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OCR for page 205
APPENDIX C
CAPACITY CHARACTERISTICS
FOR
INTERCHANGES AND CLOSELY-SPACED INTERSECTIONS
This appendix describes the development, calibration, and application of models that
collectively can be used to predict the capacity of traffic movements at signalized interchange ramp
terminals and other closely-spacedintersections. Specifically,ffiese models predict three important
capacity characteristics: saturation flow rate, start-up lost time, and end lost time. The mode! for
each characteristic is developed using theoretic constructs that incorporate Me factors that have an
influence on the characters tic's magnitude or duration. These models were calibrated with data
collected at twelve interchanges (the field studies are described in Appendix B). It should be noted
that the traffic characteristics described in this appendix reflect passenger car performance as all
heavy vehicles were excluded from the database.
This appendix includes five main sections. The first four sections describe the development
and calibration of models of saturation flow rate, start-up lost time, clearance lost tone, and large
utilization. The last section describes the proposed form of these models and their application to
capacity analysis.
C.1 SATURATION FLOW RATE
This section describes the development and calibration of a saturation flow rate mode}
applicable to the signalized movements at interchanges and closely-spaced intersections. Separate
models are developed for the left-turn and through movements at these junctions because of their
unique operational character. In each case, the saturation flow rate mode} is denved from a mode]
of the discharge headway process for queued vehicles. The saturation flow rate is defined as the
minimum discharge headway reached and sustained by the discharging queue. The models are
sensitive to factors that effect the discharge process at interchanges end closely-spacedintersections
such as distance to the downstream queue and signal timing.
Three topics are discussedin the remainder ofthis section. First, the basic issues related to
the measurement of the minimum discharge headway and start-up lost time are descnbed. Then, the
minimum discharge headway and resulting saturation flow rate models are developed for through
movements. Finally, the minimum discharge headway and saturation flow rate models are
developed for the left-turn movements. The development of start-up lost time models for each of
these movements is described in the next section.
C.. Minimum Discharge Headway and Start-up Lost Time
Methods of Computation. The discharge of a traffic queue, upon presentationofthe green
indication, is charactenzedby the reaction time of the queued Divers to the indication followed by
C-l
OCR for page 206
their steady acceleration to a desired discharge speed. As a result, the first few vehicles have
relatively long headways; however, Me headways of subsequent vehicles aradualIv decrease as Rev
1 ~1 1 · 1 it- ~1 T To, · , ~,1 ~ - ~-
~ =~ ~ J - J
approach ule Desired Discharge speed. u~umate~y, the a~scnarg~ng stream converges to a relatively
constant headway In He range of I.8 to 2.0 seconds per vehicle. This constant value represents the
minimum discharge headway H of He queue; its Inverse represents the saturation flow rate s.
Typical discharge headways observed for each of He first ten queue positions are shown in
Figure Cal. As this figure suggests, the minimum discharge headway is electively reached by the
sixth queue position; however, it should be noted Hat He headways for each subsequent queue
position continue to decrease slightly. The first four or five queue positions. having headwords larder
_ _ J. ~ ~ ~ _, ~_7
. ~ ~ · ~ · · ~ ~ r ~- ~ ~ · o ~
than me m~n''num, each Incur art Increment or lost-t~me due to the react~on-tune process and the
subsequent acceleration to the desired discharge speed (i.e., the speed at saturation flow). The sum
of these lost-time Increments for each vehicle represents the total start-up lost time Is of He queue.
3 5 Discharge Headway, sec
Measured Headway
3.0
2.5
2.0
1.5
11
~ Is~ 1106~----
12:
' Avg. Minimum Discharge Headway (Ha)
1 1 1 1 !
5 6 7 8 9
0 1 2 3 4
Queue Position
Figure C-1. Discharge headway by queue position.
10
In recognition of Be relatively constant headway achieved by He higher queue positions, He
1994 Highway Capacity Manual (TICK) (1) recommends that the minimum discharge headway be
estimated as the total headway of He combined higher queue positions divided by He number of
vehicles observed in these queue positions. In this context, the total headway is computed as He
difference between the discharge time of He last vehicle to discharge TO d that ofthe last vehicle
to Incur some start-up lost time Tat p. Thus, the minimum discharge headway can be computed as:
C-2
OCR for page 207
T - T
_ d(J) d0-l)
H
J J-(J-1)
1
J
J-(~-1) Ash,
(C-1)
where:
Hj = minimum discharge headway based on specification of the jth queue position as the first to
achieve the minimum discharge headway, sec/veh;
Td(i) = discharge time of the ~ queued vehicle (i = j-l to by, see,
h, = headway of the vehicle in the id queue position' see;
j = ``specified'' first queue position to discharge at the minimum discharge headway; and
J= last queue position to discharge.
The HCM (1J indicates Hat the saturation flow rate s can be computed from He minnnum discharge
headway using He following equation:
3,600
s. =
J HJ
(C-2)
where:
Sj = saturation flow rate for the subject lane based on specification of the jth queue position as
the first to achieve the minimum discharge headway vphgpl.
By definition, He start-up lost time can be estimated from the discharge time of He mth
vehicle and the average minimum discharge headway as follows:
~ o.' = Too.' - j H.
=
where:
Is I = start-up lost time based on Hj, sec.
