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OCR for page 111
APPENDIX D
DEVELOPMENT OF UNIFORM DELAY EQUATIONS
FOR TRAFFIC-ACTUATED CONTROL
An analytical model for the estimation of phase times, cycle lengths and volume/capacity ratios was
described in detail In Appendix C. This information is used by the HCM Chapter 9 delay model to
assess the level of service ~OS) for each movement. The computational structure of the Appendix
C timing estimation model introduces an improved method for delay estimation at a traffic-actuated
intersection. Potential improvements to the HCM Chapter 9 delay model are presented In this
appendix.
The traditional delay formulation used by virtually all analytical models is based on two components,
or terms, which are added together to produce the computed delay per vehicle:
I. The un~forrn delay term, which determines the delay that would occur if all vehicles arrived
in a completely uniform manner. This term is computed as the area under the queue accumu-
lation polygon (QAP).
2. The, incremental delay term, which adds a correction factor to compensate for randomness
in the amval patterns and the occasional oversaturation that may result.
Since the QAP's must be developed in detail by the signal timing estimation model described in
Appendix C, the value of the uniform delay term (i.e., the area contained within the QAP) may be
computed by a simple extension to the signal timing estimation procedure. The detailed development
of the equations for computing the QAP areas is presented here.
There are nine distinctive shapes that may be taken by the QAP, each of which is associated with a
specific phasing alternative. Each of these cases requires a different computational formulation, or
set of equations. Each case is developed as a separate figure in this report. Each figure shows the
shape of the QAP and presents the derivation of an equation that determines the uniform delay by
computing the area under the associated polygon. A summary of the phasing alternatives illustrated
in Figures D-] through D-9 is presented in Table D-~.
The derivations are presented in a format similar to the format used in the supplemental worksheet
now contained in HCM Chapter 9 for uniform delay computations with compound left turn protec-
tion. In all cases, the uniform delays may be determined using only those variables (volumes, satura-
tion flow rates and signal timing) that are used by the existing HCM Chapter 9 methodology.
Each of the figures conforms to a common terminology with respect to its labeling. Intervals are
illustrated along the horizontal axis as follows:
Appendix D: Page
OCR for page 112
r indicates the effective red time
g indicates the elective green time
gq indicates the portion of the permitted green time blocked by a queue of opposing
vehicles
gu indicates the portion of the permitted green time not blocked by a queue of opposing
vehicles
indicates the portion of the protected green time required to service the queue of vehi-
cles accumulated on the previous phases
indicates the extension to the protected green time that occurs while the controller
waits for a gap in the amving traffic long enough to terminate the phase
gf indicates the portion of the green time in which a through vehicle in a shared lane
would not be blocked by a led turn vehicle waiting for the opposed movement to
clear. This condition occurs ondy at the beginning of the permitted green when one
or more through vehicles are at the front of the queue.
Points in the cycle at which the queue size is important to the computations are also identified as fol
lows:
Qr indicates the queue size at the end of the elective red
Qq indicates the queue size at the end of the interval gq
Qp indicates the queue size at the end of the permitted green period
Q'p indicates the queue size at the end of the perrnined Been period, adjusted for sneakers
Qg,, indicates the queue size at the beginning ofthe protected green (green arrow) period
Qf indicates the queue size at the end of the interval gf
The flow rates that determine the slopes ofthe various sections of the QAPs are indicated es follows:
qa indicates the arrival flow rate' which is assumed constant over the cycle
s indicates the saturation flow rate for protected movements
sp indicates the saturation flow rate for permitted movements.
Appendix D: Page 2
OCR for page 113
The slope ofthe lines representing the departure of vehicles Tom the queue generally represents the
net departure rate, i.e., the saturation flow rate minus the amval rate.
Note that, in cases where protected phases exist, these are arranged to be the last to occur. The
length of these phases will be determined by their detector actuations. The actual length will be the
sum of the time required to service the queue that exists at the beginning of the phase plus the
extension time.
Table D-~. Summary of phasing alternatives for the computation of uniform delay
Figure l Description of Phasing Alternative 1
D-1 | Single protected base l
D-2 | Permitted left fur s from an exclusive lane. l
D-3 | Permitted left tur i from a shared lane (gq > g]
Dot | Permitted leR tur i from a shared lane (gq ~ g)
D-5 | Compound left to n protection: HCM Chapter 9 Case 1
D-6 | Compound left to n protection: HCM Chapter 9 Case 2
D-7 | Compound left tu n protection: HCM Chapter 9 Case 3
D-8 | Compound left tu I protection: HCM Chapter 9 Case 4
. .
