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OCR for page 279
APPENDIX D
CLOSELY-SPACED INTERSECTION FLOW MODELS
D.1 OVERVIEW
The following appendix presents research that addresses traffic operations on the arterial
street connections (links) between He traffic signals at signalized interchange ramp terminals, and
also on the connecting links to adjacent intersections downstream from the interchange. Various
Arc models are applied to Be complex traffic operational conditions. The fundamental problems
of queue spilIback arid flow blockage are addressed. Guidelines for identifying "closely-spaced"
intersections are provided. A detailed computer-based aigorithm,basedon the Prosser-Dunn model,
for assessing the impacts of queue spilIback was developed arid applied to a wide range of traffic
conditions, including oversaturation.
D.2 LINK FLOW CONDITIONS
Traffic flow on a signalized cross arterial link traveling through an interchange is very
complex due to several factors. The downstream traffic signal routinely interrupts the flow forming
queues behind the signal which must be subsequently dissipated Junng the next cycle. The amount
of queue formation depends on the amount of upstream mining traffic, the quality of signal
progression, and the resulting total traffic demand that arrives on the movement.
Traffic operations on the link ultimately depend on whether the link's arrival demand
volume for any movement exceeds the link's capacity to service it. If the link becomes
oversaturated, due to demand exceeding output capacity, then the link urine initially experience severe
queue spilIback and soon watt become flooded with cars. The degree of resulting traffic flow and
congestion depends almost entirely on the ability of the link to discharge vehicles downstream,
otherwise total stoppage (grid lock) mI} occur. During heavily congested conditions, signal
coordination primarily determines which feeding movements get to proceed, but usually not the
Overall quantity of flow on the link.
Assuming that traffic conditions on a link are urldersaturated, then flows on the link may be
assumed to cycle through a series of states shown in Figure D-} where the arrival flows from the
upstream movements proceed downstream and some are routinely stopped, accumulated in queue
behind the downstream signal displaying red, and then subsequently serviced on the following green.
Whereas arrival flows to the interchange ramps and some minor cross streets may be random at some
average arrival flow rate, most arrival demands to the head of a link are, in reality, the output flow
profile from an upstream signal modified by some platoon dispersion, depending on the distance
"raveled downstream. Platoons are usually not dispersed Critic they travel about two (2.0) minutes,
so platooned flow along the crossing arterial is common for urban interchange operations.
D - ~
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L
u
TV ~ q.=
Flow, q (vph)
_ Vm (qm+ql+qr)
Saturation
q -S
Arrival Flow
qm = V
"d"
qm= V
No Flow
qm= 0
Green
Red
Green
Time in Cycle
Figure D-1. Traffic Flow Conditions on a Link.
D - 2
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D.2.l HCM Arrival Flow Profile
Anival flow to a downstream closely-spaced arterial signal is inherently dependent on
upstream traffic and signal conditions which may not be well known. Modeling assumptions may
have to be made to provide tractable solutions. For Highway Capacity Manual fly level of analysis,
one assumption usually made is Hat He anival flow to an arterial signal is composed of two
component flows: one arriving during the downstream green and one during He red, or
Va = Vg (g/CJ + Vr (r/C)
where:
v =
a
Vg =
vr
g/C =
r/C
=
=
average arrival volume during cycle, vph,
anival volume on green, vph;
arrival volume on red, vph;
green ratio; and
red ratio.
D.2.2 PASSER I! Arrival Flow Profile
(D-~)
The next higher level of arrival flow profile found in traffic signal timing software is in the
PASSER I} - 90 software developed by IT] for arterial signal timing optimization (29. PASSER I!
assumes a two-flow model somewhat like the HCM except that one flow, the larger arrival flow, is
defined as being the flow of the larger arriving platoon, in time, space, and rate of flow, and all other
flows are combined into a single secondary flow region. This flow model is
va = vp (s/CJ + vnp (C p)/C (D-2)
where:
v =
a
Vp =
~=
no
average arrival volume during cycle, vph,
maximum arrival volume in largest platoon, vph, and
average arrival volume during remainder of cycle, vph.
Flow profiles depend on the progression arid are calculated based on the resulting time-space
diagram for the arterial. Delay is calculated as being the area of the queue polygon resulting from
the piece-w~se integration of the input-output flow profiles over a representative cycle.
