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OCR for page 11
11
Chapler Three
THEORY AND MODELS
This chapter desonbes the operation of an AWSC
intersection and documents how this descriptor can be
translated into computational procedures for senice time,
capambr, and delay. Candidates models are identified and
evaluated. A recommended procedure is descnbed.
CONCEPTS OF CAPACITY AND DELAY
Capacity
The capacitor mode] for signalized intersections in Be
1994 HCM Update is based on He saturation headway
Tomato for a given approach. The saturation headway
is computed from an ideal value (~.9 seconds) that is
modified based on intersection geometry, traffic control
parameters, and traffic flow conditions. The estimation
of the saturation headway is often complex. For
example, several models have been developed to forecast
the saturation headway for a left turn stream at a
signalized intersection depending on whether the steam
movement is permitted or protected, and whether it
occurs from a shared lane or an exclusive lane. The
capacity flow rate is computed from the saturation flow
rate and tile proportion of the signal cycle that is allocated
to this stream, as Shown in Equation 5.
g
c = s C
(A
where c is He approach capacity, s is He saturation flow
rate, g is the effective green time for the subject
approach, and C is the cycle length.
The capacity of a lane at an AWSC intersection is also
dependent on the saturation headway of that lane. Since
there is no traffic signal controlling the stream
a. . ~.. ~. ~
The HCM defines He capacitor of a transportation facility
as
the maximum hourly rate at which persons or vehicles
can reasonably be expected to traverse a point or a
uniform section of a lone or roadway during a given time
period under prevailing roadway, tragic, and control
conditions.
.
For an AWSC intersection prevailing conditions mean the
geometry of the intersection and the distribution of flow
rates on each of He intersection approaches. Because of
the interaction between the traffic streams on each
approach, and because it is this interaction Hat governs the
maximum flow rate on each approach, two capacity
concepts must be considered:
what is the capacity of a given lane or approach
given He flow rates on the other intersection
approaches. Here the question is: how much can
the flow on the subject approach be increased, if
the flows on He other approaches remain fixed.
what is the capacity of each lane or approach,
given a distribution of flows on all of the
intersection approaches. Here is the question is:
how much can the flows on each approach be
increased if He initial distnbution remains fixed.
Delay
Delay is defined as He time that a motorist spends
traveling at less than his or her desired free flow speed.
Slowing or stopping at an intersection, usually the result of
the red phase of a signal, the presence of a stop sign, or the
presence of a queue, is counted as delay. At an AWSC
intersection, all vehicles must stop before proceeding
through the intersection. The delay inherent in the
operation of an AWSC intersection includes four
components: deceleration, time in the moving queue,
service time, and acceleration time. The deceleration and
acceleration components of delay are often referred to as
geometric delay.
movement, or allocating the n~tofway to each
conflicting traffic stream, He rate of departure is
controlled instead by the interactions between the traffic
streams ~emseIves. There is a degree of conflict that
can be observed Hat increases with the number of
approaches Cat are loaded simultaneously. To a lesser
extent, He geometry of the in~rsechon itself condors this Standard queueing models assume that vehicles form a
rate of departure. vertical stack as they wait In queue for service at the stop
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12
line. While the service Patton at some traffic facilities can
be modeled as either deterministic or random, the pattern
of departure from the stop line at an AWSC intersection is
controlled by the degree of conflict exerted by He presence
of vehicles on the opposing and conflicting approaches.
The results presented later in this report show that the
service times can be assumed to be a finite collection of
values, based on the intersection geometry, He
composition of vehicles in the traffic stream, end the flow
Dies on each intersection approach, with the probability of
each value based on a specific degree of conflict or
likelihood of vehicles on the opposing and conflicting
approaches.
.
Stopped delay has been used as the primary measure of
effectiveness for detennnung the level of senice at
signalized intersections since He introduction of the 1985
edition of the HEM. Stopped delay provided several
advantages over the previous measure, load factor. It was
directly measly able in the field and it related directly to the
motonst's expenence, thus providing a strong link to
perceived level of senice. The 1994 HCM Update
included, for the first time, a delaybased measure of
effectiveness for both TWSC and AWSC intersections.
When accounting for the difference between total delay
and stopped delay, this provides a common measure that
can tee used to compare the performance and operation of
an intersection under different kinds of control.
