such an estimation procedure should be consistent. If the
minor street Livers within a specified composition of
traffic streams have a given distribution of cntical gaps,
We procedure should be able to reproduce this distribution
closely. The procedure should reproduce the average
critical gap reliably, without being dependent on other
parameters such as:
.
.
.
traffic volumes on the major or minor street,
delay experienced by Me drivers, and
over external influences
OD]Y if the results from a procedure are consistent can it be
used to study the influence of external parameters on the
critical gap. Othe~w~se the influences identified through
emp~ncal studies might result from the inconsistency of the
estimation procedure, in which case they are not really
related to He extemal parameters being investigated. Tests
of consistency are reported In chapter three.
Several previous studies have addressed critical gap
estimation methods and procedures. Miller (1972) has
documented some of the early procedures, and additional
procedures have been developed more recently. The most
commonly used procedures include the following:
maximum likelihood procedure (e.g., Troutbeck,
1992)
SiegIoch (1973)
Meshwork (1970)
Raff (1950)
Harders (1976)
Hewitt (1992)
Roget mode] (e.g., Cassidy, 1994)
SiegIoch's method is quite simple and reliable for
estimating critical gap (tc) and follow-up time (t) from
saturated conditions (i.e. continuous queueing on the minor
street). His procedure includes He following steps (see
Figure I):
.
.
.
.
Observe a traffic situation during times when
Here is, Handout interruption, at least one vehicle
queueing in the minor street.
Record die number of vehicles, "i", entering each
main stream gap of duration "I".
For each of the gaps accepted by "i" drivers,
compute He average of He accepted gaps (shown
as x's in Figure I).
Find the linear regression of these averages
(average gap as a function of i).
The increase of this regression line Mom i to i+!
iStf.
The intersection of He regression line wig He
honzontal axis gives- to = tc - tf /2.
OCR for page 7
1
9t
8t
an
a) 7
cay
·_
a) 6
>
a
as
of
5
4
3
2
1
O
-
I I I~ 1 1 1 1 1 1 11 1 1
0 5 10 15
Gap Size, see
20 25 30
Eo Average Values - Regression Line t=ftn)
Figure 1. Siegloch's Method of Estimating Critical Gap and Follow-up Time
SiegIoch's method is easy to apply and reliable, since the
memos used to estimate cntical gap and follow-up time is
exactly compatible with the derivation of the
corresponding SiegIoch capacity formula. However, the
method is only suitable for use with data dunng
oversaturated conditions. Traffic operations in
undersaturated conditions can also provide information
about to and If, however, the SiegIoch procedure cannot be
applied to undersaturated conditions.
It is more complicated to estimate the critical gap to from
traffic observations of undersaturated conditions because
a critical gap cannot be directly measured; however, it can
be assumed that the critical gap for a driver on the minor
street is greater than the Diverts maximum rejected gap
and smaDer~an~e accepts gap. This is hue if Me driver
behaves consistently. A series of accepted gaps ta (gaps in
the priority stream accepted by minor street vehicles) can
be descnbed by an emp~ncal statistical distribution
Unction (see Figure 2~; however, We distribution function
of critical gaps to must be to the left of the ta distnbution.
O
-
>0.5
. ~
E
O
3
4 5 6 7
Crap, sea
Figure 2. Cumulative Distribution Functions of Accepted Gaps F,(t)
and Critical Gaps Fcff)
By assuming exponentially distributed priority stream gaps
and a normal distributions for ta and to, Ashworth
(196S,1970) found that the average critical gap to can be
estimated Tom {a (mean of the accepted gaps ta in
OCR for page 8
8
seconds) using the following equation:
t = ~ - V X S2
(1)
where Vp is the priority traffic volume, veh/sec, and sa2 is
Me variance of the distribution of tat sec2.
Hewitt (1985,1992) also developed a procedure for
estimating to under more realistic conditions than those
assumed by Ashworth
Troutbeck (1992) describes a maximum likelihood
estimation procedure for cntical gaps. This procedure can
be used to estimate the critical gap under traffic conditions
that are not necessarily oversaturated. The details of the
maximum likelihood procedure are discussed below.
The maximum likelihood method of estimating the critical
gap dishibudon is based on the fact that a driver's critical
gap is greater then his largess rejected gap and smeller then
his accepted gap. The first step is to assume a probabilistic
distribution for the critical gaps. For most cases this can be
assumed to be log-nor~nal. This distribution is skewed to
the right and has non-negative values, as would be
expected In these circumstances. The distribution is
reasonably general and is acceptable for most studies.
The foHow~ng notations are used for subsequent equations:
Yi is the logarithm of the gap accepted by the ith
driver
Yi is ~ if no gap was accepted
xi is the logarithm of We largest gap rejected by
the ith Liver
xi is zero if no gap was rejected
~ is the mean ofthe distribution of the logarithms
of the individual drivels critical gaps
02 iS Me variance of the distribution of the
logan~ns of Me individual Diverts cntical gaps
fit ~ is the probability density function for the
normal distribution
Fit ~ is the cumulative distribution function for the
normal distribution
The maximum likelihood of a sample of n drivers having
accepted and largest rejected gaps of (ye x) is
i.!
II [Fluff) - F(X`~]
n
The logarithm, L, of Me likelihood is Men
n
L = ~ ~ [~f) ~ AXED]
f-1
(2)
(3)
The maximum likelihood estimators, ~ and 02 Mat
maximize ~ are the solutions to the two equations:
BL = o
aL = o
~2
That is, they are solutions of:
(4)
(I
Iffy`) FOXY)
-
BL ~B~ BM (O
= , = 0
8p f-1 [if) ~ [(X`)
AL
_:
~2
It can then be shown Mat
This then leads to the two equations Mat must be solved
iteratively using numerical methods:
at) FOXY)
-
n ~`s2 ~2 = 0 (7)
f-1 [if) ~ F(X`)
annex' = -~X)
arty = X 2p J(X)
(8)
(9)
`) it) O
f-1 Fief) ~ F(X`)
(10)
OCR for page 9
9
n (X' ~ it) ,tt,X`) ~ (A' ~ 4) ](Yt)
~ pity`) ~ jinx) = 0 (11)
where fix), ffy3,F(x) and F(5r~) are also functions of ,0 and
~2
A computer program was developed to solve these
equations. The mean cntical gap to and the variance s2 can
then be computed by:
~ = en ~ °.so2
S2 = [C2 (e°2 -I)
It is this mean cntical gap that has been used In venous
gap acceptance capacity and delay models.
