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APPENDIX C
DEVELOPMENT OF AN ANALYTICAL MODEL FOR PREDICTING
PHASE TIMES AT TRAFFIC-ACTUATED INTERSECTIONS
An analytical model for estimating the average cycle length and phase times generated by a traffic-
actuated controller was one of the principal products of NCHRP Project 3-48. In this appendix, the
mode! will be summarized and examples of its application wait be presented. The entire procedure
developed under this project encompasses both the analytical model and a computational structure
for implementation of the model. The computational structure win be described in Appendix E.
TYPES OF TRAFFIC CONTROL
Traffic engineering textbooks describe three types of traffic signal controllers:
· Pretimed Controllers, In which a preset sequence of phases is displayed in repetitive order.
Each phase has a fixed green time and change interval that are repeated in each cycle to
produce a constant cycle length.
· Fully-Actuated Controllers, in which the timing on all of the approaches to an intersection
is influenced by vehicle detectors. Each phase is subject to a minimum and maximum green
time, and some phases may be skipped if no demand is detected. The cycle length for fillly-
actuated control win vary from cycle to cycle.
Semi-Actuated Controllers, in which some approaches (typically on the minor street) have
detectors, arid some do not. The earliest fonn of sern~-actuated control was designed to keep
the green on the major street in the absence of a minor street actuation. Once actuated, the
minor street green is displayed for a period just long enough to accommodate the traffic
demand.
While these equipment-based definitions have persisted in traffic engineering terminology, the evolu-
tion of traffic control technology has complicated their function from the analyst's perspective. For
purposes of capacity andlevel of service analysis, it is no longer sufficient to consider the controller
type as a global descriptor of the intersection operation. Instead, an expanded set of these definitions
must be applied individually to each lane group because each lane group could fall into any of the
above control categories.
Each lane group may be served by a phase that is either actuated or non-actuated. Non-actuated
phases may be coordinated with neighboring signals on the same route, or they may function in an
isolated mode without any influence Dom other signals. Non-actuated phases generally operate with
fixed minimum green times which may be extended by reassigning unused green time from actuated
phases with low demand, if such phases exist.
Appendix C: Page 1
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Actuated phases, on the other hand, may be used at intersections where coordinated non-actuated
phases exist, but they may not be coordinated themselves. In such cases, the actuated phases are
subject to early termination (force off) to provide for system progression. Actuated phases are
subject to being shortened on cycles with low demand. On cycles with no demand, they may be
skipped entirely, or they may be displayed for their minimum duration.
For purposes of analysis, the length of each phase and consequently the cycle length willbe fixed at
intersections where all approaches are non-actuated. This denotes the condition ofpretimea, opera-
tion mentioned previously. In current practice, one or more phases under this type of control will
usually be coordinated. In general, if the intersection is sufficiently removed from its neighbors to
operate in an isolated mode, then actuated operation will produce lower delays and a better level of
service. The analysis procedures prescribed previously in this report will indicate the degree to which
the delays may be reduced by actuated control on any phase.
Where all phases at an intersection are actuated, then the length of each phase, and consequently the
cycle length, win vary with each cycle. This denotes the condition of fi`lly-actuated operation men-
tioned previously. Coordination with neighboring signals is not possible under this control mode.
Fully-actuated signals are generally used only at intersections where distances are such that
coordination would not be expected to be beneficial. The analysis procedures prescribed previously
in this report will support an evaluation of the comparative benefits of coordinated operation versus
actuated operation.
The semimctuated control mode includes all of the cases that do not fall under either operation,
pretimed nor fully-actuated. The majority of coordinated arterial systems must be treated as systems
of semi-actuated controllers with coordinated non-actuated phases serving the arterial approaches
and isolated actuated phases serving the cross street approaches. The cross street approaches include
minor movements such as protected left turns from all approaches. The cycle length is constant at
coordinated sem~-actuated intersections and variable at isolated sem~-actuated intersections.
The analysis procedures presented in HEM Chapter 9 are based on the assumption of a fixed
sequence of phases, each of which is displayed for a predictable time. In the case of pretimed control
(i.e., no actuated phases), the length of each phase is assumed to be fixed, and constant from cycle
to cycle. Actuated phases must be approximated for analysis purposes by their average green time,
recognizing that the actual time may differ from cycle to cycle. For a given timing plan (i.e., constant
or average green times), the differences between actuated and non-actuated phases are recognized
by the parameters used in the incremental tea ofthe delay equation.