~ hi - ~ j H ~
(C-3)
For most practical applications, the HCM (1) recotrunends that the fifth and higher queue
positions can be used to estimate the minimum discharge headway (i.e., j = 51. In general, these
averages are sufficiently accurate as to pelt the estimation of the capacity of an intersection traffic
movement; however, they are likely to be higher than the true minimum discharge headway H. As
C-3
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was discussed at He start of this section, H represents the average headway converged upon by the
highest queue positions (say, those above position 10~.
In practice, it is difficult to quantify H because of the difficulty of obtaining an adequate
number of headway observations at the higher queue positions. As a result of these sampling
difficulties, Equation C-l is generally used to compute H; (as an estimate of H) for the purpose of
.. . . . ,. ., ~ ,.. . . ... .. . . . . .. ...
capacity analysis. As the results ot this research share this purpose, the models developed in this
appendix are based on H5 rawer Man H. lithe implications ofthis approach are discussed in the next
section, however, subsequent to that discussion, all references to the m~nnnum discharge headway,
saturation flow rate, or start-up lost the variables do not include the "5" subscript
Equations C-l, C-2, and C-3 were used to compute the average discharge headway,
saturation flow rate, and start-up lost time for the Trough movements included in the field studies.
These averages are reported ~ Table C-~. Examination of He data In this table indicates Hat Here
is some vanation in perfonnarlce among the various interchange configurations, although these
differences are not statistically signtficar~t. Moreover, the limited number of sites studied for each
configuration type (! to 3 sites each) requires caution In any extrapolation of the observed Rends to
generalities about He discharge characteristics of various interchange configuration types.
Table C-~. Through movement discharge characteristics by junction types
| Min. Discharge Headway, N SaL Flow | Start-up Lost Time, to
Junction Configuration Rate, s
Type Obs. Mean Std. Dev. (pcphgpl) Obs. Mean
(sectveh) (sec/veh) (see)
nterchange |CDI I 3,027 1 1.93 1 0.53 1,865 1 687 2.48
IPUCIOA]B 1 667 1 1.~8 1 0.53 1,915 1 226 2.66 .
|ParCIO B 1 1,291 1 1.90 1 0.51 1,895 1 261 2.30
I SPUI I 1,945 I 1.86 1 0.53 1,935 r 443 2.52
IllnDI3 1 1,257 I I 67 r 047 2,156 1 147 2.88
1 ImDI w/f 1 1,105 1 1.88 1 0.53 1,915 1 293 3.45
I TOtal / AVera 7e 1 9~92 1 1.87 1 0.52 1,925 1 2,057 2.65
l IterSeCtiOn |AGI 1 5,971 T 1.88 1 0.52 1,915 1 1,474 2.46
.
Std. Dev.
(see)
1.07
0.96
1.09
o.ss
0.81
.03
.08
.07
Notes:
1
AGI - at-grade intersection; TUD! - tight urban diamond interchange (no frontage road or U-tu~n lanes); SPIJI -
single-point urban interchange; CDI - compressed diamond interchange; TUDI w/f - tight urban diamond
interchange (with frontage road and U-turn lanes); Parclo B - partial cloverleaf with off-ramps beyond overpass;
Parclo AB - partial cloverleaf with both off-ramps on the same side of the overpass.
2 - Averages reported do not include cycles Hat incurred spillback from a downstream intersection. Flow rates are in
passenger car units (i.e., pcphgpl) as no heavy vehicles were included in He database.
3 - This interchange type represents one site. The Trough headway is low for this site because of a -3.6 percent
. .. . .. . . . . . . . ~
downgrade on the approach studied; all other sites had approach grades of less than ~Z.5 percent.
C-4
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Potential Bias in Headway and Lost Time Estimation. A closer examination of the data
was undertaken to explore the causes of the variability in the tabulated headways shown in
Table C-~. This examination focussed on the possibility that the method used to calculate the
minimum discharge headway introduces some bias. Specifically, the specification of queue position
five as He first to achieve the minimum discharge headway is likely to introduce some bias into the
resulting average (as obtained from Equation C-~) if the headway s recorded for position five at a
given site are not, on average, equal to the average value of the higher queue positrons.
Further exam~nahon of the data at the study sites indicated that this bias does exist and that
the amount of bias is dependent on two factors. The first factor relates to the inclusion of headway s
from lower queue positions, that are not discharging at the minimum headway, in the estimate of the
average minimum discharge headway. Specifically, the average headway for a Tower queue position
can be larger than the minimum discharge headway and, if included in its estimate, wall bias the
estimate to values larger than the true minimum discharge headway. This effect can be seen in
Figure C-l where the line representing average minimum discharge headway is "pulled" upward
slightly by headway observations in queue positions five and six.
The second factor affecting the amount of bias due to the use of headway s not at the
m~nun~n value relates to the frequency of headway observation at each queue position. In general,
the number of observations is highest for the lowest queue position and decreases with each
increasing queue position. This van ation in observations can amplify the effect noted above (i.e.,
differences in average headway between queue positrons) by giving greater "weight" in the overall
average to the more frequently observed lower queue positrons.
The following example is posed to illustrate the effect of the bias due to the use of lower
numbered queue positions in the average minimum discharge headway computation. Consider a
traffic lane at each of two different intersections. Each large has a demand volume of 200 vphpT and
art identical queue discharge character that yields the headway s shown in Figure C-2. These
headways represent the "true" headway s that are known in advance for the purpose of this example.
Inspection of this figure Indicates that the hue minimum discharge headway H of each lane is I.~S
sec/veh (1,915 vphgpl).
A field study is conducted to estimate the true minimum discharge headway H for each
intersechon through the use of Equation C-! (i.e., Ha. Headway s are measured for Tree hours at
each ~ntersechon. The cycle length is 120 seconds at one intersection and 60 seconds at the other.