These cntical gap estimation procedures, along with the
Ram Harders, and loft mode} procedures, were evaluated
regarding their consistency to produce estimates of critical
gap. The results of these tests are reported In chapter three.
t~2'
(13) where
Obtaining Follow-up Time
Unlike the cntical gap, Me follow-up time can be directly
observed. For such an evaluation, the times between
vehicles from the minor street entering the same gap of the
priority stream should be measured, during periods of
continuous queueing. For practical purposes, at least nf
observations should be used to get an estimate of sufficient
reliability(S%probability that We estimate is in a range of
rf around Me Rue estimated.
Assuming Of= 0.4 x If (Herders, 1976), the following
equation can be obtained from sampling theory:
n =a x ~
f S of
(14)
of is the necessary number of observations
rf is the relative error = ef / If
of is the absolute error
At is Me standard deviation of Me statistical
distribution of Me If
at is a function of S. and is given as follows: If S
= 90°/0, at= 0.4; If S = 95%, at= 0.6; If S = 99%,
al= ~
OCR for page 10
10
CAPACITY MODELS
Two general types of capacity analysis models have
been used for TWSC intersections, gap acceptance
models and emp~ncal models. Stance unsignal~zed
intersections give no positive indication or control to
the driver regarding when he or she can enter the
intersection, the driver alone must decide when it is
safe to enter the intersection by looking for a safe
opportunity, gap or headway in the major stream
traffic. This is the basis of Me gap acceptance
process that is used in most analysis models of
capacity and level of service at unsignalized
intersections. At TWSC intersections, a driver must
also respect the priority of other drivers. Other
vehicles may have priority over the Diver byingto
enter the traffic stream and the driver must yield to
these drivers. Various models have been developed
based on the gap acceptance theory and different
assumptions of Me gap acceptance process. Models
developed based on gap acceptance theory are
closely related to queuing theory. Emp~ncal models
were developed using regression techniques. This
section documents these available models as well as
Me related theories.
Gap Acceptance Models
This sub-section presents the basic premise of two
stream capacity models based on gap acceptance.
Then the hierarchy of the different traDic streams at
a TWSC intersection and therefore Me definition of
conflichug volumes for a given movement are
discussed.
Capacity with Two Traffic Streams. To understand
traffic operations at a TWSC intersection, it is
useful to concentrate on the simplest case first
(Figure 3~.
AD gap acceptance methods for TWSC intersections
are derived from a simple queuing mode! in which
the crossing of two traffic streams is considered. A
pnonbr traffic stream (major stream) of volume vp
(veh~r) and a non-priority traffic stream (minor
stream) of volume vn (vestry are used in this
queuing model. Vehicles from Me major stream can
cross the conflict area without any delay. Vehicles
Dom He minor stream are only allowed to enter the
conflict area if the next vehicle from the major
stream is still at least to seconds away (tc is the
critical gap). Otherwise, they must wait. Moreover,
vehicles from the minor stream can only enter the
~ntersechon at least If seconds after the departure of
the previous minor stream vehicle (tf is the follow-
up timed.
I TV
Noe~pnor~ ~/
1 , ~
v
P
Priory dram
Figure 3. Illustration of the Basic Queueing Theory
The mathematical derivation of the cap acid en for
the minor stream is as follows. Let gets be the
n~nberofminor stream vehicles that can enter into
a major stream gap of duration I. The elected
number offer t-gapsperhow~s3600 x up x fell,
where fits is the statistical density function of the
gaps m the major stream, an~vp is (he To We olthe
major steam in vps. Therefore, We capacity
provided by t-gaps each hour is 3600 x vp x fit) x
g(t).
To determine the total capacity, expressed in veh/s,
the following must be Antedated over the whole
range of major stream gaps:
~0
an = Vp Jo At).git) dt
t ~ O
(1~
where en is the maximum traffic volume that can
depart Tom the stop line in the minor steam in
OCR for page 11
11
veh/sec, vp is the major steam volume in veh/sec,
fate is the density function for the distribution of
gaps in the major steam, and gets is the number of
minor stream vehicles that can enter into a major
stream gap of size t.
Two types of headway distributions for Me major
stream have been used In gap acceptance models:
Me negative exponential distribution, and Cowan's
M3 (Cowan, 1975) distnbution. The probability
density function of Me negative exponential
distribution is given in Equation 16:
J(t) = Ae~'` (16)
where A is ~earT~val rate or traffic volume on the
major stream, veh/sec.
Cowan's M3 distribution modifies the negative
exponential distribution by introducing a
"bunching" factor a. The resulting probability
density function is given in Equation 17:
J(fl= 1 -a
O
Ale -l(t - 'A
t> t
m
t = t
m
t < t
m
(1~
where tm is Me minimum inter-vehicle Tacking
headway, a is the proportion of free vehicles
Raveling wad headways greater Man tm seconds, and
is a decay constant given in Equation I8:
A = ~ '' V (~)
Based on the gap acceptance model, the capacity of
the simple two-stream situation (Figure 3) can be
evaluated using elementary probability theory
methods if the following assumptions are used:
constant to and if values (driver population
is homogenous and consistent),
negative exponential distribution for
priority stream headways, and
· constant traffic flows for each traffic
stream.
For first assumption, two different formulations for
the term gate must be distinguished. This is the
reason for two different families of capacity
equations. The first fancily assumes a stepwise
constant function for gets, resulting In the following
integer values for gate:
g(t) = ~ n.pn~t) (~19)
n ~ O
where putt) is the probability that n minor stream
vehicles enter a gap in the major stream of duration
t.
Pn (t)= ~ ~ for arc+ (n-l) x tf ~ t C tc + n x If
The second family of capacity equations assumes a
continuous linear function for gate, which may result
in non-integer values for gate. This approach was
first used by Siegioch (1973) and later by
McDonald and Armitage (1978~.
O for to<
g(t)= ~ t° for trio (21)
tf
where
to = tc ~ 2
Once again it should be emphasized that, in both
Equations 19 and 21, to and If are assumed to have
constant values for all drivers.