DESCRIPTION OF INTERSECTION OPERATIONS
between certain turning maneuvers (such as a northbound
led ton vehicle and a southbound through vehicles, but In
generalthe northsouth shams alternate rightofway with
the eastwest streams. A fourphase pattern emerges at
multilane Dogleg intersections where the development of
the r~ghtofway consensus is more difficult. Here drivers
from each approach enter the intersection together as
nghtofway passes from one approach to the next and
each is served In turn.
AWSC Intersections require Divers on ad approaches to
stop before proceeding into the intersection. While the
priority to the driver on the right is a recognized mle in
some areas, it is not a sufficient descriptor of intersection
operations. Whalen fact happens is the development of a
consensus of rightofway that alternates between the
Divers on the intersection approaches, a consensus that is
dependent primarily on the intersection geometry and the
arrival patterns at He stoplme.
Consider an in~s=hon composed of two oneway sheets.
Here, Divers alternately proceed into the intersection, one
vehicle from one approach followed by one vehicle from
the other approach. This same twophase pattern is
observed at a standard fourleg AWSC intersection where
Divers from opposing approaches enter the intersection at
roughly the same time during capacity operations. Some
in~ruptionofthis pattern occurs when there are conflicts
While these patterns are useful to describe the overall
intersection operation, we must next consider how the
pawns affect the capacity of an approach, which we win
describe as He subject approach. At the intersection of
two oneway streets, the headways of vehicles departing
from the subject approach fall into one of two cases. If
there are no vehicles on any of He other approaches,
subject approach vehicles can enter the intersection
immediately after stopping However, if there are vehicles
waiting on the conflicting approach, a vehicle from the
subject approach cannot enter the intersection immediately
after the previous subject vehicle but must wait for
consensus writhe next conflicting vehicle. The headways
between consecutively departing subject approach vehicles
win be shorter for the first case than for the second. Thus
the headway for a departing subject approach vehicle is
dependent on He degree of conflict experienced In
interacting with vehicles on the over intersection
approaches. This degree of conflict increases with two
factors: He number of vehicles on the other approaches
and the complexity of the intersection geometry.
Two other factors affect the departure headway of a
subject approach vehicle: vehicle type and turning
movement The headway for a heavy vehicle will be longer
than for a passenger car. Further, the headway for a left
turn vehicle wid be longer Han for a through vehicle,
which in turn will be longer than for a right turn vehicle.
~ summary:
.
.
AWSC intersections operate un either twophase
or fourphase patterns, based primarily on the
complexity of He intersection geometry. Flows
are determined by a consensus of right of way that
alternates between He northsouth and eastwest
streams (for a singlelane approach) or proceeds
In ton to each intersection approach (for a multi
lane approach intersection).
The headways between consecutively departing
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13
data set and a more extensive measurement of saturation
headways. The model, published as an interim procedure
in TRC 373 (1991) and now included in the 1994 HCM
Update, relates capacity to the relative distribution of
volumes on each of the intersection approaches.
c = E~a~,vp`+E,Joc L,l+~a3LTp~+~a4RTp ~(~)
.
subject approach vehicles is dependent on We
degree of conflict between these vehicles and the
vehicles on the other~nter~ion approaches. The
degree of conflict is a function of the number of
vehicles faced by the subject approach vehicle
(with whom he or she is competing for righto£
way) and by the number of lanes on the
intersection approaches.
The headway of a subject approach vehicle is also
dependent on its vehicle type and its horning
maneuver.
This description of intersection operations must be
translated into computational models or procedures that
can be used to calculate the service lime, capacity, and
delay for given conditions of traffic flow rates and
intersection geometry. A search of the literature yielded
six potential models Hat were considered as candidates for
a new version of the. HEM include both emp~ncal and
queueing models.
CANDIDATE CAPACITY MODELS
Capacity Model 1
Hebert (1963) proposed an emp~ncal model for the
capacity of an AWSC intersection. This mode! is included
as part of~el985HCM.
7200
c = ,^
v (10.15  sv ) i",
pm pm
where vp', is the proportion of volume once main street.
Capacity Mode] 2
Richardson (1987) developed a mode! for estimating
capacity using an M/G/l queuing mode! based on the
service times measured by Hebert
3600
c  ~
[~l  PC1) (}  PC2 )3 bm  Tc] + TC JO
where p is He traffic intensity (the ratio of the arrival rate
to the service rate), tm is He minimum saturation headway,
and Tc is He saturation headway for vehicles faced by a
conflicting approach vehicle.