PlIASE PLANS
Two-phase control is the most straightforward and simplest ofthe available phase plans. Each oftwo
intersecting streets is given a green phase during which all movements on the street are allowed to
proceed. All left and right turns are made on a permitted basis against an opposing vehicle or
pedestrian flow. This phase plan is generally used wherever the led turn movements do not require
Appendix C: Page 2
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.
protected phasing. Multi-phase control, on the other hand, is applied at intersections where one or
more led turns are determined to require protected phasing. Multi-phase control can be provided in
a wide variety of ways, depending on the number of turns requiring protected phasing and the
sequence and overlaps used.
It is common in capacity and level of seance analysis to use a "single-ring, sequential" representation
of the phase plan in which a single phase is used to indicate the combination of aD movements that
are proceeding at a given point in time. Modern tra~c-actuated controllers do not use this scheme.
Instead, they implement a "dual-ring, concurrent" phasing in which each phase controls only one
movement, but two phases are generally being displayed concurrently.
The dual-nng, concurrent concept is illustrated In Figure C-~. Note that eight phases are shown, each
of which accommodates one of the through or leg turning movements. A "barrier" separates the
north-south phases Dom the east-west phases. Any phase in the top group (ring I) may be displayed
with any phase in the bottom group (ring 2) on the same side of the barrier without introducing any
traffic convicts. For simplicity, the right turns are omitted and assumed to proceed with the through
movements.
{in E~ ~
5 EBL
6 WET
. -
Left side of barrier
E - W Movements
Ring 2
Barrier
7 I SOL
8
in.
4
J
SAT
1
Right side of barrier
( N - S Movements )
Figure C-1. Dual-ring concurrent phasing scheme with assigned movements
~1
Appendix C: Page 3
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The definition of a "phase" as presented in Figure C-1 is not consistent with the definition accepted
by most traffic analysts nor with the definition given in the introduction to HCM Chapter 9. It is,
however, a definition that is universally applied in the traffic control industry. It is the responsibility
of the analyst to recognize which definition is applicable to any given situation. For purposes ofthe
capacity and delay analysis procedures presented in the HCM Chapter 9, each lane group is con-
sidered to be controlled separately by a phase with specified red and green times, so either definition
could apply. The examples shown throughout the HCM Chapter 9 are based on the single ring
sequential concept. However, the dual ring definition must be used for estimating the timing plan at
traffic-actuated intersections using the mode! described in this appendix.
The advantage ofthe dual-ring concept is its ability to generate the optimal phase plan for each cycle
in response to the traffic demand. Pretimed controllers and earlier versions of traffic-actuated
controllers are more constrained in this regard. The maximum flexibility is provided by allowing the
first (usually led turns phases in ring ~ and 2 to terminate independently after their respective demands
have been satisfied.
It is also possible to constrain these phases to terminate simultaneously to emulate the older, and less
efficient equipment. For example, independent termination of the two left turns would introduce an
"overlap" between the left turn phase and the through movement phase. The overlap phase would
accommodate the heavier of the two left turns together with the concurrent through movement,
thereby making more effective use ofthe green time. Simultaneous termination of these phases would
eliminate the benefits of the phase overlap. The degree of benefit obtained from phase overlaps of
this nature depends on the dissimilarity of opposing leD turn volumes.
ALLOCATION OF GREEN TIME
The allocation of green time is an important input to the methodology presented in the HCM for the
estimation of delay. It is necessary to know the average cycle length and effective green time for each
lane group to be analyzed. The most desirable way to obtain these values is by field measurement,
however, there are many cases when field measurement is not possible. For example, the comparison
of hypothetical alternatives precludes field measurements. Even for the evaluation of existing condi-
tions' the required data collection is beyond the resources of many agencies.
A procedure for estimating the signal timing characteristics is therefore an important traffic analysis
tool. Such a procedure is also useful in designing timing plans that will optimize some aspect of the
signal operation. In this respect' pretimed and actuated control must be treated differently' because
the design and analysis objectives are different. For pretimed control, the objective is to design an
implementable timing plan as an end product. With traffic-actuated control, the timing plan is
generated by the controller itself, based on operating parameters that are established for each phase.