D.2.3 PASSER Ill Arrival Flow Profile
PASSER Ill is a computer model developed by TT! that is widely used to evaluate arid
optimized traffic operations at signalized diamond interchanges (3~. Its arrival flow model provides
the next higher level of sensitivity to variation in predictable arrival patterns. PASSER Ill models
D-3
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three flow regions of the cycle, having two flows for each of the three upstream protected phases
(i.e., A, B. C). The resulting downstream flow profiles are
va VAP (Ap/C) + VAa fA a/C) + VBp (Bp/C) + Vga (Ba/C) + vcp (Ca/C) + VCa (Ca/C) (D-3)
where:
average arrival volume during cycle, vph;
platoon flow downstream from Phase A, vph;
average arrival flow to/fiom Phase A, vph;
platoon flow downstream from Phase B. vph;
average arrival flow to/from Phase B. vph;
platoon flow downstream from Phase C, vph; and
average arrival flow to/from Phase C, vph.
For diarnondinterchanges,arrival flows comingirom an upstream Phase C signal are normally zero
because this phase is the outbound left turn phase to the on-ramp.
D.2.4 TRANSYT 7F Arrival Flow Profile
An even higher-level arrival flow model is employed in TRANSYT 7F, another signal
timing optimization and analysis program supported by Federal Highway Administration (4). A
discretized output flow profile is developed for each upstream input flow which, when adjusted for
platoon dispersion, becomes a component to the arrival flow profile of interest. The individual
arrival flow profiles are summed to generate the total arrival flow profile. As in the above programs,
the upstream phasing sequence must be defined together with all elements ofthe time-space diagram,
signal progression, and platoon dispersion The discretizationtime slice in TRANSYT 7F can be as
small as I/60 of a cycle. Thus, there can be up to sixty discrete arriva] flows calculated for a
movement, which are the sums of all upstream feeding flows for that time slice (T+K), or
v&(T+K&) = ~vm(T+K=)
(D-4)
where T is the travel time between the intersections. This high level of detail can only be achieved
by automating the analytics to make the volume estimation process practically feasible.
D.2.5 Arr~val Flow Mode! Recommended
The selection of arrival flow modeling sets the standard for the level of precision for all ~e
level of service modeling to follow. It is presumed that the average flows that would occur during
the cycle, peek period, and wi~in the hour of interest can be estimated with suff~cient accuracy. For
external approaches or isolated approaches that do not have a signal within two minutes travel time
to the subject approach, the assumption of random arrival flows (Poisson) is sufficient. For
traditional coordinated arterial signal systems having modest turning traffic (say 20%) and generous
D -4
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intersection spacings, a two-flow model (flow during red, flow during green) as employed in
Chapter 1 1 (Arterials) of the HCM is probably sufficient in most cases. An obvious improvement
would be to transition to a two-flow model like used in PASSER II (flow during platoon, flow not
in platoon). For interchanges operating with high turning traffic and within a system of closely-
spaced intersections, arrival modeling should be at least as detailed as used in PASSER III wherein
six arrival flows are possible, two (platoon and non platoon) for each of three feeding movements
during the cycle. While coordination in interchange systems is probably not as effective (due to
higher turning traffic and more balanced flows) as in arterial systems having predominate through
traffic, at least the features of platoons arriving at interchanges could be readily identified so that
queuing and delays could be reduced somewhat over solutions assuming random flow.
D.3 MACROSCOPIC FLOW MODELING
Models of traffic flow routinely used to describe continuous flow traffic facilities, like
freeways, can also be used to describe the dyna~nicsof flow attraff~c signals (5). These models can
be used to describe the nature of operational problems experienced. Traffic signal operation
routinely creates brief interruptions to continuous flow, forming bottlenecks where the arrival
demand exceeds output capacity during the red interval. Shock waves form behind the signal due
to the queuing and spillback that occurs (6).
D.3.! Flow Models
When traffic flow can be assumed to be in a steady-state condition, even for a fairly brief
period of time (t) and space (x) such as for random arrival flow or platoon flow at saturation, then
the average flow rate during this period, v (vpt) can be thought of as being produced by a traffic
stream having an average density k (vpx) traveling at an average speed u (xpt)' or
v =ku
where:
v = average traffic flow rate' vpt,
k = average traffic stream density, vpx; and
z' = average traffic speed, xpt.
(D-5)
In undersaturated,signalized arterial operations,the flow rate on a short section of roadway
just upstream of a traffic signal is usually in one of three states: v = v, v = 0, or v = s. That is, the
flow is characterized as being either the arrival flow, stopped in queue, or the queue has been
transformed into a platoon with saturation flow, s. These three flow states were identified in Figure
D-1 . Whether these changes from one state to another occur almost instantaneously(creating shock
waves) or transition over a brief period of time (forming characteristic waves) is more of a
theoretical issue and presumed herein to be of minimal importance. For convenience, rapid response
to signal change is assumed so that conventional shock wave analysis can be employed.