D-9 | Compound left tu ~
Appendix D: Page 3
OCR for page 114
- ~
d
._
U)
-
C,
._
o
lo
Qr = r qa
Qq Qr
Qp = Qp/ = 0
1
qa~
/
r
A Single Protected Phase
Qr
. .. .. ..
l
i
s-q
\QP=
Qp =0
Time (seconds)
0.5 ~ r Qr + Qr2 / (s - qa) ~ forisolated operation
D1 - qa C
D _ 0 5 [ r Qr + Qr2 / (s - qa) ~ (1 - P) f for coordinated
1 - qa C 1 - (g / C) phases 2 & 6 only
Figure D-1. Uniform delay computation for a single protected phase
Appendix D: Page 4
OCR for page 115
Permitted Left Turns from an Exclusive Lane
,
~I
~ I
~I
._
cn
-
._
;>
o
Qr r qa
Qq Qr gq qa
Condition 1: QST < gu
Qp = Qp/ = 0
D. =
Condition 2 QST ~ gu
Q = Q ~
P q
Qpt = 0
D1 =
Qq
. _.._. ._ .._
Qr /
l
/
qa ~
~/;1
-/W
.
1 _ I
~, P qa
~Qp=QP' .Qp'
Time (seconds)
O.5[(r+ gq)Qq+Qq /(sp qa)]
qa C
gu (sp qa)
05 [ (r + gq)Qq + gu (Qq + Qp)]
qa C
QST= Qq
S. ~ q
Figure D-2. Uniform delay computation for permitted left turns
from an exclusive lane
Appendix D: Page 5
OCR for page 116
Permitted Left Turns from a Shared Lane (:q > ~f)
-
._
U)
-
C~
._
~C
o
o
z
Qr r qa
Qr
76.
r e
~ '
Time (seconds)
Qf = Qr gf (S qa)
Qq = Qf + (gq gf) qa
Q = Qf + (gq gf) (qa 1 + PL (EL2
Condition 1: QST < gu
QST = Qq
S. ~ q
~pp > 1
if n0pp = 1
Qp = Qpl = 0
D = 0 5 [r Qr gf (Qr Qf) + (gq gf) (Qf + Qq) + Qq / (Sp ~ qa):
qa C
Condition 2: QST ~ gu
Qp = Qq ~ gu (Sp qa)
Qpl = 0
[ Qr gf (Qr Qf) (gq gf)(Qf Qq ) gu (Qq QP )]
qa C
Figure D-3. Uniform delay computation for permitted left turns
from a shared lane (gq ' gr)
Appendix D: Page 6
OCR for page 117
Permitted Left Turns from a Shared Lane (:q < gf)
o~
._
u,
._
o
o
z
Qr = r qa
Qf =
Qr gf (S ~ qa)
Qq = Qr ~ gq (S qa)
Condition 1: QST < gu
Qp = Qp/ = 0
Qr
~/7 S-qa
WF;7~
r
\: Qq
l;;''-''-''-''
\e Qf
1
' :~a Q
gqll I Qp=Q/\ ~ Qp'
Time (seconds)
QST= Qq
S. ~ q
0 5 ~ r Q + gf (Qr + Qf) + Qf2 / (Sp - qa):
Condition 2: QST > gu
Qp = Qf gu (Sp qa)
Qp' = 0
[ Qr gf (Qr Qf) gu (Qf + QP )3
D1 = qa C
Figure D-4. Uniform delay computation for permitted left turns
from a shared lane (gq < gf3
Appendix D: Page 7
OCR for page 118
Compound Left Turn Protection (Case 1)
~1
c: i
._
u,
-
c~
._
~:
o
o
z
Qr r qa
Qq = gq (qa sq)
Sq = 0
s
s =
q E
L2
Condition 1: QST ~ gu
Qp = Qp/ = 0
Qr
/ i\
1 \ 1
I ~ S ~ qa
Qq
~-- ..