These cycle lengths are found to yield average queues of 6.7 and 3.3 vpcpI, respectively, and the
frequencies of observation by queue position shown in Figure C-2. The average ~ninimllm discharge
headway is computed for each lane and shown in Table C-2. As this table indicates, the two
estimates are different solely because of the bias effects described previously. Moreover, neither
estimate yields the true minimum discharge headway of I.~S sec/veh (although, the lane with the
longer average queue length yields a closer estimate).
C-S
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Discharge Headway, see
2.5 .
Frequency of Observation
2.3
2.1
1.9
1.7
1.5
Untrue Discharge Headways
06.67 vpcpl
\ - ' ~ APE
\~ - Frequency of Observation an
_ , B
: . _
-it-: . it,
---I I--~--n--
7 8 9
n
I
10 11
.. ~ ~
2 3
4 5 6
150
120
90
60
30
Queue Position
13
o
Figure C-2. Data used to demonstrate potential bias in the estimate of minimum discharge headway.
Bonneson (29 has investigated the amount of bias observed in average minimum discharge
headway estimates for several leh-turn and through traffic movements at several sites From this
investigation, he fourld that minimum discharge headway estimates obtained from the "average of
the fifth Trough last position" H5 were typically 0.13 and 0.06 sec/veh higher for Me left arid
through movements, respectively, than unbiased estimates of Hobtained from other methods. This
increase translates into the saturation flow rate being ur~derestunated by about ~ ~ O and 50 vphgp}
for the respective movements. From a practical standpoint, these differences may be considered
small. However, from the standpoint of statistically quantifying the effect of venous factors (e.g.'
lane widths on minimum discharge headway or saturation flow rate, these differences can be very
important.
There are several implications that stem from He use of a biased estimate of the true
magnum discharge headway. First, the estimate always exceeds the true minimum discharge
headway, however, as mentioned in the previous paragraph, this effect may be small for practical
purposes. Second, He estimate of the start-up lost time will be biased toward a value that is smaller
Han the hue value because it is computed using the biased estimate of magnum discharge headway
(as per Equation C-3). Third, and most important, the bias will likely cloud any statistical analysis
of cause and eKect by introducing added variability in He data set. As a result, the Due effect of a
treatment or factor (e.g., lane width, grade, interchange configuration, etc.) may be obscured by data
from sites having different degrees of bias. In fact, it is this type of bias that causes most of the
differences between the estimates of minimum discharge headway for the interchange configurations
shown in Table C-.
C-6
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Table C-2. Effect of headway observation frequency on the estimate of minimum discharge
headway
Number of Vehicles Observed: 600
.. . ...
Cycle Length, see: ~120 ~60
Average Vehicles / Cycle, vpcpl: ~6.7 ~3.3
.
Queue Position True Headway Observations hT; * n Observations
(hTj ) sec/veh (nO,i) see (n)
3.63 180
2 2.43 83 148
3 2.18 75 116
2.05 68 85
5 1.97 61 120.36 53
1.93 54 104.36 21
7 1.91 46 87.88 0
~
8 1.90 39 74.03 0
1.89 32 60.52
_.
10 1.89 24 45.30 0
11 1.89 17 32.06
12 1.88 10 18.85 0
13 1.88 3.77
14 1.88- 0 - 0 ~0
15 H= 1.88 0 0 0
Sum: 285 547.13 74
.
Avg. Headway (Hs = E(hT i*no i)/En): 1.92 see (1,875 vphgpl) 1.96 see (1,835 vphg~l) l
hTj*n
see
104.58
40.59
o
o
o
° 1
° 1
O
O
. O
145.17
C.~.2 Saturation Flow Rate Mode! for Through Movements
Factors Affecting Discharge Headway. In addition to the biases descnbed in the preceding
section, the differences in headway and lost-time among He interchange configurations shown
Table C-} can be partly explained by differences among He mdividua1 study sites. A review of the
literature on the topic of through movement headways suggests that several site-specific factors exist
that can have an elect on the discharge process. For example, Bonneson (by, in a previous study
of headways at intersections arid single-point urban interchanges for NCHRP Project 340 (3), found
that the number of vehicles served per cycle had an effect on the minimum discharge headway.
Specifically, he found that the headways observed for each queue position were lower when there
were more vehicles queued behind that position. He called this elect that of "traffic pressure." In
this context, traf Eic pressure is believed to result from the presence of aggressive drivers (e.g.,
corrunuters) that are anxious to minimize their Gavel time In otherwise high-volume conditions. As
these drivers are typically traveling during the morning and evening peak traffic periods, they are
C-7
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typically found to be concentrated in the large queues associated with these periods. It should be
. . _ . , _ , . .
~7 ~. . ~_ _
~ ~ ~ ~ _ _ ~ ~ ~ ~ _ ~ _ _ _ ~ A ~ ~ ~ ~ ~ ~ ~ ~ ~ - ~ · · 1 ^^ . ~ , ^~
novel Teal Blokes' 1vlesser, and clover (fly round a similar effect or trattlc queues on headways; they
termed this effect "headway compression."
Bonneson 62J recommended the following equation for predicting the minimum discharge
headway of a single-point urb art interchange through movement as a function of traffic pressure:
Hrh 1.57 0.0086 vat' (C-4)
s
where:
Huh = through movement minimum discharge headway, sec/veh;
us = speed at saturation flow, m/s; and
I' = demand flow rate per lane (i.e., traffic pressure), vpcpl.