Capacity Model 3
Kyte and Marek (1989) and Kyte (1990) extended
Hebert's empirical approach for capacity based on a larger
where vpi is He volume on tiLe ith approach, Ej is the
number of lanes once jib approach, LTpk is He proportion
of leRtuming traffic, and RTp,: is the proportion of rigl~t
bwming tragic, and It's are He regression coefficients.
CANDIDATE DELAY MODELS
Delay Model 1
Tro~beck and Akcelik (1991) developed a mode} for delay
based on queuing theory. This mode} includes an
approximation for the timedependence of delay on queue
formation and clearance during a peak period, where T is
He length of He peak penod. It is shown in Equation 9.
d = s + gOOT [ (x  I) + 4(x I) ~ 4soT3]
where T is He length of the study period or congested
penod, x is the degree of saturation, and s is the service
time.
Delay Mode! 2
Kyte (1990) developed an empirically based delay model
teased on the vol~e/capacilyraiio. This model,published
as an interim procedure in TRC 373 and now included in
He 1994 HCM Update, relates He delay on an approach to
He degree of saturation. The model is given in Equation
10.
d = e3sx `103
Delay Mode! 3
Richardson (1987) proposed a delay estimation model for
AWSC intersections based on He MlG/1 queueing theory
model.
[2A(lp)] (11)
where p is the traffic intensity, ~ is the arrival rate, and
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14
is He standard deviation of the service time.
MODEL EVALUATION AND SELECTION
Framework for Mode! Evaluation
A primary objective of NCHRP 346 is to develop new
methodologies for computing capacity and level of
service for s~conkoDed intersections based on data Hat
are representative of U.S. conditions. At the heart of
these methodologies must be models that produce
reasonably accurate forecasts of capacity and delay
(assuming Hat delay is He basis for determining level of
serviced ailing input data normally available to practicing
traffic engineers.
How do we determine that a mode] is able to meet this
objective? Consider He following statements:
· Specification. We must be able to specie a
mode! using standard traffic engineering
parameters.
Theory. The model specification must represent
a sound underlying theory of traffic flow.
Calibration. We must be able to estimate the
mode} parameters using the data that have been
collected. This process of determining He
~ O
numencal values of the mode] parameters is
caned mode! calibration.
Range. The mode] must be able to account for
a wide range of traffic flow and geometric
conditions likely to be encountered by the
practicing traffic engineer.
Validation. We must be able to verily the
accuracy ~ of file mode] forecast over a uncle
range of operating conditions. When forecasts
are verified using data that was not used to
calibrate the model, this process is called moa7e!
valid.
Quality. The mode! must produce better
forecasts than other competing models.
.
.
If the standards inherent ~ these statements can be met,
a mode} can be put forward as the core of a new
methodology to forecast capacitor 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 mode! is listed below.
Ideally, a mode} should be (~) theoretically sound, (2)
easily validated with field data, (3) practical and easily
applied by the practitioner, (4) produce sufficient and
appropriate measures of effectiveness as output, and (5)
relevant~n terms of common situations encountered by the
practitioner.
Each of the models described above were evaluated using
these cnteria. The results of this evaluation are given In
Table 12.
Table 12. Model Evaluation
=....................................
, . ............ ........... . .;
............................................................................................ ..................... .................. ..................... ................. ...................
......................................................................................................
., , ., , , , ~ ~ ~ ~ ~
P Y Y Y Y
Y Y P Y Y
P Y Y Y Y
P Y Y Y Y
Y Y Y __
AWSC: Capacity Models
Model 1
Model 2
Model 3
AWSC Delay Models
Model 1
Model 2
Model 3
Notation
Y = Yes, meets criterion
N = No, does not meet criterion
P = Partially meets criterion
AWSC Intersection Capacity Models
Cap acid Mode} ~ considered only two cases faced by the
subject approach driver and is thus somewhat limited In its
ability to deal with a broad range of traBic conditions.
Capacity Model2 provides a sound theoretical base but is
limited to the two cases of Capacity Mode! I, Capacity
Mode} 3 is empiricalEybased and includes a number of
practical conditions faced by a Diver at a stopcontrolled
approach While the present version of Capacity Mode} 2
is limited to Only two cases, it has one distinct advantage
overate other two models: if it can tee extended to consider
a large number of cases, it can provide capacity estimates
based on a sound theory of ~ntersechon operations.
Capacity Mode} 2 was recommended for final testing.