Combining Equations 15 and 19 results in Me
capacity equation used by Drew (1968) and by
Harders (1968~. These authors denved the equation
in Me following form:
OCR for page 12
12
or
v e-vp.tc
~-e P r (22)
Vpe~~P(tC - tr)
n e vp i, 1 (23)
Combining Equations 15 and 21 results In
SiegIoch's (1973) formula:
en = -.e P ° (24)
J
These formulae result In a relationship of capacity to
conflicting flow as illustrated In Figure 4. Both
approaches for gets produce useful capacity
formulae where the resulting differences are small
and can usually be ignored for practical
applications.
1200
_ 1 000 '
.='
600
400
200
O
. ~ v
. =~
==
_=
200 400 600 800 1000 1200 1400 1600
Conflicting Priority Volume, veh~hr
Slegloch --- Harders
Figure 4. Relationship Between Capacitor (C=) and Priority
Street Volume (vp) for the Two Streams (for this example, tc
= 6 see, and h= 3 see)
More general solutions have been obtained by
replacing We exponential headway distnbution for
Me pnor'~,r stream with a more realistic one (e. g., a
dichotomised distribution). If the stepw~se constant
function for gate is used (Equation 19), this more
general equation is
av e~A("~'~
- = P At (25)
1-e~ r
This equation is illustrated In Figure 6. This is also
similar to equations reported by Tanner (1967),
Troutbeck (1986), Cowan (1987), and others. If ~
is set to 1 and tm to 0, Men Harders' equation
(Equation 22) is obtained.
If Me Linear relationship for gets according to
Equation 21 is used, Men the associated cap acid
equation is
av e~A('°~'~
c = -P
n Atf
or
(2o
(1-Vptm) e ° is (27)
of
This was proposed by Jacobs (1979) and has also
been referred to as the Troutbeck modification to the
Siegioch equation. Changing the a term has a
pronounced effect on capacity, as shown In Figure 5.
1200
_ 1000
c)
~ 800
·= 400
is'
2CiO
0
`i I
1 ~-
r~
-Tanned Fourth ~
1 1
a= 1
-~=Q~ ~-
~-I=
~ ~ ~-
0 200 400 600 800 1000 1200 1400 1600
Conflicting Priority Volume, veh/hr
Figure 5. The Effect of Changing cc in Equation 26
OCR for page 13
13
Equations 25 and 26 require two additional
parameters, ~ and tm, as defined in Equation 17.
Sullivan and Troutbeck (1994) have developed a
computer program to calibrate these two terms
based on field measured headway date. The program
has been modified dunug this research project to
calibrate a and tm for each site based on Me traffic
flow charactenstics.
Unfortunately, the three assumptions discussed
earlier are idealized, and do not reflect actual
operations. To address more realistic conditions,
researchers have tried dropping one or more of these
assumptions. SiegIoch (1973) studied different
types of gap distributions for the priority stream
(Figure 6) using analytical methods. S~m~lar studies
have also been perfonned by Troutbeck (1986),
Catchpole and Plank (1986~. Grossmann (1991)
investigated these effects using simulations. These
studies showed ~at:
.
Capacity decreases if the constant to and if
values are replaced by realistic distributions
(Grossmann, 1988~.
Drivers may be inconsistent ~ i.e. one driver
can have different cntical gaps at different
times). A ~iverm~ghtreject a gap one time
and accept it at other times. This effect
results In Increased capacity.
The use of consistent and homogeneous
drivers in the analysis produces much the
same result as the use of inconsistent and
heterogenous drivers.
If the exponential distribution of major
stream gaps is replaced by more realistic
headway distributions, capacity increases
by about We same order of magnitude as
the decreasing capacity effect of using a
distribution for to and if values (Grossmann
1991 and Troutbeck 1986).
The elect of a driver's valving behavior in relation
to opposing flow was also investigated during this
research. For example, drivers may adjust their
critical gap and foDow-up time values according to
Me magnitude of the opposing flow. Drivers may
tend to accept smaller gaps when the opposing flow
is high and they experience higher delays. Therefore,
Me critical gap and follow-up time may no longer be
constant values, but flow-dependent values. Also, a
minor stream vehicle may enter a small gap and
force Me major stream vehicle to be delayed. This
situationis referred toas"limitedprior~ty,'.Figure
7 shows Me hypothetical capacity results using flow
dependent cntical gap and follow-up time values
and considering limited priority situations. In this
figure, the bunched headway distnbution for Me
major stream was used, and Me following linear
functions were assumed for the critical gap and
foDow-up time: to= 5.15 - 6.3 x vp, and If= 3.0 - 3.0
x vp.
1200
1000
~ 800-
.~
2t 600
.~`,', 400- .
m
200
O
~ /
I
.
_ ~
__
=_
_ 200 400 600 800 1000 1200 1400 1600
Conflicting Priority Volume, veh/hr
1 1 1 1 1
Jacobs(Siegloc}^routbeclc) |
Figure 6. Comparison of Capacities for Different Types of
Headway Distributions in the Major Street Traffic Flow (for
this example, tc = 6 see, tf = 3 see, and t, ~ = 2 see)
1 1 1 1
1 1 1
Flow Dada
_~
0 200 400 600 800 1000 1200 1400 1600 18 30
Major Stream Flow, veh/hr
-
Figure 7. Capacities with Flow Dependent tc and tf and
Limited Priority
OCR for page 14
14
When to and tf are made flow dependent and/or
"limited pnority?' is assumed capacity is increased
at the lower flows and reduced at the higher flows
compared to the case assuming a constant critical
gap, follow-up time, and random headway
distribution. The Emoted priority case produces
results that are close to being linear relationship,
providing good reason to believe that the
approaches used by Kimber (1980) and Kyte (1991)
are reasonable.
in summary, many unsignalized intersections have
complicated driver behavior patterns, and there is
often lithe to be gained Dom using a distribution for
the variables to and tf or complicated headway
distributions. Moreover, using simulation
techniques, Grossmann showed that these effects
compensate for each other, so that the simple
capacity Equations 22, 23, and 24 give realistic
results In practice.
Hierarchy of Traffic Streams. Unlike We simplest
case (depicted in Figure 3) from which the basic gap
acceptance capacity formulae are derived, a
hierarchy of traffic streams exists at all TWSC
intersections. These different levels of priority are
established by traffic ndes as follows:
.