The speed in Equation C-4 represents the maximum speed drivers tend to reach as they
discharge from a traffic queue. In theory, it represents the speed associated with a traffic stream
flowing at its saturation flow rate. This speed was found to vary between 12 and 15 m/s in the sites
studied by Bonneson 62) (it was denoted by the variable "imp," in Bonneson's work). One reason
offered for this variation was the proximity of some sites to adjacent intersections. Specifically,
Bonneson noted that lower speeds were associated web those sites where the distance to the
downstream ~ntersechon (and its associated queue) was relatively short. This suggests that discharge
headway s may be lower because of lower discharge speeds that result from the impending
downstream stop faced by the discharging Divers.
The HCM (1J describes many additional factors that can affect discharge headway. These
factors include: lane width, vehicle classification, local bus frequency, parking activity, approach
grade, and area type. To avoid confounding the effect of these factors with those specifically being
considered in this study (e.g., distance to back of queue), several steps were taken to avoid or
remove We aforementioned factors from the data collected for this project. Specifically, the study
sites all had lane widths of about 3.6 meters, approach grades of less than +2.5 percent, no local
busses, and no parking activity. In addition, all heavy vehicles (i.e., vehicles wad more than two
axles) and all queued vehicles that followed heavy vehicles were removed from the data base.
Analysis Approach. The analysis of the through movement headway s focussed on an
examination of the headway s for passenger cars in the fifth and higher queue positrons attweIve
study sites. Several precautions were taken to eliminate Me elects of bias and to account for factors
that might confound the analysis. Specifically, analysis of variance (ANOVA) techniques were used
to control for differences in sample size (i.e., unbalanced data) and to account for extraneous
differences among otherwise similar sites (e.g., bias by queue position). The ANOVA was
implemented with the Statistical Analysis System's (SAS) (5) general linear model (GEM) because
of its ability to provide stahshcs corrected for unbalanced data. The distnbubon of the residual
errors was checked graphically to verify: (~) that they did not deviate significantly from a normal
C-8
OCR for page 213
.
distnbuhon, and (2) that their variance was essentially constant over the range of the independent
variables. All significance tests were conducted at a 95 percent confidence level (i.e., ~ = 0.05~.
The effects of bias in the examination of factors affecting headway were eliminated by
~ ~ - · . · (~' ~ ~ · 1~ , '' · . ~ · ~ Tow A · ~ ~ ~ ~ (t . · '' ·, ·, ~ -
includlng queue position as a -~blocklng t actor In the AN()VA model anti by Nesting it within
each potentially influential factor (e.g., traffic pressure or phase duration) being considered. By
blocking and nesting on queue position, all of the ANOVA comparisons and parameter estimates
are made on a queue-position-by-queue-positron basis, thereby eliminating any bias by differing
sample sizes or queue positions. In modeling terms, blocking gives each queue position its own
intercept whereas nesting allows the predicted headway to vary independently with each influential
factor. If the factor is continuous, this latter effect is equivalent to allowing each factor to have a
unique slope for each queue position. Using these techniques, the effects of potentially influential
factors were examined for each queue position on an individual basis with the effect of bias
eliminated.
If the effect of a factor was found to be the same for each queue positron (i.e., it had the same
sIope), then the nesting technique was eliminated and a common slope fitted to all positions.
Similarly, if any queue positions were found to have the same coefficient (i.e., the same intercept),
then they were combined, or pooled, to determine a single representative intercept. These types of
combinations are desirable because they yield better estimates of the model parameter coefficients.
Once the influential factors were identified from the ANOVA, regression techniques were
used to fit the data (via these factors) to the proposed model. Linear or nonlinear regression
techniques were used to quantify the calibration parameters, depending on the model formulation.
The linear regression was implemented with the SAS 65) regression model (REG3; the nonlinear
regression was implemented with the nonlinear model - Ad.
Effect of Traffic Pressure. The ANOVA analysis indicated that two factors had a
significant influence on Me headway s at each queue position. These factors include: traffic pressure
and distance to the back of the downstream queue at the start of the upstream/subject phase. The
first factor is described in this section, the second factor is described in the next section.
The average headway recorded for each queue position at each site was found to decrease
slightly as the number of vehicles served per cycle increased. This trend is consistent wad the
findings of Bon neson 629 and Stokes ef al 649. The effect of traffic pressure on discharge headway
is shown in Figure C-3.
~. ~' . .
Figure C-3 shows the effect of traffic pressure as it exists in one queue position at all sites
and at all sites and queue positions combined. The analysis indicated that there were no statistically
significant differences In We elect of traffic pressure among sites and queue positions. Comparison
of Figures C-3a and C-3 b support this finding.
C-9
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Minimum Discharge Headway, sev/veh
2.2
2.0
(Each data point represents the average of 35 obs)
A
A
A
B A A
A A
A
A A A A
A A A
A A
A A A
A A
A
1.8 + A A
A A
1.6 ~
_+___________+___________~___________~___________+_
5 10 15 20 25
Through Demand Flow Rate per Lane (Traffic Pressure), vpcpl
a) Tra~c pressure e~ectfor queue posifion seven af aRstu~ sifes.
Minimum Discharge Headway, sec/veh
2.2 ~
2.0 +
(Legend: A = 1 obs, B = 2 obs, etc.)