AWSC Intersection Delay Models
Delay Mode! 3 is based on sound theory but it has two
limitations. It does not directly account for oversaturated
conditions and one of its parameters, the variance of the
service time, may be difficult to estimate for complex
conditions. Delay Mode} 2 is empirically based but does
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~ r
not directly consider oversaturated conditions. Delay
Mode} ~ meets ad of the established criteria and was
recommended for final testing.
DESCRIPTION OF RECOMMENDED CAPACITY
MODEL
Capacity Mode} 2 is the basis for the procedure that is
recommended for the computation of capacity of an
AWSC intersection ~ this section, the mode] is descnbed
In detail. Then, the required extensions to the mode! are
explained and illustrated with a numerical example.
Formulation I. Intersection of Two Oneway Streets
The first formulation of the mode} is based on the
intersection of two oneway streets, each stopcontrolled.
Vehicles on either approach travel only straight through
the~ntersection. Seeiigurel.
_ _.
Figure 1. ConfiguradonFormulabon 1
, Conflicting Approach
Subject Approach
The service time for a vehicle assumes one of two values,
based onHebert's headway measurements: so is the service
time if no vehicle is waiting on the conflicting approach
and s2 is the service time if a vehicle is waiting on He
conflicting approach. The mean service time for vehicles
on an approach is the expected value of this bivalued
distnbution. For the northbound approach, the mean
· · 
service time Is
sAr = s! (1  Ply) + s2 Pw
(12)
where Pw is the traffic intensity for the westbound
approach and is equal to the probability of finding at least
one vehicle on that approach Thus ]~ s the probability
of finding no vehicle on the westbound approach.
By symmetry, the mean service time for He westbound
approach is
sW = sl, (}  Pat) ~ S2 PN
(13)
When Equation 12 is substituted into Equation 13, and
noting that the traffic intensity p is the product of the
arrival rate ~ and He mean service time s, the service times
for each approach can be solved directly in terms of He bi
valued service times and the arrival rates on each
approach, as in Equations 14 and 15.
S], [1 ~ AN (S] + shy]
[1 ~ AI! AW (S]  sit]
(14)
S] [1  AW (Sit + s2~3
w [1  AW AN (sl2  sit] ~ 5)
Formulation 2. Intersection of Two Twoway Streets
As before, He service time for a vehicle assumes one of
two values, so or s2. The mean service time for vehicles on
an approach is again the expected value of this bivalued
OlstnOutlon. AS exl?ected In tms case, computing the
service time is more complex than In formulation 1. A
northbound vehicle win have a service dine of s1 if both
the eastbound and westbound approaches are empty
simultaneously. The probability of tills event is the
product of the probability of an empty westbound
approach and the probability of an empty eastbound
approach. The mean service time for the northbound
vehicle is given in Equation 16. See Figure 2.
sn sit (1  pE) (1  PW3 + s2 ~  (1  pay (1  PA
Unlike formuladon I, it is not possible to directly solve for
He mean service time in terms of a combination of arrival
rates end the bivalued senice times. The service time on
any approach is dependent upon or directly coupled web
the traffic intensity on the two conflicting approaches.
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16
This coupling prevents a direct solution. However, it is
possible to solve iteratively for the service time on each
approach, based on a system of equations for each
intersection in the form shown in Equation 16, above.
Opposing Approach
Cow  gapped
1
., .
Figure 2. ConfigurabonFo~mulation 2
I Conflicting Approach
Tom Right
Subject Approach
Extended Mode] for Single Lane Sites
There are four problems USA these two formulations of the
mode} that must now be rectified. First, the limitation of
only to cases is severe. Drivers face much more complex
condidons based on the interaction of two or more vehicles
simultaneously Eying to enter the intersection. Second,
vehicle type significantly affects the saturation headway.
Third, the drrver~ntheright rule often does not correctly
describe actual intersection operations. It is more likely
Table 13. Probabilitr of Degree of Conflict Case
that, during conditions of continuous queueing, vehicles on
opposing approaches wid enter the intersection at the same
time, regardless of which vehicle arrives first at the stop
line. Fourth, there is a difference In the parameter
ford by the mode} and He parameter that is measured
In the field The first two problems reflect the limitation in
Hebert's database. The third problem reflects an incorrect
description of He operation of traffic at AWSC
intone containedin~e 1985 HEM, namely that each
approach is served iteratively, proceeding
counterclockwise to the next vehicle on the right. The
fourth problem requires a cIanficabon of the
correspondence between the measured serv~c;*e time values
and the forecasted values of headway. This will be
discussed in greater depth later ~ this section.