Rank ~ steam has absolute pnorits,r and
does not need to yield right of way to
another stream,
Rank 2 stream must yield only to a rank
stream,
Rank 3 stream must yield to a rank 2
stream and In turn to a rank ~ stream, and
Rank 4 stream must yield to a rank 3
stream and in turn to rank 2 and rank ~
streams (minor sheet left turn movement at
a 4-leg intersection).
These trailic streams are illustrated In Figure 8. The
figure also illustrates that the major street left turn
movement has to yield to the through traffic on the
major road. The left turning traffic from the minor
road has to yield to all other streams, including We
queuing tragic In the rank 2 stream.
Rank 1: 2.3,~6
2: 1,4,E,12
3: ~11
4: 7,10
a)
b)
CroN; intcrsoct;on
T-lat Lion
1 1~1~
Rank 1: ~.5
2: 4,.
3: 7
Figure 8. Traffic Streams and Their Level of Ranking (note:
the numbers adjacent to the arrows indicate the enumeration
of streams given in the 1994 HCM)
No rigorous analytical solution is known for the
derivation of We capacity of rank 3 movements such
as the minor street left turn movement at a T-
junction (movement 7 in Figure S). In this case,
models using gap acceptance theory typically use
impedance factors as an approximation. Impedance
is a concept introduced by Harders (1968), which
has since been reconsidered wig some significant
refinements by Briton and Grossmann (199 I). For
each movement the probability that no vehicle is
queuing et the entry is given bypO. This probability
of a queue-tree state is given with sufficient
accuracy by Equation 28 (BnIon, 1988~:
Pot = 1 - x
(28)
where Po2 Is the probability of a queue-free state for
a rank 2 movement, x is the degree of saturation,
v/en, v is We traffic volume of a rank 2 stream, and
en is the capacity of the rank 2 stream.
As a practical approximation of impedance to rank
3 vehicles by opposing rank 2 vehicles, the
following considerations apply. Only during the
portion pO 2 of the total time, can vehicles of rank 3
enter the intersection due to highway code
regulations. Therefore, for rank 3 movements, the
basic value en for the potential capacity must be
reduced to pO 2 X On to get the real potential capacity
ce
cc,3 Po,2 Cn,3
(29)
OCR for page 15
The foDow~ng examples refer to turning movements
numbered in Figure 8.
For a Tjunction, this means
C,7 = po4 X C,,,7 (30)
For a crossjunction, this means
c,, = Px x cat
CC.II Px Call
with
PA = PHI X po'4
01)
(32)
(33
The subscripts refer to the index of the movements
according to Figure 8. Next, the values of poll and
Poll can also be calculated according to Equation
33.
For rank 4 movements (minor street left turn at a 4-
leg intersections, the statistical dependence between
the Poi values in rank 2 and rank 3 movements
cannot be calculated from analytical relations and
must be emp~ncally derived. They have been
evaluated in numerous simulations by Grossmann
(1991; cf. Brilon and Grossmann 1991~. Figure 9
shows adjustments to account for the statistical
dependence between queues in steams of ranks 2
and 3.
To calculate the maximum capacity for the rank 4
movements (numbers 7 and 10 in Figure S), the
auxiliary factors pzg and Pelt should be calculated
first:
Pa,! = 0~65py~
PYJ + 0.6 ~(34)
e
15
09 1 1 1 1 1 1 1 1 1~
of ===== 1 1 ~1
_0.7 --_-- :~ ~
0O2 __ - -- LLII
O.!
O _~ ~ ~ ~ ~ ~ __~_
O 0.1 02 03 0.4 05 0.6 0.7 0.8 0 ~1
Pyti
Figure 9. Reduction Factor to Account for Me S~6shcal
Dependence Between Streams of Ranks 2 and 3 (E3rilon,
Grossman, 1991)
where i equals g or 11 (~e movement numbers
according to Figure 8) and Pyi ~Px POi (the product
Pyi is used as Me entry value on the horizontal axis
of Figure g).
The maximum cap acid of the minor street lefiturn
movement at a 4-leg intersection is calculated as
C~7 = (pan x Po,~2) Cn,7 (35)
c. lo = (Pal x pO.9) c,,.~O (36)
Conflicting Traffic Volume. For each traffic
stream, the maxi n potential capacitor en should
be calculated using the method shown In chapter
To using We sum of all conflicting traffic volumes
with higher rank Man the rank of the traffic stream
in question. To aid in correct calculations, Table 1
can tee used This table basically corresponds to the
German guidelines from bow 1972 and 1991 as
well as to the 1985 and 1994 HEM.
OCR for page 18
18
of different ranks of priority.
Although this general equation may represent an
adequate solution, no investigations of such
comprehensive regression analysis are known. In
addition to the influence of priority stream traffic
volumes on the minor street capacity, Me influence
of the geometric layout of the intersection should
also be included. To do this, the constant values
could be related to road widths or visibility or even
over characteristic values of the intersection layout
by using another set of I~near regression analysis.
investigation of TWSC intersection capacity has a
number of advantages and disadvantages:
The advantages include:
.
.
.
.
Using a similar approach, Kyte (1991) also
developed empirical models using We regression
technique:
Cn = ~ Af V'
t4~'
In addition, Kyte proposed another method for We
direct estimation of ~ntersechon capacity In We field.
The method is based on the fact that the capacity of
a single-channel queueing system is We inverse of
the average service time. The service time at an
unsignalized Intersection is He time that a vehicle
spends in the first position of a queue. This capacity
estimation method is discussed In detail in chapter
four.
The estimation of delays and queue length using the
empincal approach is again derived using queueing
Peony. Here, however, these equations use the
maxims entry Bow as an input, and so delay is not
calculated Tom to and If values. In practice, these
empincal regression delay equations are always
combined wad the Kimber-Hollis capacity.
Using the empirical regression technique for the
-
There is no need to establish a dleoretical
model.
Reported emp~ncal capacities are used.
The influence of geometric design
elements can be taken into account.
The effects of prionty reversal and forced
priority are taken into account
automatically.
There is no need to descnbe driver
behavior in detail.
The disadvantages include:
Transferability to other countries or other
times (driver behavior might change over
time) is quite limited. For application
under different conditions, a large sample
must always be evaluated.