(Each data point represents the average of 100 obs)
A
AA
AB A
A A
A AA A
B A BA A AA A A
ABAA BA A A
AAB AA AA
AA A A
I A BBA B ;
I AA A A A A
1.8 ~A A A
I AAAA A
A
A
1.6 +
AB A A
A
_~___________~___~ ____+_ __________+_______ ____+_
0 10 20 30 40
Through Demand Flow Rate per Lane (Traff~c Pressure), vpcpl
b) Tra~ic press~ure effect for aR queue posifions af ad sfa'~ sites.
Figllre C-3. Effecf of tra~qc press~re on fArough movemenf minimz~m ctlischarge headway.
C-IO
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The data shown In Figure C-3 represent the average of several observations. The number of
individual observations was so large that, when they were plotted, they tended to become a black
mass of ink which obscured the examination of trend-w~se effects. To overcome this problem, the
data were first sorted by tane volume, segregated into contiguous groups of 35 (or 100) observations,
and then used to compute average headways and lane volumes for each group.
· . ~ . . ~ ~ ~ ~ . ~ · . . .
Effect of Distance to the Back of Downstream Queue. As mentioned previously, the
ANOVA indicated that Me distance to Me back of downstream queue had a significant effect on the
queue discharge headways. This distance is measured from the subject movement stop line to the
"effective" back of queue at the start of the subject (or upstream) phase. The effective back of queue
represents the location of He back of queue if all vehicles on the downstream street segment (moving
Or stopped) at the start of the subject phase were joined into a stopped queue. If there are no moving
vehicles at the start of the phase, then the effective and actual distance to queue are the same. If
there are no vehicles on the downstream segment at the start of the phase, then the effective distance
to queue would equal the distance to the through movement stop tine at the downstream intersection.
Figure C-4 shows the relationship between the discharge headway of the fifth and higher
. . . . .. .. . . . .
. .. . .
queue positions and the corresponding distance to the back of downstream queue. As the headways
for these queue positrons are considered to be effectively at their minimum value, the data shown
represents individual estimates of the minimum discharge headway H. The trend shown in this
figure indicates that the minimum discharge headway decreases with increasing distance to queue.
Minimum Discharge Headway, sec/veh
2.5 +
1.5 +
1.0
(Legend: A = 1 obs, B = 2 obs, etc.)
(Each data point represents the average of 50 obs)
A A
AA
B BAA A A
A AA B BAA A BA A A AN
A A BCDDBBACA B BAA A
A AD CAB CEEDBACAC GAB B ABA
A B BA AB AACA ABB BB
A A A BCBAA BA
A CB
A
A
0 75 150 225
Distance to the Downstream Queue, m
300 375
Figure C-4. Effect of disfance-fo-queue on through movement minimum discharge headway.
C-11
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Table C-14. Lane utilization factors for lane groups with random lane choice (U.)
Lane Group Flow | Number of Lanes in the Lane Group
Rate, vpc ~2 ~3
5.0 ~1.00 1.32 1.67
0.0 1.00 1.22 1.45
15.0 1.00 1.17 1.36
20.0 1.00 1.15 1.3
25.0 1.00 1.13 1.28
30.0 1 1.00 1.12 ~1.25
35.0 1.00 1.1 t -- 1.23
40.0 I l.oo 1 1.10 1 1.22
4
2.08
.74
.59
.51
.45
.41
.38
1.35
Preposition~ng Model. The large utilization factor for interchanges arid associated closely-
spaced intersections can be strongly influenced by drivers propositioning for downstream turns. The
magnitude of the effect is largely a site-specific characteristic, depending on the number of vehicles
mining left or right at the downstream intersection (or ramp terminals. If this information is
available (such as hom a turn movement count where downstream destination is also recorded), We
lane utilization factor can be computed as:
U =
p
Max (v 'ill ~ V 'd) ~
Y'
(C-54)
where:
Up = lane utilization factor for propositioning;
vie= number of vehicles In the subject lane group that will be mining left at the downstream
intersection, vpc;
vies = number of vehicles in Me subject lane group Cat will be turning right at Me downstream
Intersection, vpc; aIld
Max(v 'fib V'dJ = luger of v a, "d v dr.
Inane Utilization Factor. The possibility of propositioning must be evaluated to determine
whether to use Equation C-53 or C-54 to estimate the large utilization factor U. This possibility can
be determined Mom Me following test:
Max(v','`, v'`~)
V' - : prepositioning
Max f v'd', v'd,) ~
< _
v' N
no preposifioning
C-64
(C-SS)
OCR for page 269
Based on the outcome of this test, the lane utilization factor is computed as:
U=
U : propositioning
U
r
.
· · -
no preposltlonzng
(C-56)
The distance to the downstream intersection could effect the propensity of drivers to preposition.
In recognition of this effect, the test equation is recommend only for intersections in interchange
areas or other closely-spaced Intersections where the ~nter-signal distances are less than 300 meters.
C.5.6 Predicting Queue SpiBback
Two conditions must occur to precipitate spillback during a saturated flow state. First, the
discharging stream from the subject traffic movement must arrive at the back of a stopped
downstream queue. Second, the subject phase must be sufficiently long as to permit the discharge
of enough vehicles to fill the available downstream distance under jam density conditions.