D - 5
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Basic macroscopic traffic flow modeling for steady-state conditions presumes (hoary car-
follow~ng laws and empirical observations) that the speed of operation is a function of the average
density of the section of road (immediately ahead) and the current speed (5) such that
~U~-m i- = 1
u k
f q
and solving Equation D-6 for u =f(k) and substituting into v = k u, ~ = kf~J yields
v = k Uf t1 _ ~ k yI~1~1/~1-m)
q
for the section x of interest, where:
v = traffic flow in section x at t, vpt;
k = traffic density, vpx;
Uf = free speed, xpt;
kq = jammed queue density, vpx; and
I, m = shape coefficients.
(D-6)
(D-7)
The traffic flow graphs (for Case I) shown in Figure D-2 were drawn for the following
assumed arterial operating conditions: Uf = 64 km/in (40 mph); kq = 143 vpkmpI, (7m/veh, 230
vpmpI, 23 It /veh), s = 1800 vphp} at a saturation flow speed of 37 km/in (23 mph). Under these
assumptions,] = 2.645 and m = 0.666. The three flow vanables (v, k and u) are all interrelated and
can be calculated given one of them together with the section parameters arid coefficients.
D.3.2 Saturation Flow
Saturation flow can be assumed to be the "capacitor" flow region of the flow curves,
suggesting that the saturation flow may increase as the platoon speeds up, or flows faster if the
platoon can travel faster for a given vehicular spacing (density). Research shows that saturation flow
increases with increases roadway quality and operating speed. Over several editions of the HCM,
freeway capacity has increased from about ~ 800 vphpT at 50 km/in to 2000 vphp! at 70 kmAl, and is
now approaching 2300 vphp! at 90 larch. This NCHRP research combined with the results of
NCHRP 3-40 (~) suggests similar bends for sianalizedintersections end interchanges when flowing
- - - ~ ~ ~ - -7
~ ~ ~ - ~ ~ ~ o ~ ~ . . ~ . ~ ~ ~ a, 1
at equal loadings and pressure. Figure ~-;z also Sates now the ~low-censlty curves would
expand with increasing quality of operations. Case I] assumes a saturation flow of 2,000 vphp} at
40 km/in (25 mph). The loci to the family of curves would represent the expected saturation flow at
the signal when We platoon is flowing at its maximum flow for increasing quality of operating
conditions.
D - 6
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Flow vs Density
2000
1800
1600
1400
-
~ 1200
- 1000
0 800
-
E~
-
600
400
200
O
2000 - .
18~
, _
, '.
\
Saturation Flow
stopped Quelle
\'
Case I ~
\
"+ Case II
:'..
1 1 1 11~
2S SO 7S 10012S lSO
Density (vpkm)
Speed vs Flow
1600
1400
1200
1000
800
600
400
200
o
~ ~ .
~ ' ~
,"~ """
;;~/ Samration Flow \
/ \ '
.'/ \ '%
~HI ~ ~II ~
it"
~ ERIC ~
lk ~'`
~ 1 1 1 1 1 1 ~1 ~
0 10 20 30 40 50 60 70 80 90
Speed (kmph)
S=1800 vph
S=2000 vph
Uf= 64 kInph
Uf= 88 kmph
.
Figure D-2. Characteristics of Traffic Flow for Two Capacity Conditions.
D-7
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D.3.3 Shock Wave Speed
When arterial arrival flow passing through a green signal is suddenly stopped by the onset
of red, the output flow suddenly drops from ~ = v to v = 0. When this charge in output flow of the
section occurs, the storage begins to queue behind the signal at a storage density k,, the queuing Clam
density'' of about 7.0 meters per vehicle (23 feet/vehicle). The speed at which the storing queue
propagates (spills back) upstream can be estimated from shock wave theory 66) as
W l~v v (D-8)
q
where:
Wr = shock wave spillback speed due to red onset, xpt,
v = arrival volume (dunng red), vpt; and
k = traffic stream density, vpx.
The speed at which the shock wave propagates upstream increases with increasing arrival
volume. For an approach having average flow conditions described as Case ~ in Figure D-2, and a
green ratio of 30% which yields a phase capacity of 540 vphpl, then for arrival volumes of 20, 60
and 100 % of signal capacity, the shock wave speed progressing upstream during red would be
estimated by Equation D-8 to be 0.22, 0.67, and 1.12 mps, respectively. However, if the signal
became oversaturated or poorly timed such Mat the start of red ended platoon motion while at
saturation flow v = s, then the shock wave speed would rise to 5.35 mps. However, as long as the
signal is undersaturated, the maximum queue length per cycle would remain the same.