This creates two separate objectives for traff~c-actuated control. The first is to determine how the
controller will respond to a specified combination of operating parameters and traffic conditions. The
second is to provide some indication of the optimal values for the key operating parameters.
Appendix C: Page 4
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Functional Requirements of the Model
A practical traffic-actuated control mode} must be functionally capable of providing reasonable
estimates of the operating characteristics of traff~c-actuated controllers under the normal range of
design configurations at both isolated and coordinated intersections. It must also be sensitive to com-
mon variations in design parameters. Examples of design parameters include:
Tra~c-actuated controller settings (initial interval, allowable gap, maximum green
time)
2. Conventional actuated vs. volume-density control strategies
3. Detector configuration (length and setback)
4. Pedestrian timing (Walk, and Flashing Don't Walk)
5. LeR-turn treatment (permitted, protected, permitted and protected, not opposed)
6. LJefc-turn phase position (leading or lagging)
Data Requirements
The information that is already required by the HEM Chapter 9 procedure is used to the extent
possible to avoid the need for new data. Most of the additional data items relate to the operation of
the controller itself The model structure is based on the standard eight-phase dual-r~ng control
scheme previously illustrated in Figure C-1. This scheme is more or less universally applied in the
U.S.A. From a capacity and level of service point of view, less complex phasing concepts (including
simple two-phase operation) may generally be represented adequately as a subset of the dual-ring
scheme. For purposes of this discussion' the scheme for assignment of movements to phases
presented in Figure C-1 will be adopted. This will greatly simplify the illustration of all modeling
procedures without affecting the generality of the results.
Appendage presents worksheets that describe each step ofthe computational process. The primary
objective of these worksheets is to provide a clear and concise description of the computations for
software development purposes. The process is highly iterative, and productive application ofthe
worksheets is not possible. The computational process has therefore been implemented in software.
The following data are required as input to the model described in this appendix. A detailed
explanation of each of these data items is presented in Appendix E.
Appendix C: Page 5
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Approach-Specific Data
The following items are specific to each of the approaches to the intersection:
· Left Turn ~T) Treatment Codes
· Position Codes
· Sneakers
· Free Queue
· Approach Speed, SP
· Termination of Rings ~ and 2
Phasing and Detector Design Parameters
The following data items are specific to each phase controlled by a traffic-actuated controller:
· Phase Type
Phase Reversal
· Detector Length, DL
· Detector Setback, DS
Controller Settings
The controller itself has several operating parameters that must be specified for each
phase. Collectively, these will be referred to as the "controller settings," because they
must be physically set in the controller with switches, keypads or some other electrical
means. The following settings will exert a significant influence on the operation of the
intersection and must therefore be recognized by the analysis methodology:
Maximum Initial Interval, MxI
· Added Initial Per Actuation, AI
· Minimum Allowable Gap, MnA
· Gap Reduction Rate, OR
Appendix C: Page 6
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· Pedestrian Walk plus Don't Walk, WDW
· Maximum Green, MxG
Intergreen Time, I
· Recall Mode
Minimum Phase Time for Vehicles, MnV
A Basic Green Time Estimation Mode!
The determination of required green time is a relatively straightforward process when the cycle length
is given. However, traffic-actuated controllers do not recognize specified cycle lengths. Instead, they
determine, by a mechanical analogy, the required green time given the length of the previous red
interval and the arrival rate. They do this by holding the right-of-way until the accumulated queue
has been serviced.
The basic principle underlying all signal timing analysis is the queue accumulation polygon (QAP),
which plots the number of vehicles queued at the stop line over the cycle. The QAP for a simple
protected movement is illustrated in Figure C-2
represented in this very simple case as a triangle.
The queue accumulation and discharge is
The accumulation takes place on the left side of the
triangle (i.e., effective red) and the discharge takes place on the right side of the triangle (i.e. effective
green). More complex polygons are generated when permitted movements occur and when a
movement proceeds on more than one phase. Chapter 9 of the HEM includes an extensive discussion
on this subject.