To resolve these problems, an extension of the mode! has
been developed. The new mode! is based on five
Satan headway values, each reflecting a different level
or degree of conflict faced by the subject approach driver.
Table 13 specifies the conditions for each case and He
probability of occurrence of each. The probability of
occurrence Is based on He traffic intensity on the opposing
end conflicting approaches. The essence ofthemodel,and
its complexity, is evident when one realizes that the traffic
intensity for one approach is computed Tom its capacity,
which in turn depends on He traffic intensity on the other
approaches. This circularity is based on the
interdependence ofthe traffic flow on all of the intersection
approaches, and shows the need for iterative calculations
to obtain stable estimates of departure headway and
service time, and thus, capacity.
,
...................................................................... 1 ~
. If ................... ....... . i""'''"" P ''''' '"'I'""''''''' ''"'"""''I''''''''' ~""''''''""'~""''''''"''"'']
1 Y N N
. . .. . .
2 Y Y N
~
3 N ~
~N
4 Y Y Y
4 ~Y N
5 Y Y Y
N
N
N
y
(lpoXl~p=XlPCR)
(PoXlPa)(l~PcR)
(l~PoXPaXl~P<
(lpoXl~P=xp~
y
N
y
y
(PoXlPcs)(Pce)
(Pod)
(lPoXP~(P`~)
(POX1PC:LXP~
Note: Sub ifs the subject approach Opp is the opposing approach ConL is the conflicting approach from the left Conit is the conflicting approach firm the right
OCR for page 11
17
From Table 13, the probabili~, P[CJ, for each degreeof
conflict case can be computed. The ~a~c ~ntensities on
dle opposing approach, ~e conflictirlg approach ilom ~e
lefl;, and the conflichng approach fLom ~e nght are g~ven
by PO, PGL, and PCR' respectively.
P[C1] = (1  po) (1  PCL) (1  PCR) (17)
P[C2] = (po) (1  PCI) (1  Pc~.) (~)
P[C3] = (1  Po) (PC:L) (1  PCR) 19
+ (1  po) (1  PCE,) (PCR)
P[C~ = (po) (]  P=) (P=)
+ (Po) (Pcz) (1 ~ PCR) + (l  Po) (PCL) (Pc~) (20)
P[C~ = (po) (p=) (p~) (2~)
The departure headway for an approach is ~e expected
value of the saturation headway distribution, or
h~ = ~ P[Ci] h$' (22)
f ~ ~
where P[C] is the probability of the degree of conflict case
Ciandh~, is ~e sah~ion headway for that case, g~ven the
~ffic skeam and geometnc conditions of the ~ntersection
approach
The service time requ~red for the calculation of delay is
computed based on the departure headway and the move
up dme.
s = hd ~ m (23)
where s is the service bme, h~ is the departure headway and
m is the moveup Ome.
The computation of capacity is based on two different
concepts. If it is desired to compute the capacity of the
subject approach, g~ven the flows on~e other approaches,
the volume on the subject approach is ~ncreased
incrementally unti! ~e degree of saturation on any one
approach exceeds one. This flow rate is tiLe maximum
possible flow or throughput on the subject approach under
prevailing conditions. Or, it may be desired to compute
the capacity of the approach assuming a fixed proportion
of h~icon all approaches based on~e g~ven conditions.
Here, the volumes on all approaches are increased
incrementally (maintaining ~e same proportion of traffic
on each approach) unti! the degree of saturation on any one
approach exceeds 100 percent. Aga~n, this flow is ~e
max~mum throughput under prevailing conditions. Bo~
capacity calculations can be considered correct, depending
on the question that ~e engineer or analyst must answer.
The computational method is descnbed below.
Step ]. For each approach, detennine flow rates for each
turning movement, proportion of heavy vehicles, and
geometnc configuration.
Step 2. Compute the base saturation headway for each
degree of conflict case for each approach, g~ven ~e
proportion of left anfd nght turns, proportion of heavy
vehicles, and geometnc configuration.
Step 3. Establish the starting value of ~e departure
headway for each approach. A value of 4.0 seconds is
typically used (note: almost any stardng value w~!
converge).
Step 4. Compute the degree of saturation for each
approach based on ~e product of the arrival flow rate (in
veh/sec) for the approach and ~e initial value of the
departure headway.