- 7 .A '' ~
D.3.4 Platoon Wave Speed
When the signal turns green following an extended red time and subsequent queue buildup,
the platoon responds and begins to move forward, reaching saturation flow conditions in a few
seconds after green onset. For HCM-level of analysis, the platoon is assumed to reach saturation
flow almost immediately once the queue-platoon transformation begins at any point in the queue.
Under these simplifying assumptions, the platoon's green wave speed would be given by
W =
where:
Wg
k -
h _
Ant
=
REV S
g Ak
k - k
q s
platoon start-up green wave, apt;
saturation flow during green, vptpl;
platoon density during saturation flow, vpxpl; and
queue storage density, vpxpl.
D - 8
(D-9)
OCR for page 287
The queue's transformation into a moving platoon would propagate upstream following
green onset at 5.35 mps (12 mph). If the output saturation flow is unimpeded on the downstream
link, then the green wave speed is fairly constant over all arrival volume conditions, simplifying
the analysis. The main problem would be to determine how far upstream queue spilIback has
progressed each cycle before it begins clearing.
D.3.5 Clearing Wave Speed
Once the platoon begins to move forward on green, flow in the platoon is assumed to be
saturation until the platoon clears the stopline. The arrival volume, v, is entering the upstream end
ofthe platoon; whereas, saturation flow, s, presumably is occurring downstream to the stopline, or
downstream boundary. During this cleanng penod, the back of the platoon is traveling downstream
from its maximum backup at the platoon clearing wave speed of
W = Av = s - v
C Ak ks - k
(D-10)
Continuing with the data of Figure D-2 where kq = 143 vpkmpI, s = I 800 vphpI, us = 37 km/il, then
ks = 48.6 vpkmp] from k = v/u. An examination of the above wave speeds follows.
Figure D-3 presents the resulting wave speeds for the above conditions for volume-to-
capacityratiosofO.2to I.O for aselectedg/C ratio of 0.3 end a LOO-second cycle. The speed ofthe
shock wave, W., is very slow (about ~ nips) and only increases slightly with increasing v/c ratios of
the signal. The platoon start-up wave, Wg, is noted to be a constant of 5.35 mps, and the platoon
clearing wave is high (about ~ 6 mps) and only decreases slightly with increasing arrival volumes.
Thus, because the wave speeds are fairly insensitive to arrival volumes at traffic signals, analyses
based on the wave speeds are relatively stable as long as traffic flow on the link is undersaturated.
D.3.6 Queue SpilIback
The duration and extent of queue spilIback determines whether an upstream intersection wall
be severely affected by downstream operations. In essence, the characterization of adjacent
intersections being too "closely spaced" can be defined for undersaturated conditions where X<=
1 . Using the above shock wave theory, the maximum length of queue spilIback can be determined
in time arid space by algebraic solution of the Wr and Wg wave intercepts for a given red time. r The
elapsed time following green onset before Wg catches Wr is
~7
W
T = r r (D-1 1)
g W- W
where Tg is the elapsed time since the onset of (effective) green.
D-9
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we
we
2s
1 = 2.645
20 - . m = 0.666
E ~--
- IS
C
~ 10
3
s-
O- ~1
0 0.2 0.4 0.6
-
Wr
o.g l 1.2
Phase Volume-t - Capacity Ratio, X
Figure D-3. Wave Speeds at Traffic Signals During Undersaturated Conditions.
The maximum queue backup for undersaturated conditions, [n'' is equal to A, = Wg*T,, or
W W
L = g r r
m W ~ W
g r
subject to the restriction that the downstream phase is not oversaturated.
(D-12)
Figure D-4 presents the queue spilIback for an approach having random flow-(uniformly
distributed over the cycle) for v/c ratios up to ~ .0, or saturation. The traffic and control conditions
are as above (s = ~ 800 vphpI, C = ~ 00 see, g/C = 0.31. The capacity of the approach is 540 vphp!
(c = 0.3 * ~ 800~. The maximum queue spilIback distance upstream from the stopline would be S2
meters and 99 meters for v/c ratios of 0.6 and ~ .0, respectively, using Equation D-12. These results
indicate that storage links less than ~ 00 m may often experience spilIback problems on entry flows
where good progression is not provided, even when the downstream flow is undersaturated.