Two methods ofdetermining the required green time, given thelength ofthe previous red, areillus-
trated in Figure C-2. The first employs a "Target v/c" approach. This is the basis for the planning
method described in HCM Chapter 9, and for some pretimed control timing plan designs. Under this
approach, the green time required is determined by the slope of the line representing the target v/c
ratio. The target v/c ratio will be achieved if the phase terminates when the queue has dissipated.
The second method recognizes the way a traff~c-actuated controller really works. It does not deal
explicitly with v/c ratios; in fact it has no way of determining the v/c ratio. Instead, it terminates each
phase when a gap of a particular length is encountered at the detector. Good practice dictates that
the gap threshold must be longer than the gap that would be encountered while the queue is being
serviced. Assuming that gaps large enough to terminate the phase can only occur after the queue
service time (based on v/c = 1.0), the average green time may be estimated as the sum of the queue
service time and the phase extension time as shown on Figure C-2. Each of these components will
be discussed separately.
Appendix C: Page 7
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8
3
CD
-
C~
._ 4
o
o
At
2
Green time based on
phase extension time
i_
it,
Green time based on
target v/c ratio
Time (seconds)
Green extension time:
Figure C-2. Queue accumulation polygon illustrating two methods of green
time computation
Queue Service Time
The queue service time, as, can be estimated as
where
qr r
g = f
qr, qg = red arrival rate (veh/sec) and green arrival rate, respectively (veh/sec);
r = effective red time (see);
saturation flow rate (vehlsec), and
fq = I.oS O.] (G/G~)2
Appendix C: Page 8
(2)
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The queue calibration factor, fq' was described by Ak~elik [1] as a factor required to account for
randomness in arrivals in determining the average queue service time.
Green Extension Time
To estimate the extension time analytically for a particular phase, it is necessary to determine the
expected waiting time for a gap of a specific length, given the average inter-vehicular headway, and
some assumptions about the headway distribution. An analytical model for this purpose was des-
cnbed by Ak,celik [1,2]. This model made use of Lin's earlier work [3~4]. The average green exten-
sion time is estimated by the following formula which is based on the bunched exponential arrival
headway distribution:
e A(eO+tO-~)
(pq ,~
where
(3)
e0 = the unit extension time setting, MnA on Worksheet 1.
to = the time during which the detector is occupied by a passing vehicle
to = (Ld, + Ev) / V
(4)
where
L`, = vehicle length, assumed for purposes of this discussion to be 25 it.
Ld = detector length, DL on Worksheet 1, and
v = vehicle approach speed, SP on Worksheet 1
= minimum amval (intra-bunch) headway (seconds),
(p = proportion of free (unbunched) vehicles, and
A=
a parameter calculated as:
i_ ~q
-
1 -Aq
(5)
where q is the total arrival flow (veh/s) for all lane groups that actuate the phase
under consideration.
Appendix C: Page 9
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The bunched exponential distribution of amval headways was originally proposed by Cowan [51. A
detailed discussion of this mode! and the results of its calibration using real-life data for single-lane
traffic streams and simulation data for multi-lane streams are given in Ak~celik and Chung [63. The
following relationship was originally proposed by Briton t7] for estimating the proportion of free
(unbunched) vehicles in the traffic stream (`p):
~ = e_b~q
(6)
where b is a bunching factor. The recommended parameter values based on the calibration of the
bunched exponential mode} using real-life and simulation data are:
Single-lane case:
Multi-lane case (number of lanes = 2~:
Multi-lane case (number of lanes > 2~:
= 0.5 s and b = 0.5
= 0.5 s and b = 0.8
Computational Structure for Green Time Estimation
= 1.5 s and b = 0.6 (7)
(8)
(9)
Although this green time estimation mode} is not difficult to implement, it does not lead directly to
the determination of an average cycle length or green times, since the green time required for each
phase is dependent on the green time required by the other phases. Thus, a circular dependency is
established which is solved by an iterative process. With each iteration, the green time required by
each phase, given all the green times required by the other phases, may be determined.
The logical starting point for the iterative process is the minimum times specified for each phase. If
these times turn out to be adequate for all phases, the cycle length will simply be the sum of the
minimum phase times for the critical phases. If a particular phase demands more than its minimum
time, then more time must be given to that phase. Thus, a longer red time must be imposed on all of
the other phases. This, in turn, will increase the green time required for the subject phase.