No real understanding of traffic operations
atone intersection is achieved byte user.
Generally, He derivations are based on
driver behavior under oversaturated
conditions.
Each situation to be described with the
capacity formulae must first be observed.
On the one hand, this requires a large
effort for data collection. On the other
hand, many of He desired situations are
found infrequently, since these
oversaturated intersections are often
changed to signal control or modified
using over measures before data can be
collected.
.
.
.
OCR for page 19
19
DELAY MODELS
In the calculation of the capacity of a two stream TWSC
intersection, it is assigned that He headways In the major
stream have a specified distribution. This distribution can
be changed to another distribution and a new equation for
capacity can be developed.
The capacity estimates are Independent of the order in
which the major stream headways arrive. The Intersection
would have the same capacity if the gaps offered to the
minor stream drivers in one hour, were ordered so that the
smallest gap was presented first then the next largest, the
next largest and so on. However, the same ability to re-
order gaps is not permissible for delay estimates.
Using these usual gap acceptance assumptions, and
assumptions about the amval patterns of the minor stream
drivers and the order of the arrival of the major steam
headways, the delays can be estimated. Delays are a
function of all of these stream and driver attnbutes.
There are generaDytwo~nds of delay models: steady state
solutions and time dependent solutions. This section
documents these delay estimation models, including the
definition of delay, the underlying theory, and their
formulae.
Definitions of Delay
In queuing theory, delays are estimated using a vertical
stacking model. Vehicles are assumed to all stop at the
stop line and are stacked vertically. Accelerations and
decelerations are instantaneous. This delay, ceded
q~eueing delay, includes some accelerations, such as In the
Diverts move-up process, but not ad of them. The
remainder of the delay is geometric delay. Total delay is
the sum of the queueing and the geometric delays.
Kimber et al. (1986) discuss the role of queuing and
geometric delay and introduce the term "pure geometric
delay", which is the delay caused by the geometry of the
intersection when the driver is certain that no other
vehicles are approaching. If the driver needs to check for
He presence of other vehicles then his negotiation speed is
lower. In the analysis presented here, He geometric delay
includes three components:
· the "pure geometric delay";
· the extra delay associated with He presence of
other vehicles; and
· the delay from stopping or slowing doom.
Some of the delay calculated as geometric delay has
already been included In the queuing delay. Figure 10 is
an example showing the factors associated with the
geometric delay. Here, die minor street is illustrated as
yield controlled.
Consider a gap with a duration of exactly the critical
acceptance gap, le. Under gap acceptance rules, only one
vehicle can enter in this gap and it may enter at only one
instant. It is assumed for this discussion that this instant is
to after the passage of the last major stream vehicle (and to
- to before the next major stream ones. If the entry queue is
empty, then a minor stream alTiving at this instant would
not be delayed It is assumed that the minor stream vehicle
would arrive at the stop line at t, traveling at its approach
speed. If it arrived either shortly after or shortly before t,
then it would have been delayed. Figure 10 illustrates the
idealized trajectory of this undelayed vehicle based on
these gap acceptance assumptions. More realistically, an
entering driver would need to slow down to negotiate the
intersection and would incur a delay of dock dI,eg and daccell
before being able to resume the departure speed. The
vehicle would cross the stop line at fir s after He last major
stream vehicle, and would then have a geometric delay of
the sum ofthese three lesser delays, as shown In Figure 10.
dgeom = ddeCat + dun +d~t
(42)
where dgeom is the geometric delay, d, is He delay that
occurs when decelerating Dom He approach speed to the
speed at which He driver is able to negotiate the
intersection, dog is the delay Hat occurs when traveling
through the conflict point at the negotiation speed, and
daeCe,~ is the delay that occurs when accelerating Dom the
negotiation speed to the departure speed.
The negotiation speed is dependent upon the maneuver
type and can also be dependent on the cntical gap. The
negotiation speed is the minimum speed at which He
vehicle would Gavel when driving Trough the intersection
when there is a long gap in the major streams. This driving
behavior is assigned to be exhibited when a gap is equal to
the critical gap and the driver has arrived at the specified
time shown In Figure 10.
At a TWSC intersection, He negotiation speed is zero and
the minor stream vehicles always stop. The delay dreg
OCR for page 20
20
would also be zero andante ddoo ~1 and d, would be longer
than shown in Figure 10. Similarly Me negotiation speed
may not be constant. For a night ton movement at a
TWSC intersection, the negotiation speed might start from
zero, Men increase to 10 mph before Me vehicle rounds
the corner. The vehicle would then accelerate to a higher
speed.
There are other cases of geometric delay resulting from
various arrival patterns. This has been more fully
documented in the NCHRP 3~6 Working Paper 19
(NCHRP 346, 1995~. In this study, queueing delay was
measured Tom Me field videos.
Major
Stream
Vehicles
-
Distance Trajectory Trajectoty
~ I4 dale
W/~/:se~abon
/
Figure 10. Geometric Delay for an Undelayed Vehicle Arriving in a
Gap of Tc
Underlying Theory
This section is a review of several of the melons available
to estimate delays. For this discussion, it is assumed that
Here are only two streams - one stream has priority and the
other steam has to yield right of way to the priority
steam. The priority traffic stream (major stream) has a
volume of vp (veh/hr). The non-priority traffic stream
(minor stream) has a volume of vn (veh/hr3.
Steady-State Delays Based on the Gap Acceptance
Assumptions. A general form of the equation for the
average delay per vehicle is
d = Did,, (~1 + Y £ ~ (43)
where y and ~ are constants, x is the degree of saturation
(vn / en), en is He capacity of the minor stream to enter the
intersection, given a major stream flow of vp, and DO is
Adams' delay (Adams, 1936~.
Arlamq' dela~r is the average delay to minor steam vehicles
when minor stream flow is very low. It is also the
minimum average delay experienced by minor steam
vehicles.
Troutheck (1990) gives equations for fly, £, and Dmjr. based
on formulations by Cowan (1987~. If minor stream
vehicles are assumed to arrive at random, then y is equal
to 0. On the other hand, if there is platoon~ng In the minor
stream, Den ~ is greater than 0. These equations include
the gap acceptance parameters: the critical gap and the-
follow-up time.