Occurrence of the first condition can be predicted by comparison of the actual and ideal
signal offset between the subject phase and that of the clown stream through movement. In general,
if the difference between the actual and ideal offsets is positive, then there is a possibility of
spillback. A negative difference indicates that spillback is unlikely. This offset difference can be
computed as:
~ = 095a Hi
with,
where:
0// = L
u
s
nS 3,600 h
N s c
d n
(C-5
(C-58)
~ = difference between the actual and ideal signal offset, see;
ok = actual offset between the subject phase and that ofthe downstream through movement (phase
start time downstream minus phase start time upstream), see,
off= ideal offset to ensure progression without speed disruption, see;
us= speed at saturation flow, m/s (say, 14 m/s);
he = clearance headway between Me last queued vehicle and We first arriving vehicle, see (say,
2.0 see);
Sn= saturation flow rate for the subject lane under prevailing conditions assuming the "no
spillback'' condition, vphgpl;
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L = distance between the subject and downstream intersection stop lines (i.e., link length), m;
nS = number of vehicles on the downstream street segment (moving or queued) at the start of the
subject phase, vein; and
Nit = number of through lanes on the downstream segment, cartes.
Occurrence of Me second condition is dependent on the first condition. If We first condition
occurs (i.e., the offset difference is positive), spillback we occur if the phase is long enough to
discharge enough vehicles such that Me resulting queue exceeds that of the downstream street
segment. If the phase is short, such that the discharged vehicles are Insufficient in number to fill the
downstream distar~ce-to-queue, then spillback will not occur. The corresponding maximum green
interval duration that, if exceeded, would lead to spilIback is computed as:
G = D ~ 3,600) ~
max v ~w J s
(C-59)
where:
G,~'a, = maximum green signal Interval duration for the subject (or upstream) phase that is allowable
without spiliback during saturated flows, see;
D = elective distance to the back of downstream queue (or stop line if no queue) at the start of
the subject (or upstream) phase, m;
so= saturation flow rate for the subject lane under prevailing conditions assuming the "w~th-
spiliback" condition, vphgpl;
[, = average lane length occupied by a queued vehicle (see Equation C - 5), m/vein; and
Is = start-up lost fume, sec.
Equations C-58 and C-59 were used to develop two figures that can be used to predict the
likely occurrence of spilIback for the subject phase. Specifically, the relationship between distance-
to-queue D and ideal offset ok (i.e., Equation C-58) is shown in Figure C-30. The relationship
between distance-to-queue D and the maximum green interval duration G,~ (i.e., Equation C-59)
is shown in Figure C-31 .
As Figure C-30 Indicates, the ideal offset is often negative for most Intersection pairs when
there is a downstream queue. A negative value indicates that the downstream through phase must
start prior to the subject upstream phase to provide smooth traffic progression.
Figure C-30 can be used instead of Equation C-58 to determine whether Me first condition
is satisfied. Specifically, Me actual offset is compared with Me ideal offset for a given sheet segment
length arid distance-to-queue. If Me Intersection of the actual offset (entered Dom the y-axis) and
the known distance-to-queue (entered from the x-axis) falls below the line corresponding to the
segment length, then Me difference In offset wall be negative and spiliback is not likely to occur. If
Me intersection occurs above the appropriate segment length line, then the difference In offset wit}
be positive and spiliback is likely to occur during saturated flow conditions. In this latter case, Me
second condition should be checked to venty whether spiliback watt occur.
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40
20
Ideal Signal Offset, see
Ideal offset with no queue
40
-60
-80
-100
-120
· 200~
~ 400-m segment length
No spillback for actual offset & distance combinations
tbaLfaU beloYv low.
-140 ~ ~ ~ ~ ~ i
15 45 75 105 135 165 195 225 255 285 315 345 375
Distance to the Downstream Queue, m
Figure C-30. Relationship between distance-to-queue and ideal signal onset.
100
90
80
70
60
50
40
30
10
lo
Maximum Green Interval, see
~ ~spillb~ckfor~tual yr~errS dista, ,ct:-=mbir~ations- - ~ ~ ~ - ~ ~ ~
that fall below line.
! 1 1 1 1 1 1 1 ! 1 1 1
15 45 75 105 135 165 195 225 255 285 315 345 375
Distance to the Downstream Queue, m
Figure C-31. Relationship between distance-to-queue and maximum upstream green.
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Figure C-3 ~ indicates whether the second condition occurs. Specifically, it Indicates the
maximum green interval duration that would not lead to spilIback; green intervals in excess of this
maximum would likely lead to spilIback. In general, this figure indicates that the maximum green
interval increases linearly with increasing distance-to-queue. In fact, a first-order estimate of the
max~murn green duration is about one second of green for every three meters of distance.
If the offset difference (USA ~ Off) is determined to be positive, then Figure C-3 ~ can be
consulted to determine if spiliback will occur for saturated flow conditions arid the known green
interval duration. If the intersection of the actual green interval (entered from the y-ax~s) and the
known distar~ce-to-queue (entered from the x-axis) falls below the line, then the green Interval is
sufficiently short as to prevent spillback. If the intersection occurs above the trend line, then the
green interval is sufficiently long as to precipitate spillback whenever traffic demands are high
enough to fully utilize the signal phase. Therefore, when the intersection point is above the trend
line and demands are expected to fully use the phase, the "with spillback" form of the distance-to-
queue adjustment factor ED should be used.
The following example is presented to illustrate the use of Figures C-30 and C-31 to
determine if spillback is likely to occur under saturated flow conditions. Consider a 400-meter street
segment bounded by signalized Intersections. The average distance-to-queue to queue at the start
ofthe upstream Intersection through phase is 135 meters. This phase has 50 seconds of green. The
Offset to the downstream through movement phase is -10 seconds (i.e., Me downstream through
phase starts 10 seconds before the subject phase). Consultation wad Figure C-30 indicates that an
ideal offset for this distar~ce-to-queue is about -45 seconds and, more importantly, that the actual
offset-dist~ce combination intersects above the "400-meter" line. This point of Intersection
indicates that the difference in offset is positive arid Mat the first condition is satisfied. Therefore,
spilIback is possible, depending on the duration of upstream green.