qa~ S~: qa
i (~: QP/
V i
1
1
l
Time (seconds)
~nOPP > I
(where s'=s/0.95) -ifnOpp=1
QST = Qq
S. ~ q
D _ 05 [ r Qr + Qr / (s ~ qa) + gq Qq + Qq2 / (sp - qa)]
1 qa C
Condition 2 QST > gu
Q = Qq ~ gu (Sp
Qp~ = 0
D, =
qa)
O.S ~ r Qr + Qr / (S ~ qa) + gq Qq + gu (Qq QP) ~
qa C
Figure D-S. Uniform delay computation for compound left turn protection (Case 1)
Appendix D: Page 8
OCR for page 119
Compound Left Turn Protection (Case 2)
0D
._
v~
-
._
o
o
z
Qr r qa
Qq = Qp 1/ + gq (qa sq)
Qp'/ = Qr g (s qa)
Sq = 0
s (where s'=s/0.95) - ifn0pp=
S -
q E
L2
Condition 1: QST ~ gu
Qr
~ \x,: Sp- q /
QP i I ~ QP/
I~ QP=Q~ ~S)P/'
r
\
~S ~ qa
Time (seconds)
~nOpp > 1
1
QST = Qq
S. ~ q
Qp = Qp/ = 0
D1 = [ r g (Qr QPN) gq (QP/I + Qq) + Qq / (sp ~ qa) ]
qa C
Condition 2 QST ~ gu
QP = Qq gu (sp qa)
Qp/ = 0
D = 0 5 [ r Qr + g (Qr + Qp//) + gq (Qp// + Qq) + gu (Qq Qp) ]
qa C
Figure D-6. Uniform delay computation for compound left turn protection (Case 2)
Appendix D: Page 9
OCR for page 120
Compound Left Turn Protection (Case 3)
:S
._
v,
-
v
._
o
o
z
Qr = Qp / + r qa
Qq = gq (qa
sq = 0
s =
q
Qp = Qfq
Qp / = 0Qp ~ s a
j~Sp- qa~
qa- S. q V
1
Qr
.. ..
.. ~-
Qp
', r
~ S ~ qa
T~me (seconds)
q)
sl
EL2
gu (Sp qa)
if nOpp > 1
(where s' = s / 0.95) if nOpp
where Sa = Min (Sneakersmax ~ Qp)
D _ 05 [ r (Qp' + Qr) + Qr / (s ~ qa) + gq Qq + gu (Qq + Qp)]
1
qa C
Figure D-7. Uniform delay computation for compounc! left turn protection (Case 3)
Appendix D: Page 10
OCR for page 121
Compound Left Turn Protection (Case 4)
:~
5
._
U)
C,
._
5:
o
o
z
Qr r qa
Qq = Qr + gq (qa sq)
sq = 0
s =
q
s/
EL2
Condition 1: QST < gu
Qp = Qp/ = 0
Qq
q S. L~
/
Qr
~ ~1 Sp-qa
I N:
I Qp- Qp~
Time (seconds)
QST= Qq
S. ~ q
if nOpp > 1
(where s' = s / 0.95) if nOPP 1
=
0.5 [ r Qr + gq (Qr + Qq) + Qq / (sp qa) ]
qa C
Condition 2 QST ~ gu
Qp = Qq ~ gu (sp qa)
Qp/ = 0
D = 0 5 ~ r Qr gq (Qr Qq) + gu (Qq + Qp) ]
Figure D-8. Uniform delay computation for compound left turn protection (Case 4)
Appendix D: Page 11
OCR for page 122
Compound Left Turn Protection (Case 5)
:-
-
~:
._
u,
c~
·_
~:
c~
o
o
z
Qr r qa
Qq = Qr + gq (qa sq)
sq = 0
Sq =
Qq
qa- Sq ~W 1
W: Sp - qa
~ .......
i ~S~qa
Qr -
.. .. .. .. ,~
/
q~7/
Qp
Q~
Time (seconds)
~nO" >!
Es (where s'=s/0.95) ifnOpp=T
L2
Qp = Qq gu (Sp qa)
Qp / = 0Qp ~ Sa
where Sa = Min (Sneakers~:, Qp)
D = 0 5 [ r Qr gq (Qr Qq) gu (Qq Qp) Qp~ / (s qa) ]
qa C
Figure D-9. Uniform delay computation for compound left turn protection (Case 5)
Appendix D: Page 12
Representative terms from entire chapter:
uniform delay