For random minor stream arrivals, £ iS given by
eVP if - V [f - ~ + V (e VP " _ By)
v (e VP tr ly)
Note Hat ~ is approximately equal to I.0. Dan, depends on
the platoon~ng characteristics in He major stream. If the
platoon size distribution is geometnc, then Troutbeck
(19X6) found Hat
sac - t~ 1 At2 _ 2t + 2t a
D . = e _ t _ _ + m (45)
mm avp c A 2(tmA + a)
where this the ass~nedmin~mu~n headway between major
stream vehicles and a is the proportion of tree vehicles, or
He bunching factor.
The blanching factor ~ is in effect a platoon size parameter.
The more vehicles that travel In bunches, He longer the
platoons. The average platoon size is 1/~ although He
variance of the platoon size distribution can vary
independen~dy of lo.
Differences in He platoon size distribution affect He
average delay per vehicle, as shown in Figure Il. Here, He
critical gap was 4 see, He follow-up time was 2 see, and
He priority steam flow was 1000 veer. To emphasize
the point, tibe average delay for a displaced exponential
sonority stream was 4,120 seconds, when He minor stream
flow was 400 Whir. This is much greater Man He values
OCR for page 21
21
for Be Tanner and exponential headway examples, which
were around ~ I.5 seconds for the same major stream Bow.
The two curves for Tanner headways have a different
ordenng of the headways. The curve with the "geometric
bunches" is where the headways are assumed to be
independent of each other. The curve path the Borel-
Tanner bunches has the same distribution of headways but
there are more longer platoons or bunches. The vertical
difference between these curves is a direct result of this
ordering. The average delay is also dependent on the
average platoon size, as shown In Figure 12. The
differences In delays are dramatically different when Be
platoon size is changed.
50 , Displaced 1 Legate /B~ = /
40- - / }Bengal / Bomb //
/ _~ Tanncr Headway
<10~. ~ ~ ~ ~ ~
O- . . . . . . . .
. . . . . . .
O 200 400 600 800 1000
Major Stream Flow, vehlhr
Figure 11. Average Steady-State Delay Calculated Using Different
Headway Distributions
so
40
30
In
Cal 20
'a 1 0
o
2
0 200 400 600 800 1000
Major Stream Flow, vehlbr
Figure 12. Average Steady-State Delay Based on Geometric Platoon
Size Distribution and Different Mean
Tanners (1962) mode} has a different equation for Adams'
delay, because the platoon size distribution In the major
stream has a Borel-Tanner distribution. This equation is
D = ev' ('e - ·,J ~ ~ ~ opt" ~ 2ta'Vp A) (46)
Another solution for average delay has been given by
Harders (1968~. It is not based on rigorous queuing
theory. However, as a first approximation, the following
equation for the average delay to non-pnonty vehicles is
quite useful.
d=
- e -(V' ·c ~ very
+ t'
n _ v (47)
3600
M/G/! Queueing System. Queuing theory can also be
used to approximate He gap acceptance process by
assuming the simple two-stream system (Figure 3) can
be represented by a M/G/l queue. The sewer is the first
queueing position on the minor street. The input into the
system is formed by the vehicles approaching from He
minor street which are assumed to arrive at random (i.e.
exponentially distnbuted) arrival headways ("My. The
time spent in the first position of the queue is the service
time. This service time is controlled by He priority steam,
with an unlalown service time distribution. The "G"
represents a general service time. Finally, the " I" in MIG/l
stands for a single service channel, for example, one lane
in the minor street.
For He MIG/l queueing system, in general, the PolIaczek-
Khintchine formula is valid for the average delay of
customers in He queue
x W(1 + C2)
D = ~ (48)
g 2 (1 - X)
where W is the average service time (the average time a
minor street vehicle spends in the first position of He
queue near the intersection), and Cw is the coefficient of
variation of service times,
OCR for page 22
22
Cw = ~ (49)
and Vary is the variance of service times.
The total average delay of minor street vehicles is then
d=Dq+ W.
In general, Me average service time for a s~ngle-channe}
queuing system is the reciprocal of the capacity. If the
capacitor is derived from Equations 25 and following, and
includes We service fine W In the total delay, We following
results are obtained:
Cn ~ x C] t50'
where
~1 + C2
2
Up to this point, the denvabons are generally valid. The
real problem now is to evaluate C. Only the extremes can
be defined, which are:
Regular service: Each vehicle spends the same
dme~ntihe first position. This gives Vary = 0,
C = 0, and C=0.5 This is the solution for the
M/DA queue.
Random service: The times vehicles spend in the
first position are exponentially distributed. This
gives Vary = E~,C=1 and, C=l.O. This
gives the solution for the ~1 queue.
Unfortunately, neither of these simple solutions applies
exactly to He unsignali~d intersection problem However,
as an approximation, some authors recommend He
application of Equation 50 with C = 1.
Equation 43 can be further transformed to
d = Den, (1 + y) (1 + ~ + y 1 - x) (51)
where £ Andy are documented ~ Troutbeck (1990).
This is similar to the PoDaczek-Khintchine fonnula
(Equation 48~. The randomness constant C is given by
(~+£~/~+~) md the hum 1~ x 0~) c" be considered
to be an equivalent capacity or service rate. Both terms are
afimchonofthe critical gap parameters to and If and of the
headway distributions. However, C, it, and £ values are
not available for all conditions.
As a generalproperty of the M/G/1 system, the probability
pa of the empty queue is given by
Pa = ~ - X ~s2'
This formula is sufficiently realistic for practical use at
unsignalized Intersections.
MAG2/] Queueing System. Other researchers have found
Bathe service lime distribution in the queuing system is
better described by two types of service times, each of
which has a specific dis~ibudon:
We is the service time for vehicles entering He
emptr system (no vehicle is queueing on the
vehicle's arrival)
W2 is Be service time for vehicles joining He
queue when other vehicles are already queuing
Again, id bode cases the service time is the time the vehicle
spends waiting in tiLe first position at Be stop line. The
Initial concepts for this solution were Produced by
Denser (1962, 19643 and in a comparable way by Tanner
(1962) as well as by Yea and Weesalml (1964~.