Consultation with Figure C-3 ~ checks the second condition. Specifically, it mbicates that
the maximum green interval duration is about 41 seconds (or about 45 seconds using the ~ :3 rule)
and, more unportantly, that Me actual green-distar~ce combination intersects above the trend line
~ndicat~g Me strong likelihood of spilIback during saturated flow conditions. This finding indicates
that the "win spilIback" fob of the distance-to-queue adjustment factor should be used to compute
the capacity of the phase based on the assumption that traffic demands are sufficiently high as to
fully use the signal phase.
C.5.7 Predictive Ability of the Proposed Models
As a verification of the calibrated saturation flow rate and start-up lost time models, the
predicted and measured discharge times of several last-in-queue vehicles were compared. The
predicted discharge time was computed using the following equation:
T = J H + I (C-60)
where:
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Tow = discharge time of He J~ queued vehicle, see;
J= last queue position to discharge;
H= miIiimum discharge headway, sec/veh; arid
Is = start-up lost the, sec.
For this verification, Equation C-60 was used (with the saturation flow rates predicted by
Equation C41 arid the start-up lost times predicted by Equation C49) to predict He discharge time
of the last queued vehicle observed dunng several hur~dred signal cycles.
evaluation are shown In Figure C-32.
Predicted Discharge Time, sec
(Legend: A = 1 obs, B = 2 obs, etc.)
60
40 ~
20 +
A
A
A
AA A
AAA
~ JBAI
BABAA
AABBBA
AGACA
~7
The results of this
(Each data point represents the average of 25 offs.)
A
10 20 30 40 50
Measured Discharge Time, sec
Figure C-32. Comparison of predicted and measured discharge times.
As the trends In Figure C-32 indicate, the recommended models are able to accurately predict
the discharge time of the last queued vehicle. Moreover, the fit is equally good over the range of
discharge times, indicating no time-based bias in the prediction process. The R2 for the fit of the
predicted to measured discharge times is 0.95.
It should be noted Hat He database used for this evaluation was the same as that used for the
mode] calibration, thus, this evaluation is a mode} verification rather than a model validation.
Nevertheless, it still provides strong evidence ofthe accuracy of the recommended models. It should
be noted that the data points shown in Figure C-32 represent the average of 25 observations each.
This averaging was ~d~ to facilitate the graphical presentation of the model's trend-wise fit
to the data, the ~ndividu~ data points were used in He assessment of model fit.
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C.5.S Effect of Saturation Flow Rate and Lost Time on Capacity
The ideal saturation flow rate recommended in Chapter 9 the 1985 Highway Capacity
Manual 699 was 1,800 pcphgpl. Chapter 2 of this manual recommended a start-up lost time of 2.0
seconds. More recently, the ~ 994 HCM (~) included a recommendation of 1,900 pcphgp! for Me
ideal saturation flow rate but made no recommendation for a specific start-up lost time value Comer
than to note that appropriate values for this lost time were in a range of i.0 to 2.0 seconds). The
ideal saturation flow rate recommended In this report is 2,000 pcphgpl which represents art increase
beyond He current HCM value. However, the "cost" associated with this increase is an increase In
the starbup lost time. Fortunately, the increase in start-up lost time is generally small and does not
totally offset the gains in capacity obtained from the higher flow rates.
The relative unpact of alternative combinations of saturation flow rate and start-up lost time
on capacity are shown In Figure C-33. The capacity shown In this figure was computed using
Equation C-39 with a 90-second cycle length and a green extension of 2.5 seconds. In general, Here
is an ~ ~ percent increase In capacity Implied by the increase in saturation flow rate Tom I,SOO to
2,000 pcphgpI, for the same I.S-second start up lost time. However, if a more accurate start-up lost
time of 2.82 seconds is used with He 2,000 pcphgp! flow rate, then the increase is reduced by about
20 to 40 percent (corresponding to a true increase in capacity of only about 6 to 9 percent). The
point to be made here is that start-up lost time is not a true "lost time;" rawer, it is a "cost" of
associated wad accelerating to the speed at saturation. Hence, the higher the saturation flow rate,
He higher the speed wait be at saturation flow and He longer He acceleration (or staxt-up lost) time.
1600
1400
1200
1 000
800
600
Capacity, vphpl
r
Sat. Flow / Lost Time
_1,800 /1.50
+2,000 /1.50
. `2,000 / 2.82 an_ _
~_
,~
~ I- - - - - - - - - - - - - - - - - - - gO~ec. cyctelengtr - - - - - - - - - -
400
20
- - - - - - - -
- - - - - ~ - - - ~
1
-
30 40 50 60 70
Phase Duration, see
Figure C-33. Elect of saturation flow rate and start-up lost time on capacity.
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C.S.9 Conclusions and Recommendations
This section summarizes the conclusion and recommendations resulting from the analysis
of capacity characteristics associated with signalized interchange ramp terminals and closely-spaced
intersections.
The ideal saturation flow rate recommended In this report is 2,000 pcphgpl. In Me context
ofthe factors studied for this research, this ideal flow rate applies to through traffic movements that
have art infinite distance to the back of downstream queue, operate under non-spilIback conditions,
and have traffic volumes that are relatively high (reflecting those fond during peak demand
periods).