A Simple Two-Phase Example
This circular dependency will converge quite reliably through a series of repeated iterations. The
convergence may be demonstrated easily using a trivial example. More complex examples will be
Appendix C: Page 10
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introduced later to examine the effects of controller settings and traffic volumes in a practical
situation.
Consider an intersection of two streets with a single lane in each direction. Each approach has
identical characteristics, and cames 675 vehicles per hour with no leD or right turns. The average
headway is 2.0 seconds per vehicle and the lost time per phase is 3.0 seconds. The actuated
controller settings are:
Initial interval:
Unit extension:
Maximum green:
Intergreen:
10 seconds
3 seconds
46 seconds
4 seconds
The maximum phase time for each phase will be (46+4) = 50 seconds. The rn~n~mum phase time will
be (10 + 3+ 4) = 17 seconds, which will be the starting point for the timing computations. So, the
first iteration will use a 34 second cycle with 17 seconds of green time on each approach. Allowing
for lost time, the elective red time win be 20 seconds, and the effective green time will be 14 seconds
for each phase.
The total lost time is the sum oftwo components, including the starting lost time and the ending lost
time. In the HCM Chapter 9 procedure for estimation of capacity and delay, all of the lost time is
assumed to be concentrated at the beginning of the green. This is a valid approximation for delay
estimation because the lost time is only used in the computation of effective green time, and its
position in the phase is irrelevant.
However, for purposes oftrafflc-actuated timing estimation, the apportionment of lost time between
the begging and the end of the phase will have a definite influence on the results. The lost time at
the beginning ofthe phase willinfluence the length ofthe phase. The lost time at the end ofthe phase
will influence the delay, but it will have no effect on the phase duration. For purposes of this
discussion' assuming a specified lost time of n seconds, ~ second is assigned to the end of the phase;
n-l seconds is assigned to the beginning.
The computational process may be described as follows:
I. Compute the arrival throughout the cycle, q:
q = 675 / 3600 = 0.IS75 vehicles per second.
2. Compute the net departure rate (departure headway - arrival rate):
(s-q) = 0.5 - 0.~875 = 0.3125 vehicles per second.
Appendix C: Page 11
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1
a
o
c,
-
· -
E"
-
w
Another volume-density feature is gap reduction. The purposes of gap reduction are to reduce the
probability of "maxout" and to prevent phase termination with vehicles in the "dilemma zone". The
gap reduction feature shown In Figure C-7 is accomplished by the foDow~ng Functional settings: time
before reduction, passage time, magnum gap and time to reduce. These terms are defined below.
The time before reduction period beg as when the phase is green and there is a serviceable conflicting
cad (e.g. at time "t" In Figure C-7). The passage time is the time required for a vehicle moving at the
average approach speed to travel Dom the detector to the stop line. The average approach speed
usually is assumed to be the 85th percentile speed. Upon completion of the time before reduction
period, the linear reduction ofthe allowable gap begins from the passage time to the minimum gap.
The specified time for gap reduction is called time to reduce. Thus, the gap reduction rate is equal
to the difference between the passage time and minimum gap settings divided by the setting of time
to reduce. The purpose of gap reduction Dom the passage time to the minimum gap is to reduce the
probability of maxout. When a phase is terminated on the allowable gap, the unused portion of the
passage time is always displayed to avoid dilemma zone problems.
Time Before Reduction
1~ It;
Baa,
\,~um Gap
Retime To Reduce
t Green Time (seconds)
Figure C-7. Gap reduction feature for volume-density
operation
Appendix C: Page 22
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Modeling of Volume-Densitv Control
The previously developed analytical model is used to predict the phase time of any movement in a
fully actuated operation. However, it can also be applied to estimate the phase duration for through
vehicles and protected left turns under volume-density control if proper refinements of the analytical
model are done. Before the analytical model is refined for volume-density operation, the time before
reduction will be assumed to be zero for simplicity and the gap reduction win begin at the green
phase. The proposed refinements to the model will be introduced next and a simple example applica-
tion will follow.
The main differences between volume-density and basic actuated operations are the minimum green
settings (variable initial)' the detector configuration' the gap reduction feature and the passage time
setting for the last vehicle actuation. The refinement of the analytical model for volume-density
operation focused on these areas.