The average time that a driver spends in He queue of such
a system is given by Yeo's (1962) formula:
Cn ~E(W~) - E(W2) ~ E(W2~ (53)
g 2 v y
where Dq is the average delay of vehicles in the queue at
higher positions Mange first, E(W,) is the expectation of
We, E(W~23 is the expectation of (We x Wit, E(W2) is He
expectabonofW2, and E(W22) is He expectation of (W2 X
W21. Note also He following definitions:
OCR for page 23
23
· v = y+z
y = 1 ~n E(W2)
Z = Cn E(W1)
cn= capacity
The probabili~ pO of the empty queue is
Po = YV (54)
The application of ~is formula shows that ~e di~erences
with Equation 52 are quite small ~ < 0.03~.
Including the service time in the total delay results ~n ~e
following equation (see Brilon, 19gg):
E(W) Cn ty X E(W~ ~ Z X E(W251 (55)
~ = ~ - x
v 2 v x y
Formulae for the expectations of W. and W2 respectively
have been developed by Kremser (1962~:
E(W') = ~ (eV9tc _ ~ _ y t) + [i (56)
E(W~2) = 2 (eV'c - ~ - vpt~-~ t' - ~ ~ ~ {c ~Sn
E(W2) =-tl - e ~ ~
E(W22) = 2e 2 (e ~P`c - vptc) (1 -e ~~* ~ - vp [` e ~~,'r `59'
~p
However, Kremser (1964) showed that the validib~r of these
eq~ions is restricted to the special case of tc equal to tf,
which is unrealistic for TWSC unsignal~zed ~ntersections.
Daganzo (1977) gave an ~mproved solution for E(W2) and
E(W2), which was extended by Poeschl (1983~. These new
formulae were able to overcome Kremser's (1964)
res~ictions. However, Kremser's first approach, Equation
56, g~ves qu~te reliable approx~mate results for tc and tf
values ~at apply to realistic unsignalized ~ntersections.
The foDow3ng comments can also be made about the newer
equations.
The formulae are so complicated that they are far
from being suitable for practice. Computer
programs must be wr~tten to provide numerical
solutions.
These formulae are only valid under the three
assumptions presented earlier. That means that,
for practical purposes, the equations can only be
regarded as approximations and only apply for
undersaturated or steady-state conditions.
Time-Dependent Solution. Each of the solut~ons g~ven by
conventional queueing theory above are steady-state
solutions. These are the solutions that can be expected for
non-t~me~ependent traffic volumes aDer an infinitely long
t~me, and they are only applicable when the degree of
saturation x is less ~an I. In practical tetms, Morse
(1962) found that the results of steady state queueing
theory are only usefill approximations if the observation
dme, T, is considerably greater than the expression on the
right side of Equation 60.
T > ~ ~ ~ (60)
This equation can only be applied if cn and vn are nearly
constant during time interval T. If T is not greater that the
right side of the equation, then time~ependent solutions
should be used.
Mathematical solutions for ~e time-dependent problem
have been developed by Newell (1982) but may need to be
simplified for practic~ng analysts. There is, however, a
heur~stic approximate solution for ~e case of ~e peak
hour efEect given by Kimber and Hollis (1979) which is
based on ~e ideas of Whiting, who never published his
work. During the peak period itself, haDic volumes are
greater ~an those before and ailer that peno~ They may
even exceed capacity. For this situation, the average delay
during ~e peak penod can be estimated as:
d = D~ + E + ~ (61)
D~ = 2(~2 + G - F) (62)
OCR for page 24
24
F = - : n n en)] (63)
G = tic- - (Cn - An) E| (64)
CnO (CnO - VnO) (65)
h = en ~ CnO + Vn° (6o
h
Y = - en (6
where en is the capacity of the intersection entry during the
peak period of duration T and cnO is We capacity of the
intersection entry before and after the peak period, vn is the
minor street volume during the peak period of duration T.
and v,,, is the minor street volume before and after Me peak
period. Note: each of these terms is given in veh/s; each
delay term is in seconds).
C is again similar to Me factor C mentioned for the M/G/l
system, where C = ~ for unsignalized intersections and C
= 0.5 for signalized intersections. (Refer to Kimber and
Hollis, 1979)
This formula has proven to be quite useful in estimating
delays.
A simpler equation can be obtained by using a co-ordinate
transformation memos. This is a more approximate
method. The steady-state solution is acceptable for sites
with a low degree of saturation and the deterministic
solution is satisfactory for sites with a high degree of
saturation say, greater than 3 or 4. The co-ordinate
transformation method is one approach to estimating
delays that falls between these two extremes. The
equations do not have a theoretical basis.
The steady-state solution for the average delay to the
entering vehicle is given in Equation 6X:
S { 1 - X5 ~(68)
where the terms Dn,,r`, A, and ~ were defined previously.
The deterministic equation for delay, dd is
d, = D . + to + (Xa - 1) cnT
O , overwise
Ad > 1 (69)
where Lo is the initial queue, T is the time the system is
operating In seconds, and on is the entry capacity.
These equations are diagrammatically illustrated in Figure
13. For a given average delay, the co-ord~nate
hansfonnation method gives a new degree of saturation, xt,
which is related to the steady-state degree of saturation, X5,
and the deterministic degree of saturation, Xd, such that
x, - x, = 1 - xs = a (70)
(D
In
0.5
O
o
,
7
-- 1
. . · .
0.5 ~1.5 2
Degree of Sabura~don, x
figure 13. The Coordinate Transforrnabon Technique
Rearranging Equations 68 and 69 gives two equations for
X5 and xd as a function of the delays dd and d5 which then
gives xt as:
2(d~ - Dig ~ =;Vn do ~ Din - rid
x, = T do - Dot cDm,n (71)
Rearranging Equation 71 and cropping the t subscripts
OCR for page 25
25
gives:
where
and
D, = 1 (]A2 + B - A)
A = T(1 - x)
B = 4D~
T(1 -XXI +-Y) + TASK fir) -(1 -S (-+D~,1n
(72)
- _ - Dmm(2-~) (73)
1~1
(74)
Equation 74 ensures that the transformed equation watt
asymptote to the Feministic equation and gives a family
of relationships for different degrees of saturation and
periods of operation.