D- 10
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100
90
So
-
as
-
-
An
70
60
SO
40
30
20
10
o
. ~1
C = 100 see
g/C = 0.3
Random Flow
_ ~
/
I 1 1
0 0.2 0.4 0.6 0.8 1 1.2
Phase Volume - Capacity Ratio, X
Figure Dot. Affects of Volume on Queue SpilIback for Undersaturated Coalitions.
D.3.7 Two-Flow Arrival Models
The ~ 994 Highway Capacity Manual (HCM) and its proposed arterial enhancements (l, 79
assume that the arrival volume along an arsenal is composed of two arrival flows: a flow arriving
on He red, ~,~ and a flow arriving on the green, vg, as noted in Equation Dot. The HCM's two-flow
arrival mode} cart also be applied to the above queue spilIback equations with little change in form.
Defining the arrival volume on green, vg, to be
v = R^v
g -'P
where harp is the platoon ratio, and the arrival volume on red, or, to be
v =
r
(D-13)
C - g R p
v
r
D- 11
(D-14)
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The offset ~ between the intersections u-a7 is considered from the start of the green period
of movement ~ at the upstream intersection until the start of the through green at the downstream
intersection. Hence, Go starts a time ~ after To, equal to the offset with respect to the upstream
intersection. The end of the downstream bottleneck phase, EM, is
EM= To + B+ Go
where ~ = offset in seconds from u to al.
Step 3. Calculate To and T2
(D-41)
The beginning of the clear period, T., at the upstream intersection starts when the blocking
queue clears the upstream movement of interest after the start of through movement green time at
the downstream intersection. Note that it has been assumed that a blocking queue exists. This may
be determined later to not be true if volumes are not high enough for the length of link involved. If
true, then
T~=To +0 + tq
(D-42)
In order to evaluate whether the intersignal link length wait be completely blocked under
different traffic conditions, some related factors have to be evaluated. One of them is the volume-to-
-capaci~ ratio (X~) at the downstreamintersection. Another one is the critical link length (Lo) which
determines the occurrence of queue spillback. The critical link length is calculated as follows:
L~ = G~*Wr*Wg*6C- G~)/~(lYr+ W~)~6C/X~- G~)J
where:
Wr =
Wq
=
Shock-wave speed (mps); and
Platoon starting wave speed (mps).
(D-43)
When Xa. is greater than 1 .0, the downstream link will be oversaturated by vehicles coming
from He upstream intersection. If the link length ~ is greater than critical link length [c, it is
assumed that a proportion of the storage vehicles w~11 not be cleared from the link due to the limited
capacity at the downstream intersection. Hence, some residual queue will remain on the link when
the green ends. When the link length is shorter than the critical link length, all vehicles stored on
the link will be cleared. If X~ is less than ~ .0, queue spillback may also occur due to an inappropriate
signal timing plan for the downstream green time and offset. This problem can be solved by using
He same assumption as the one for oversaturation and by checking the results after the loop
calculation in the computer program. If some blockage is found to occur, then saturation flow
adjusunents are made to account for either impelled or blocked flow using Equation C-46 in the
·
prevlous appenc 1X.
D -26
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The time at the upstream intersection, denoted as T2, when vehicles discharged from
the upstream intersection ~} first begin queuing at the downstream intersection stop line on red,
is calculated as follows:
T2 = To + ~ + Go - If T2 2 To (Daft)
Step 4. Calculate Input Flow Rates during the Cycle
The potential flow rates from the upstream intersection during the cycle are calculated in this
step. Each upstream movement's capacity is computed based on the elective green period of that
movement (SMm + SL to EMm + KU) and the saturation flow rate s. The duration oftime, t9 that
every other movement experiences saturation flow is given by
Is = Vi (C - g; ~ / (Si - Vj ~
(D - S)
Thus, from time SMm ~ AL to tS the flow is saturation and from tS to the end of movement
green time (EMm + BL), the movement flows at its arrival flow rate, v. In the event Is is greater then
the end of green of the movement, the flow continues at saturation flow for the entire period of
green for that movement. The flow at all other times in the cycle (e.g., dunug yellow and red
clearance time) is assumed to be zero.
Step 5. FindEnd of Clear Period, T3
The next step in the process is to find T3, Me end of the clear period at the upstream
intersection. The time T3 is defined as We time when the intersignal length would be once again
completely filled with vehicles, thus blocking the upstream intersection. The related duration of time
t3 iS needed to accumulate enough vehicles to fill the link ~ after time T2. The number of seconds
required to completely fill the intersignal length depends on the output flow of vehicles from the
upstream intersection after T2.
The potential output flow for every second in the cycle is known from We previous steps.