The initial interval (II) (adjusted minimum green) for volume-density operation is equal to the
specified minimum green (Smn) plus variable initial interval (added initial) subjected to the constraint
of specified maximum initial settings where the specified minimum green here is equal to the minimum
initial interval Am) plus starting gap (SG). It is common to choose a value for the starting gap
which is equal to the passage time from the detector to the stop line. The variable initial interval is
the product of the number of vehicles arrival on the red phase and the specified time for each vehicle
actuation. It can be computed as follows:
Va~ableInitialInterval(VII) = Qr * tact (12)
where
Qr = number of arrivals during previous red and clearance intervals
taC' = specified time per actuation
Then, the initial interval (II) is equal to the sum of the specified minimum green (Smn) and variable
initial interval (Vim. However, this value cannot exceed the specified maximum initial interval (Smx).
Therefore,
Initial Interval (II) = Min. (Smn + VII, Smx) (13)
where
Smn = specified minimum green time, and
Smx = specified maximum initial interval
The analytical mode! applies the QAP to compute the queue service time and the bunched arrival
model to estimate the vehicle extension time after queue clearance. Based on the computation of
total queue service time (QST) and total vehicle extension time (ge), the final phase time can be
obtained. To estimate the phase time for the volume-density operation, the computation of the QST
needs to be modified.
Appendix C: Page 23
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Representative terms from entire chapter:
cycle length
The detector placement, for the fully-actuated operation analytical model, is assumed to be at the stop
line. However, it is common with volume-density control that the detector setback is greater than
zero. In the queue discharge process when the signal turns green, the length of the moving queue will
decrease gradually if the volume to capacity ratio (v/c) is less than I.0.
The last moving vehicle in queue will pass through the upstream detector first and then the stop line.
The queue service time is defined as the time required to serge the queue beyond the detector. There-
fore, the queue service time for volume-density control, in general, is shorter than that for conven-
tional actuated operation. The proposed model uses the total queue service time of the fully actuated
operation minus a passage time to estimate the queue service time for volume-density operation. This
will be called computed queue service time (CQST).
The initial interval (adjusted minimum green) is the assured green time that will be displayed. If, at
the end of the initial interval, the length of the queue from the stop line does not reach the upstream
detector, the final queue service time should be equal to the minimum value of the initial interval and
computed queue service time. For example, if the initial interval is less than the computed queue
service time, the final queue service time is the initial interval because, for the upstream detector, the
queue has been cleared at the end of initial interval. If the queue extends past the upstream detector,
the final queue service time is just equal to the computed queue service time.
An easy way to determine whether or not the queue at the end of the initial interval, QII, reaches the
upstream detector is to compare the QII and the maximum storage of vehicles (MSV) between the
upstream detector and stop line. A vehicle is assumed to occupy 25 ft. so MSV is equal to the
distance between the upstream detector and stop line divided by 25. The value of QII may be esti-
mated Dom the average vehicle arrivals during the red and initial interval using the following formula:
QII = q' * (R + lI)
The computation for QST may be summarized as follows:
(14)
QST = Min. (CQST, II) if QII
current HCM procedure assumes that the first waiting leR turn will block all of the following vehicles
in the shared lane. This produces pessimistic results in some cases. Both the through vehicle
equivalence of a left turn ~ ~ ~ and the lane group saturation flow rate are affected.
At this point, only the SIDRA model considers the free queue explicitly. Because of its importance
to traffic-actuated control, it is essential that the proposed analytical model recognize this phenome-
non. A set of curves was developed to illustrate the effect of the free queue parameter on the
estimated phase time as a function of the approach volume. Because NETSIM does not recognize
the free queue explicitly, it was not possible to test this model by simulation.
Computation of New Throu~h-Vehicle Equivalents
Left turning vehicles will select gaps through the opposing flow after the opposing queue clears.
During this unsaturated period, the HCM Chapter 9 procedure assigns through-vehicle equivalents,
Era, for each left turning vehicle. The maneuver time required for a left turning vehicle with "n"
through-vehicle equivalents is equivalent to that for "n" through vehicles. The permitted saturation
flow rate can be computed based on the assigned value of E~, and the proportion of left turns, Pa,
in the shared lane. However, if free queues exist, this permitted saturation flow rate will become
larger because the blocking effect of led turning vehicles is reduced. The through-vehicle equivalent
for a led turning vehicle needs to be modified to account for the effect of the free queue parameter.