A simpler equation was developed by Akcelik and
Troutbeck (1991~. The approach here is to re-arrange
Equation 68 to give:
= ~ + T Ax_) ~ ~
(78)
The average delay predicted by Equation 7g is dependent
on the initial queue length, the time of operation, He
degree of saturation and He steady state equation
coefficients. This equation is an approximation and can
then be used to estimate He average delay under over-
saturated conditions and for different initial queues. The
delay equation for unsignal~zed intersections in the HCM
1994 has He above formulation.
Reserve Capacity. ~ He 19gS HCM, reserve capacity
was used as the measure of electiveness. This decision
was based on He fact that average delays are closely
related to reserve capacity. This relationship is shown In
Figure 14. Based on this relationship, a good
approximation of He average delay can also be obtained
from reserve capacities. For instance, Equation SO with C
equal to 1 gives the equation:
d = 1 ~ 1 ~
Cn ~ 1 -x J
(79)
a = ~X D~(Y + £X,) (75y
5 mm and this reduces to
Thisis approximately equalto: d = ~= R- (so)
D~,(Y + Art)
a ~
d5 - D=,.....
(76)
Kthis Is used In Equation 70, then rearranged, the resulting
equation for the non-steady-state delay is:
~ - D ~ ~ 2 1~+ (X-~)T t]~_~(X-~ -2~ (~+y)~
(77)
A similar equation for He M/M/1 queueing system can be
obtained if £ iS set to 1, y is set to zero, and Drain is set to
I/cn. The result is:
where R is the reserve capacity, en - vn.
.
According to Figure 14, as a practical guide, a reserve
capacity R> 100 pcu~ generally ensures an average delay
of less than 45 seconds, which is He LOS F threshold in
~e1994HCMforunsignalizedintersections.
OCR for page 26
o
50 . \\ Va-W - he
(me ~ Va -12SO. 1000, 750, 500 )
~30- '/ ~
_
~ 20 ~ , Van lSOO crew
O
100 200 300 400
Reserve Capacity R. vehthr
1
Figure 14. Average Delay d in Relation to Reserve Capacity R
Briton (1995) has developed equations for average delay
using the reserve capacity concept for use when Me
intersection is oversaturated. Using a revised deterministic
equation to account for We effect of the initial queue
before the peak and assuming the equilibrium queue
length after the peak is the same as the equilibrium queue
leng~before We peek period:
d = -B +;B2 +b (81)
where
B= ~ (bR--) (82)
b = |-R ~°~ 2 (1 R1)) c 1:
(83)
and Ris the reserve capacitor and No is We lethal queue. In
addition, Rf is an arbitrary point with suggested values as
follows:
· For To ~ h, Rf= -IOOvph
· For T~ 0.5h, Rf= -200 vph
· For To 0.5 h, Rf= -300 vph or -400 vph
Also, T Is We duration of the peak period In hours, c is the
capacitor of We system cluing the peak penod in veh/s (c is
also considered to be the capacity before and after We
peak), and R. is the reserve capacity before and after We
peak period in vps.
Kyte (1991) has proposed an empirical approach directly
relating delay to reserve capacity:
d = ~ e-~(Ca-V) (84)
where ,B's are the regression coefficients, en is the minor
steam capac~W, veh/hr, and vn is the minor stream volume,
veh/hr.
Effect of Capacity Estimation on Delay Estimation. The
MIM/l queueing mode} can be used to demonstrate how
changes In the capacity estimate can influence the delay
es imate,using the approximate Briton (1995) relationship
for the delay in oversalted conditions given as Equation
g ~ and following If it is assumed that T is equal to ~
hour,NOis 0, end R. is halftime capacity, then, for the same
via, Me delay changes considerably as qm changes. Shown
In Figure 15, this illustrates the importance of using the
correct capacity estimate.
50
40
30
In
Cat 20
10
o
o
Capacity, vehlhr
200600
1000 1400 11300
~I ~I ~
500 1000 1500 2000
Minor Sbeam Flow. veh~r
Figure 15. Average Delay Against He Minor Stream Flow and for
Different Capacities
For an example of how important the estimate of capacity
is to Me delay estimate, assume Mat Me minor stream flow
is 800 vehJhr in Figure 15. The delay would be estimated
to be a function of the capacity, as shown in Table 2. No
matter how sophisticated the delay methodology used, the
quality of the forecast is dependent on obtaining a good
Mate of capacity. If the capacity estimate is inaccurate,
OCR for page 27
27
It is evident that the average delay during near-saturated
and oversaturated conditions is strongly dependent on the
peak duration T. Any increase in the peak period results
in a significant increase in the average delay.
SUMMARY
This chapter documents the theoretical background of
critical gap estimation procedures, capacitor and delay
models for TWSC intersections. SiegIoch's method for
estimating critical gap and follow-up time is easy to use,
however, it requires continuous queued condition on Me
minor street approach. The maximum likelihood method
can be used under conditions that are not necessarily
oversaturated, and it provides both Me mean cni~cal gap
and standard deviation.
250
5:
a)
`~ 200
-
~ 150
a:
100
50
o
. ~
~ Z
then Me delay estimate wall also be inaccurate, no matter
which delay mode! is used.
Table 2. Example of Average Delay and Capacity for vn = 800
veh/hr
l
1800 3.6 l
1600 4.S l
1400 6.0 l
1200 8.9 l
1000 17.3
Elector the Duration ofthe Peak Period on the Estimate
of Delays. The over term Mat has a significant influence
on Me estimate of delay is Me duration of the peak period
(or Me analysis We period). This sensitivity to T is shown
In Figure 16.
300- 1 1
. 1 1
1 1 1 1 12` it
Anal lane Penod T -+
l l l l
J~ 1 ~
F
4
= ~
,
r
0 200 400 600 800 1000 1200 1400
Minor Stream Flow, vehthr
Figure 16. Example of We Effect of Changing T in Me Delay Model
Although more generalized capacitor models have been
developed by considenng non-random major steam
headway and using vaned cntical gap and follow-up time
values, capacity models proposed by Harders and Siegioch
can generally give reasonable results, when random major
stream headway and constant critical gap and follow-up
time are used.
Various delay models were developed based on Me degree
of saturation (either volume/capacity ratio or reserve
capacity. The accuracy of delay estimation primarily
depends on accurate capacity estimation. Time~ependent
delay models should be used under oversaturated
conditions.
OCR for page 28
28
.,
Representative terms from entire chapter:
gap acceptance