The summation of the output flows (vehicles per second) from the upstream intersection over time
(seconds), gives the number of vehicles that may enter link ~ until the link completely fills to its
storage capacity, An,, over We duration t3, or
n~t3,l) = nO ~ ~`qum < nma~ o < t3 < C (D-46)
The assessment of no is critical to the algorithm. no = 0 when T. < T2. However, if the link
can store more vehicles than the downstream phase can serve (its capacity) (because the link may
be long and/or the phase relatively short)then T 0 (instead of zero, or some
other value) at time T= To. When T2 < T ,, vehicles already on the link cannot clear the next
downstream phase. Thus, the most vehicles that can enter the link during the next cycle cannot
D-27
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exceed the downstream phase capacity, cm. If it is ultimately found that all the upstream flows can
use the link for a time period T3 longer than To + C, then the link does not totally fill wing the cycle
and the upstream signal is unblocked, but it may still have some reduced output flow. When queue
blockage occurs, the end of the clear period for entry into the upstream intersection would be
T3 = T2 + t3 T2 < T3 ' TO ~ C (D-47)
In the PDX program, a simple DO-Ioop calculates t3. The loop increments the number of
seconds after t2 and for each second adds the number of vehicles entering the link. No departing
vehicles need to be considered here. When the link either fills completely or reaches Me next cycle
(T~ + C) , We incrementation process stops. We number of time steps used gives We number of
seconds elapsed after t2, which is t3.
The saturation flow-queue interactionmodel described in Appendix C has been implemented
in the PDX program. As each upstream phase begins to be analyzed dunug the cycle, the
downstream queue length is calculated. The available travel distance to the back of the queue is
then determ~ned,know~ngL. The adjusted saturation flow is determinedirom Equation C-8 for the
upstream movement of interest. Preliminary analyses based on initial pointers and Equation D-43
estimate whether queue spilIback is likely. Should the completed t~me-step analysis not support the
initial assumption, then the alternate equation is selected and the process repeated.
Sfep 6. Identify Clear Period
The clear period is defined as the time in the cycle from the end of queue blocking to the start
of blocking in the next cycle at the upstream intersection. The clear period, CP, is the duration from
T. to T3 when upstream input flow can occur, or
CP = T3 T.
(D-4g)
The values of To and T3 have not been calculated module C. Thus, they can have values greater than
the cycle length. While calculating the clear period, the values of SMm and EMm should be adjusted
for start loss and end gun, respectively, at some convenient point in the program.
Sfep 7. Compute the Modif edF Effective Green Period
The modified effective green period (get) during which the upstream movement can discharge
vehicles is the time overlap of the unblocked effective green (au) of the movement arid Me clear
period. The clear period is calculated in the previous step. Table D- ~ ~ shows the modified effective
green periods (aged) for Movement ~ for different positions of to and t3 with respect to the upstream
signal time as calculated in the PDX program. Thus, the real effective green (g = geld is
g = g rip CP
D -28
(D-49)
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Representative terms from entire chapter:
arrival flow
Table D-~. Modified Effective Green Periods for Movement
Value of I, ~Value of t3 ~gerf
SMOG ~ t' t, ~t3-t,
| EM it; t3 ~| EM' - I,
| EM, I,; mod(t3, ~ -)
Testbed Design. In order to test the above program's applicability,an experimental
teethed was designed. An arbitrary paired intersection system was set up with all the traffic and
signal timing variables affecting saturation flow being defined. This test system was analyzed using
the Prosser-Dunne FORTRAN program for a range of conditions and the movement's capacities
were computed. A design scheme of the study paired intersection was shown in Figure D-9.
Required inputs such as traffic volumes, signal timing parameters such as green times,
offsets, cycle lengths, and spacing between intersection were carefully prepared. Only a pretimed
signal system was considered for this research. The spacing between the two intersections was
considered to be 100, 200, and 300 meters, respectively. These spacings were assumed to be
representative of most closely-spaced intersections within an interchange environment. Other
parameters in the teethed are summarized in Table D-12.
Table D-12. Testing Parameters for the Program and Simulation
V/C Ratios 0.8, 0.9, 1.0, 1.1, 1.2, 1.S
| Cycle length | 100 seconds
Upstream Phase Splits ~ ?:
| Downstream Phase Splits | 50-50, 60~O, 70-30
Offsets at ~ seconds Intervals (0 to C)
| Spacing 7 100, 200, 300 meters l
Total Cases Studied 900
1 ~ .. _.