The new En, for a left turnung vehicle on a shared lane can be determined by the combination of
vehicle arrival type (through or leD turn) after the left turning vehicle and the probability of the
combination during the maneuver time required for the leg turning vehicle. If En l is equal to n, n-1
vehicles will follow the leD turning vehicle. These following vehicles can be led turns or through
vehicles including right turns. There are a total of 2
The range of En ~ values shown in the HCM Chapter 9 is from 1.05 to 16 for permuted left turns in
a shared lane. These EL} values do not consider any Dee queue parameter. The proportion of left
turns in a shared lane, PL., is an important factor to the ELI value when a free queue exists. The
reasonable range for PL is Dom O to I.0. Therefore, PL and old EL] values will be taken into account
In the computahonofnewEL~ values for Dee queue settings. The reasonable maximum value ofthe
n~rnm~ter con.~idered in this stud is 2.0 vehicles. Based on the proposed method, the
new Eat values for one and two vehicle Dee queues are shown in Tables C-] and C-2, respectively.
If the old ELI or PL. is not a specified value in the tables, the new EL! can be estimated by interpolation.
If the Dee queue value is not an integer, Interpolation may be used.
,. __ ~ _ ___ ~ _ ~ ,
Table C-1. Through-car equivalents, ELI, for permitted left turns
in a shared lane with one free queue
0
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
I 05
1.05
1.05
1.05
1.05
1.05
Appendix C: Page 26
0.1
1.05
1.10
1.29
1.56
1.90
2.31
2.78
3.30
3.87
4.49
5 14
5.82
6.54
7.29
8.06
8.85
0.2
1.05
1.20
1.56
2.05
2.64
3.31
l
4.05
4.84
.
5.67
6.54
7.43
8.34
9.27
10.22
11.18
12.14
0.3
1.05
1.30
1.81
l
2.47
l
3.23
4.06
.
4.94
5.86
6.80
7.76
8.73
9.71
10.70
11.69
.
12.68
13.68
The Proportion of Left Turns ~ the Shared Lane, Pa
~.
0.4
1.05
1.40
2.04
2.82
3.69
4.62
5.57
6.54
.
7.53
.
8.52
9.51
10.51
11.50
12.50
13.50
14.50
0.5
.05
.50
-
2.25
3.13
4.06
5.03
6.02
7.01
8.80
9.00
10.00
11.00
12.00
13.00
14.00
.
_ 15.00
0.6
1.05
1.60
2.44
3.38
4.35
5.34
6.34
7.33
8.33
9.33
l
10.33
11.33
12.33
13.33
14.33 1
1
15.33 1
I O.7
I 1.05
1
I 1.70
1
1 2.61
1
1 3.58
1
1 4.S7
I S.S7
1
1 6.57
1
1 ~ ~ -
1
1 8.57
W:;
10.57 1
1
11.57 1
12.57 1
1
13.57 1
1
14.57
15.57
I 0.8
I 1.05
I 1.80
1
1 2.76
1
1 3.7S
1 4.75 1
. ,
6.75 1
7.7S 1
8.75
9.75
10.75
11.75
12.75
13.75
14.75
15.75
o.9 1
1
1.05
1.90
2.89
3.89
4.89
.
5.89
.
6.89
7.89
.
8.89
I 9.89
1
I 10.89
1
I 11.89
1
I 12.89
I 13.89
1
I 14.89
I 15.89
.
1.05
1 2.00
1 3.OO
1
1 4.OO
1 1
1 S.00
6.00
1 1
1 7.00 1
. .
1 8.00 1
1
1 9.oo 1
~:~
. .
I 12.00 1
14.00
., I
15.00 1
Table C-2. Through-car equivalents, ELI, for permitted left turns
in a shared lane with two free queue
~ .