Step 5. Compute the revised expected value of the
departure headway for each approach based on ~e
computed degrees of saturation for all of ~e approaches,
using Equation 22.
Step 6. If ~e revised departure headways for any approach
has changed by more than a smaD increment (e.g., 0.01
seconds), go to step 4. If the revised departure headways
for all approaches have changed less than this increment,
go to step 7.
Step 7. Compute anadjus~nentto the forecasted values of
the fdeparture headways to account for ~e dependence in
the headway forecasts.
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18
Step 8. Compute Me service time based on the departure
headway and We move up time.
Step 9. Compute Me capacity based either on concepts one
or two. For concept one, increase the flow rate on the
subject approach until the degree of saturation on any
approach exceeds one, maintaining the flows constant on
the other approaches. For concept two, increase the flow
rates on each approach, maintaining the same proportion
on each approach, until the degree of saturation on any
approach exceeds one.
Example Calculation
An example calculation illustrates the application of this
procedure. To simper the explanation, Me intersection of
two oneway streets wall be assumed. See Figure 3. For
standard conditions, win no heavy vehicles and no fuming
movements, assume Mat Me saturation headways are 3.9
seconds/vein for cases ~ and 5.9 seconds/vein for case 3.
The computations for Me example are summanzed in
Table 14.
200 vph
_ 1
1 ~
1
_ _
1 :.~4
~ 3. Example calculation
Step 2. Compute the base sal;uration headways. There
are only two degree of conflict cases, for which Me
saturation headways are 3.9 sec/veh (case I) and 5.9
sec/veh (case 3~. No adjus~nents are required since Mere
are neither turning movements nor heavy vehicles.
Step 3. Starting value. Starting values for the mean
departure headways for each approach are assumed to be
4 seconds.
Step 4. Initial values of degree of saturation. The degree
of saturation for each approach is Me product of the mean
arnval rate and Me service time. For Me northbound and
westbound approaches, Me initial values are given
computed In Equations 24 and 25.
ply = ID ID = (200Xooo) = 0.22 (24)
= M M = (300X4 ~ = 0.33 <
3600
Step 5. Compute the departure headways based on the
revised degrees of saturation. The expected value for the
departure headway on each approach Is computed based on
the Melees of saturation initially estimated for each
approach, as per Equation 22. Note Mat Me geometric and
traffic flow conditions for this example I~m~t Me degree of
conflict cases to case ~ aIld case 3.
h`,,NB = ~ P[C,] he = h`, (I  P.") ~ hs3 it) (2Q
. ~ ~
he = (3.9X.78) ~ (59X22) = 43 (2~
h4" = ~ P[C,] he = ha, (1  PM) ~ he (Pa) (28)
, . ~
him = (3.9X.6 ~ (S.9X.33) = 4.6 as'
Step 4a Recompute the degree of samranon based on the
Step 1. Initial conditions. The northbound flow rate is new value of headway.
300 vph and the westbound flow rate is 200 vph. Ideal
conditions are assigned.
v`rB S`,B `200x4.6) = 0.2s (30)
3600 3600
PUB = 3~6 ~ = (3O3O6XO4 3) = 0.36 01)
Step 6. Repeat steps 4 and 5 until the departure headway
values remain unchanged. Table 14 shows that Me
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19
depart headway values have nearly stabilized after only
four iterations to 4.6 seconds for the westbound approach
and 4.4 seconds for the northbound approach. What is
important to note is that this expected value is not the
saturation headway for Me approach: rather it is the
expected value of the departure headway for We vehicles
departing from the stop line, given the flows on each of the
other approaches. Again, note that these headways cannot
be used to compute capacity flows. They are not
sa~ionheadways unless Were is a continuous queue on
the subject approach.
Here's why. Suppose each headway is converted to an
equivalent hourly flow rate. In this case, these flow rates
would be 3600/4.4, or 815 vph, for the northbound
approach and 3600/4.6, or 777 vph, for the westbound
approach K the volumes are increased to the values given
as maximum flow rate, the mean headways are re
calculated to 5.9 seconds on bow approaches! What has
been computed is the mean value of We departure headway
if there are 300 vph on the northbound approach and 200
vph on the westbound approach: 4.4 seconds on the
northbound approach and 4.6 seconds on the westbound
approach. What We expected values of the headway on
each approach do allow is the computation of He
propordonofthe hour that is consumed by He vehicles on
each approach. For He above example, He proportion of
the hour that is utilized on He northbound approach is
0.37, for He westbound approach it is 0.26. Thus there
are she some resources of He intersection that are not Ally
utilized How this translates into a calculation of capacity
wiD be shown in step 9 of this example.