Total number of NETSIM simulations ~
Results and Verification. The computer program was run by using the above operating
conditions to get Me data base. Each of the above cases was simulated 10 times using TRAF-
NETSIM to obtain average simulation results. A total of 9000 NETSIM runs were performed
during the testing process. The study results were categorized and evaluated according to different
operating conditions and are summarized in the following sections.
The throughput-offset relationship was examined in order to study the outputs from the
PDX pro gram. The effects of different volume-to-capacity ratios and intersignal spacings were
studied. A very close relationship between PDX program and NETSIM simulation was observed
dunng the comparison process. The results are shown in Figures D-12 Trough D-15. In each
figure, results from the PDX Model and TRAF-NETSIM are shown in the same dimension for
comparison purposes. Figure D-16 is the regression plot between NETSIM arid PDX Model. The
coefficient of regression between the two models was observed to be 0.85. Furler research studies
are underway to improve the PDX Model.
D - 30
. ~-
600 -
~ t _r ~
SOO - _
~ · l~h~ugh-S'm
~ t I 1=~
_ X Tdel~im
g300; - X ~oug~Model
~ ~ __ _
~ · Righ~Model
s 200 - ~ ~-M - l _
~ Td a l-l Cod cl
100
O- l l l l
0 20 40 60 80 100
Offset (see)
Figure D-12. Throughput-Offset Relationship between NETSIM and PDX Models for a spacing
of 100 meters; v/c of 0.S and Saturation Flow of 1900 vphgpl.
2000
1600
-1200
800
400
o
_ . _
~-X ~ ~ X X X X ~ _
_^ _ _ ~
i
it_
0 20 40 60
Offset (see)
80 100
-Is
Ritht-Sim
~ LcR-S~
- X Total~im
X Throug~Motel
let-Model
I L~t-Model
-- Total-Model
Figure D-13. Throughput-Offset Relationship between NETSIM and PDX Models for a spacing
of 100 meters; v/c of 1.5 and Saturation Flow of 1900 vphgpl.
D -31
2000
1600 ~ ~=Through-Sim
1200 ~ LeB-Sim
_ X Total-Sim
S ~ Through-Modet
g 800; _' \* \ · Right-Mod~t
' ~ ~i\. ~ ~-Total-Model
~ 400~
0 20 40 60 80 100
Offset (see)
Figure D-140 Throughput-Ofiset Relationship between NETSIM and PDX Models for a spacing
of 200 meters; v/c of {.5 and Saturation Flow of 1900 vphgpl.
1'
2000
=~1600 ~_ , 47 ~+ ~|=T~
41200- · Lcft-Sim
~X Total-Sim
S _ ~-- : ~ lbrou0Model
g 800- '~ ~ \ ~· Ritht-Modet
~ ~5 ~I -M-I
~I ~ ~ ~ am_ . ~Total-Model
40075~ ~
0 20 40 60 80 100
Offset (see)
Figure D-15. Throughput-Offset Relationship between NE:TSIM and PAX Models for a spacing
of 300 meters; v/c of 1.5 and Saturation Flow of 1900 vphgpl.
D-32
1200
~ t ·. ~ .-.' I
{~L6cc~ · i- 1
X t -~ :~r y=eg85~ 1
C Low at' ' ~, R2~0.8^ 1
0 200 400 600 Boo 1000 1200
NETSIM Throughput (vph)
Figure D-16. Linear Regression between NETSIM and PDX Model Throughputs.
D.7.4. Discussion of the PDX Mode!
This computer program has been extensively calibrated against simulation results obtained
from TRAF-NETSIM simulation program. The purpose of the calibration effort was to pinpoint
the best fit parameter values used in this computer program to produce reasonable results. Traffic
engineering judgement was also exercised during this process. Overall, the computer traffic
program demonstrated good flexibility and accuracy in processing different types of traffic
conditions based on the comparison results observed by the research team.
It should be noted that in the original Prosser-Dunne Model, traffic operating conditions
were always assumed to be oversaturated, therefore, blocking would always occur because of
insufficient service capacity at the downstreamintersection. It was found, however, that blocking
or queue spilIback may also occur during undersaturated conditions given the limited storage
spacing and bad offsets. Project study results have shown the most important factors that affect the
estimation of queue spilIback or blocking occurrence are downstream signal intersection's green
time, the intersignal spacing (i.e., link length) and the volume-to-capacity ratio. Besides these
factors, a critical spacing that defines the boundary condition of the occurrence of queue spiliback
was identified as a function of the downstream intersection's green time and volume to capacity
ratio and other parameters. This new methodology helps define different types of problems based
D-33
on the varying nature of different operating conditions and renders the corresponding treatments.