Based on this research, it is concluded that the distance to the downstream queue has a
significant effect on the discharge rate of upstream traffic movements. This effect is amplified when
the signal timing relationship between the two intersections allows queue spiliback to occur. As the
distance-to-queue variable is bounded to a maximum value equaling the length of the downstream
street segment, the effect of distance-to-queue also includes the effect of spacing between
Interchange ramp terminals or between a ramp terminal and a closely-spaced intersection.
Turn radius has a significant effect on the discharge rate of a turn-related traffic movement.
Saturation flow rates were much lower for turn movements with small radii than they were for An
movements with large radii. In the context of junction type (e.g., s~ngle-point urban diamond, at-
grade intersection, etc.), the saturation flow rates for the left-turn movements at s~ngle-po~nt urban
interchanges are more nearly equal to those of through movements because of We large turn radii
associated with this interchange type.
Traffic pressure, as quantified by traffic volu~ne per cycle per lane, has a significant effect
on saturation flow rate. The saturation flow rates of lower volume movements are much lower Can
those of higher volume movements.
Other factors were examined for Heir potential eject on saturation flow rate. These factors
include: g/C ratio, junction type, downstream signal indication at the start ofthe upstream phase, and
dual versus single left-turn lane. Of these factors, only g/C ratio was found to be correlated wad
saturation flow rate in a statistically significant manner. Specifically, Me saturation flow rate for
lefc-turn movements with low g/C ratios was Soured to be higher than the rates of similar movements
with larger TIC ratios. This effect was also found in the Trough movements studied, however, it was
much smaller In magnitude and not statistically significant. Therefore, it was determined that more
research is needed to verify Me significance of this trend and its magnitude before an adjusunent
factor for this effect can be recommended.
Start-up lost time was found to Increase with saturation flow rate. This increase is due to Me
increased time it takes for the discharging queue to attain the higher speed associated win a higher
saturation flow rate. Predicted start-up lost tunes range from 0.61 to 3.IS seconds for prevailing
saturation flow rates of 1,400 to 2,100 pcphgpl.
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The average amount of Me yellow interval used by drivers is termed "green extension." This
characteristic can be used to compute clearance lost torte and the effective green tune. The
recommended green extension value is 2.5 seconds. Other values are possible if the approach speed
is outside the range of 64 to 76 lymph or when the volume-to-capacity ratio for the analysis period
is larger Cart 0.~. An equation is provided for these situations.
Lane use is almost always uneven (or unbalanced) in intersection lane groups. The degree
of this imbalance is expressed In terms of the lane utilization factor. The lane utilization factor
vanes depending on the nature of drivers' lane-choice decisions (i.e., to minimize travel time or to
preposition for a downstream turn). Lane utilization factors based on have} time minimization tend
to be subject to randomness in the lane-choice decision process. The factors stemming from this
process range from I.] to 2.0, depending on the number of lanes in the larle group and its
corresponding traffic volume. Lane utilization factors based on Diver desire to preposition can vary
widely, depending on the volume of traffic Mat is preposition~ng In the subject lane group.
Capacity is dependent on the prevailing saturation flow rate, start-up lost dine, and clearance
lost time. The equations provided in this appendix should be used to estimate the saturation flow
rate, as affected by distance to downstream queue, tum radius, and traffic pressure. Start-up lost time
is not a constant value; rather, it is dependent on the prevailing saturation flow rate. Thus, We
equation provided in this appendix should be used to estimate the start-up lost time that corresponds
to the prevailing saturation flow rate.
C.6 APPENDIX C REFERENCES
1. Special Report 209: Highway Capacity Manual. TRB, National Research Council, Washington,
D.C. (1994).
2. Bonneson, I.A. "Study of Headway and Lost Time at Single-Po~nt Urban Interchanges." In
Transportation Research Record 1365. TRB, National Research Council, Washington, D.C.
(1992) pp. 30 - 39.
3. Messer, C.~., Bonneson, I.A., Anderson, S.D., and McFarland, W.F. NCHRP Report 345: Single
Point Urban Interchange Design anal Operations Analysis. TRB, National Research Council,
Washington, D.C. (1992).
4. Stokes, R.W., Messer, C.~., and Stover, V.G. "Saturation Flows of Exclusive Double Lefi-Turn
Lanes." In Transportation Research Record 1091. TRB, National Research Council,
Washington, D.C. (1986) pp. 86-95.
5. SAS/STAT User's Guide. Version 6, 4th ed. SAS Institute, Inc., Cary, Norm Carolina (1990~.
6. Kimber, R.M., McDonald, M., alla Hounsell, N.B. TRRE Research Report RR67: The
Prediction of Saturation Flowsfor Road Junctions Controlled by Tragic Signals. Transport and
Road Research Laboratory, Department of Transport, Berkshire, England (1986~.
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7. Fambro, D.B., Messer, C.J., and Andersen, D.A. "Estimation of Unprotected Lef~c-Tu~n Capacity
at Signalized Intersections." In Transportation Research Recorc! 644. TRB, National Research
Council, Washington, D.C. (1977) pp. ~ 13-] 19.
8. Messer, C.J. and Fambro, D.B. "Critical Lane Analysis for Intersection Design." In
Transportation Research Record 644. TRB, National Research Council, Washington, D.C.
(1977) pp. 26-35.
9. Special Report 209: Highway Capacity Manual. TRB, National Research Council, Washington,
D.C.'(1985~.
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Representative terms from entire chapter:
saturation flow