Table 14. Example Calculations for Headway and Traffic Intensity
Step 8. Compute the service time for each approach. If
the moveup time is 2.0 seconds, He sentence times are
given in Equations 32 and 33.
s" = 4.6  2.0 = 2.6 see `32,
An = 4.4  2.0 = 2.4 see <33'
Step 9. Compute the capacity. First it is important to
cianii the notion of capacity? The capacity for each
approach cannot be computed directly. But given the
volume on each approach, the mean headway for each
vehicle can be computed (again, not the saturation
headway, but simply the headway between consecutively
departing vehicles) and, based on this headway, the
proportion of the hour that the resources of a given
approach are utilized can also be computed.
Two alternative definitions will be offered here for
consideration.
Capacity definiffon #I: maximum volume increase on
one approach with volumes on other approaches
remaining unchanged!. For a given set of flows, how
much can the flow rate be increased on one specific
approach, with all other flows remaining unchanged.
Capacity definition #2: constant proportion on each
approach. For a given percentage distnbution of flows by
approach, what is He maximum throughput for each
approach and for the intersection.
Step 9a Compute the capacity of the northbound
approach given theflow rate on the westbound approach
and compute the capacity of the westbound approach
given the flow rate on the northbound approach.
Consider the example of two intersecting oneway streets
described above. How much can the flow on tile
northbound approach be increased, if the westbound
volume is fixed at 200 vph? Or, put another way, what
limit does a flow rate of 200 vph on the westbound
approach place on the northbound approach? The
maximuln flow rate on the northbound approach, given a
200 vph flow rate once westbound approach, is 790 vph.
Or, wad a limibug flow of 200 vph on the westbound
approach, the maximum throughput on the northbound
approach is 790 vph. The reverse is also true: if the flow
rate on He nor~bo~d approach is 790 vph, the maximum
flow on the westbound approach is 200 vph. A flow rate
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20
of 200 vph has been maintained on the westbound
approach and the flow rate of 300 vph on the northbound
approach teas been increased incrementally, until the degree
of saturation on either of Me two approaches reaches one.
The resets are shown In Table 15. Note that the seconds
consumer! is the product of the volume and the mean
headway. It is literally the time during which the approach
i~dunng the hour. This value, divided by 3600, is
equal to the degree of saturation.
Table 15. Example Capacity CalculationConcept 1
_< ~  · a_
Volume ~790 200
Mean headway 4.6 S.9
Seconds consumed 3599 1180
Degree of saturation 1.00 0.33
Step 9b. Compute the capacity of the westbound and
northbound approaches given the initial volume
distributions on each approach. In the example from
above, We volume distnbution is 0.40 for the westbound
approach and 0.60 for the northbound approach. How
much can the volume on each approach be Increased,
maintaining We given distribution, until one or both
approaches reach a degree of saturation of one? The final
values are shown In Table 16.
Table 16. Example Capacity CalculationConcept 2
.
.......................... ,, 1 my .. ...............
Volume 671
Mean headway 5.4 5.9
Seconds consumed 3597 2637
Degree of salvation 1.00 0.73 '
Again, this shows that We mean headway for We original
case does not represent the saturation headway, but only
the mean departure headway for the given flow rates.
There is a rule emerging here, looking at the two examples
presented above, Mat might help define capacity. When
the seconds consumed on any one approach reaches 3600,
the flow rates are Me maximum possible flow rates or
throughput. Even Tough Were is seeming spare capacity
on the westbound approach, increasing the flow on the
westbound approach causes the northbound approach to
consume more Can 3600 seconds and Bus Me volcanoes on
its approach must be reduced.
Suggested definition of capacityflow rate: theflow rate
on each approach such that the degree of saturation of
any of the intersection approaches exceeds one.
Extension of Degree of Conflict Cases
It is expect Mat saturation headways at multilane sites
Could be longer than at singlelane sites, ad over factors
being equal. This is the result of two factors. A larger
intersection geometry (i.e., a larger number of lanes)
requires more travel time through the ~ntersecdon, thus
Increasing He sah~radon headway. Additional lanes also
means more driver concision and an increasing degree of
conflict undo opposing and conflicting vehicles, again
increasing the saturation headway.