After applying these enhancements to the original Prosser-Dunne Model, a wide variety of real-
worId operating conditions can now be categorized and evaluated systematically by their specific
types of problems, such as queue spiliback due to inadequate storage spacing and/or oversaturation.
Several key parameters used in the PDX computer program were calibrated extensively
ureter different operating conditions. Sufficient attention has been given to the effects of the
selected values of the saturation flow density on the subsequent calculation of other variables.
Because of its direct impacts on the estimate of saturation flow speed and interacting traffic wave
speed, any change to the saturation flow density would result in different mode] outputs. So far,
the parameter values used in the model have been calibrated to produce reasonable outputs
compared with the simulation results from TRAF-NETSIM. Other calibrated parameters include
the phase lost time and the unimpeded saturation flow rate. A vehicle unit length of 7.0 meters (23
feet) at jam density was used as the result of our nationwide data collection and analysis effort.
Another major Improvement made to the computer traffic mode] was the introduction of
the Queue Estimation Model. As an important part of traffic signal operation, especially during
saturation periods, the behavior and characteristics of queuing traffic and its evaluation
methodology have been studied by various researchers recently. The study approach used by the
research team during the mode! development end calibrationwas to obtain queue traffic information
dynamically throughout the cycle. A queue calculation submodule was provided to achieve this
purpose. Similarto Me capacity analysis methodology,the queue submodule performs second-by-
second calculations to estimate the changing queue status at the downstream intersection. The
useful information provided by this program can be subsequently analyzed or used in over traffic
engineering models when studying the traffic operating conditions of a paired intersection.
The effects of available downstream travel distance to the back of queue was also provided
in the PDX Model. As presentedin Equation C-8, Me saturation flow rate on green may be reduced
by insufficient clear distance at start of green that permits platoon vehicles from accelerating to
nominal saturation flow speeds. Thus, the clear period may have impeded saturation flow, but not
blocked flow.
D.7.5 Existing Software Enhancements
Implementation of Queue-Interaction Models into internationally recognized computer
programs is highly recommended. Some work toward this objective is known to be already
underway (10,11,129.
D-34
REFERENCES
1. "Highway Capacity Manual." Special Report 209, Transportation Research Board,
Washington, D.C., Third Edition, (1994).
2.
Chang, E.C. and Messer, C.~. "PASSER Il-90 Users Manual." Texas Transportation
Institute, College Station, Texas, (19901.
Fambro, D. B., Chau&ary, N.A. and Messer, C.~. "PASSER IlI-90 User's Manual."
Texas Transportation Institute, College Station Texas, (1991~.
4. Wallace, C.E. and Courage, K.G. "TRANSYT-7F User's Mar~ual." University of Florida,
Gainesville, (1988~.
May, A.D. Traffic Flow Fundamentals. Prentice-Hall, Englewood Cliffs, New Jersey,
(1990) p.306.
6. Lighthill, M.~. and Whitham, G.B. "On Kinematic Waves: Part Il. A Theory of Traffic
Flow On Long Crowded Roads." Proceeding of the Royal Society, A2239, No. INS,
(1955~.
.
Fambro, D.B., Rouphail, N.M., SIoup. P.R., Daniel, I.R., Id, I., Anwar, M., and R.~.
Engelbrecht. "Highway Capacity Revisions for Chapters 9 and ~ 1." Report No. FHWA-
RD-96-088, Federal Highway Administration, Washington, D.C. (1996~.
8. Leiberman, E.B, McShane, M.R., and Messer, C.~. Traffic Signal Control For Saturated
Conditions. KLD Associates, Inc. NCHRP Project 3-38~3) Report, Vol 2., (1992) p. 15.
9. Prosser, N. and Dunne, M. "A Procedure for Estimating Movement Capacities at
Signalised Paired Intersections." 2nd International Symposium on Highway Capacity,
Sydney, Australia, (1994~.
10. Rouphail, N.M. and Akcelik, R. "Paired Intersections: Initial Development of Platooned
A~xivaland Queue Interaction Models." Australian Road Research Board. Working Paper
WD TE91/010, Vermont South, Australia (19911.
. Akcelik, R., Besley, M., and Shepherd, R. "SIDRA (Windows) Input Redesign for Paired
Intersection Modelling." DiscussionNoteto WD 96/008. Australian Road Research Board,
Vermont South, Australia (1996~.
12. Chaudhary, N.A. and Messer, C.~. "PASSERIV-96, Version2. l, User/ReferenceManu~."
Texas Transportation Institute, College Station, Texas. (1996~.
D-35