By contrast, some movements may not as readily conflict
with each other at multilane sites as they might at single
lane sites. For example, a northbound vehicle turning
right may be able to depart simultaneously as an
eastbound through movement, if they are able to occupy
separate receiving Am when departing to He east. This
means in some cases that the saturation headway may be
lower at multilane sites.
The theory described earlier proposed that the saturation
headway is a fimcdon of He directional movement of the
vehicle, He vehicle type, and He degree of conflict faced
by the subject vehicle. This theory is extended here for
multilane sites with respect to the concept of degree of
conflict: saturation headway is affected to a large extent
by the Amber of opposing and conflicting vehicles faced
by He subject driver. For example, In degree of conflict
case 2, a subject vehicle is faced only by a vehicle on the
opposing approach. At a twolane approach intersection,
dlere can be either one or two vehicles on the opposing
approach. It is proposed here Hat each degree of conflict
case be expanded to consider the number of vehicles
present on each of the opposing and conflicting
approaches. These cases are defined in Tables 17 and
IS, for twolane approach and threelane approach
intersections.
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21
Table 17. Degree of Conflict Cases for TwoLane Approach Intersections
or

x
x
x
.: ~
.. ~ ~ ~1
.................................................... , ............................................................ ....................................................................
..........................  . t ~. ~ 'I"""'' At" '"' 1"" " "'it'" ' 1
4 x x 2,3,4
S x x x 3,4,5,6
Table 18. Degree of Convict Cases for ThreeLane Approach intersections
::5
n
1 2
1.2
x
x
x
x
2.3.4
3,4,5,6
l
~52 ~... l.'.'''.' '.'', '.'.' ,. , . ~l[
~O ~
2 x
3
_
x am' _
5 I x I x I
Extended Mode! for Multilane Sites
For multilane sites, separate saturation headway values
have been computed for Me number of vehicles faced by
the subject vehicle for each of the degree of conflict cases.
This requires a further extension of Me service time mode!
to account for this increased number of subcases.
Table 19 lists Me 28 possible combinations of Me number
of vehicles on each approach for each degree of conflict
case for intersections with two lanes on each approach.
1,2,3
1,2,3
x
x
x
x
2,3,4,S,6
3,4,S,6,7,8,9
These combinations can tee further subdivided if a vehicle
can be on either one of the lanes on a given approach
Tables 20 and21 lists the 64 possible combinations when
alternative lane occupancies are considered, where a "1"
indicates that a vehicle is in the lane while a "O" indicates
Mat a vehicle is not in a lane.
As before, the probability of a vehicle at the stopline in a
given lane is p, the traffic intensity. The product of the six
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22
traffic intensities (encompassing each of the six lanes on
Me opposing or conflicting approaches) gives the
probability of any given case occurring
The departure headway of We approach is Me expected
value of the saturation headway distribution.
12
he = ~P[C,J ho'
f=1
where Ci represents each of Me twelve degree of conflict
subcases and h,i is Me saturation headway for that case.
The iterative procedure to compute the departure headways
and capacities for each approach as a function of Me
depar~eheadways on the other approaches is the same as
described earlier. The alimony subcases clearly increase
Me complexity of this computation, however.
Table 19. Probability of Degree of Conflict CaseMultilane AWSC Intersections (TwoLane Approach Intersections)
~ :: ~.~ :~ ~
~... : ::::::::::: ~::::::::::::::::~
1 , ig i ~: ~ ~ ~ 1  ~ S my 1 Hi i ' 1' ""~i
1/0 1 0 0 0
.
2/1 1 r 1 1 1 1 0 1 0
2/2 1 1 1 2 1 0 1 0
3/1 1 1 1 1 1 7 o 1
1 O O 1
3t2 1 1 1 0 1 1
1 O O 2
.
4/2 1 1 O 1
1 1 1 O
1 O 1 1
4/3 1 1 1 2 T 2 1 0
0 1 2 1 2
1 2 O 1
1 1 O 2
4/4 1 2 2 O
1 2 O 2
1 . O 2 2
S/3 1 1 1 1
S/4 1 1 2 1
1 2 1 . 1
1 1 1 2
5/5 1 1 1 2 1 2 1 1
1 2 1 2
1 1 2 2
.. .
l 5/6 1 1 1 2 1 2 1 2
Notes: DOC Case/lrehicles is the degree of conflict case and the number of vehicles on the opposing and conflicting approaches.
r'
~1
~1
~1
n
2
.
2
2
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23
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