O
1 05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
1.05
0.1 0.2
05 1.05
1.1 ~ 1.2
.2 14
13 16
1.4 1.8
1.5 2.0
1.6 2.2
.7 2.4
.8 2.6
1.9 2.8
20 3.0
21 3.2
2.2 3.4
2.3 3.6
2.4 3.8
2.5 4.0
Th ~
0.30.40.50.6 title
1.05 1.05 1.05 1.051.051.05 1.05 1.05
13 14 15 1617i8 i 9 2.0
1.6 1.8 2.0 2.22.42.6 2.8 3.0
1.9 2.2 ~ 2.5 2 83 13 4 3.7 4 0
2.2 1 2.6 1 3.o 1 3.41 3-81 42 1 46 1 5-0
2.5 3.0 3.5 4.04.55.0 5.5 6.0
2.8 3.4 40 465258 64 7.0
3.1 3.8 4.5 5.25.96.6 7.3 8.0
3.4 4.2 5.0 5.86.67.4 8.2 9.0
3.7 4.6 5 5 6 47 38.2 9.1 10.0
4.0 5.0 6.0 7.08.09.0 10.0 1 1.0
4.3 5.4 6.5 7.68.79.8 10.9 12.0
4.6 5.8 7.0 8.29.410.6 11.8 13.0
4.9 6.2 7.5 8.810.11 1.4 12.7 14.0
5.2 6.6 8.0 9.410.812.2 13.6 15.0
55 70 85 10.011.5130 14.5 160
When the new through-car equivalent of a lefc-turn vehicle is computed, the permitted saturation flow
of asharedlane curing the period of unsaturated opposing flow can tee computed as follows, if a free
queue exists:
s
1 PL (ELI (new) 1)
where
s = protected saturated flow rate (veh/sec)
Pi = the proportion of left turns in the shared lane.
Appendix C: Page 27
Estimation of Free Green (gf) for Free Queue Settines
The free green, gf, originally described in the HCM needs to be modified when the free queue param-
eter is considered. When the green time is initiated' the opposing queue begins to move. While the
opposing queue clears, left turns from the subject shared lane are effectively blocked. The portion
of effective green blocked by the clearance of an opposing queue of vehicles was referred to as gq in
the HCM. Until the first led turning vehicle arrives, however, the through vehicles on the shared lane
are unaffected by left sums. The portion of effective green before the arrival of the first left turning
vehicle was referred as free green, gf, in the HCM. Basically, Bee green represents the time during
which the through vehicles in the shared lane are not affected by left turns when the green begins.
Generally, signal timing models assume that the first permitted left turn at the stop line will block a
shared lane. This is not always the case, as the through vehicles in the shared lane are often able to
"squeeze" around one or more left turns or led turning vehicles waiting to turn left. A reasonable free
queue setting is between zero and two. The proper value for the free queue setting can be observed
from the field. The leD turning vehicle in the free queue will not block the through vehicle in the
shared lane, so the free green will increase. The proposed method to estimate the new free green for
the free queue setting is describe as follows.
Based on the number of opposing lanes nOpp and the relationship between gq and gf, the permitted sat-
uration flow rate sp can be determined directly from the current HCM Chapter 9 procedure:
s = _ _ if n0pp= 1 and gq>gf (17)
~ -
P 1 + PL (EL2 1 )
s
so = ~others
-
1 + P[, (EL,} - 1)
where
(18)
protected saturation flow rate,
proportion of led turns in the shared lane,
through-car equivalents for each lefc-turn vehicle during unsaturated green
through-car equivalents for each leD-turn vehicle during the period of (gq- go,
when g q greater than gf and the number of opposing lanes nOpp is equal to one.
Assume the average waiting time for a LT vehicle in the free queue to leave is x seconds. This
waiting time can be derived as follows:
~ (LOPE) + pL x
sp s
Appendix C: Page 28
(19)
Therefore.
s - s + S P
x = P P ~(20)
p L
If the number of opposing lanes is greater than one and gq is greater than gf, the required time x be-
comes
s -s +S P
s s P gq gf (21)
p L
Assuming the free queue is n vehicles, (O
During the waiting time x seconds, the probability of block effect on the through vehicles due to left-
turn arrivals is assumed P'. Then P' can be computed as:
p, = pn if no!
P' = P if O
c)
In
~ To
~ is
~o~
Figure C-~. Effect of the free queue on phase times for the example
problem
For the same free queue value, the phase times will increase due to the increasing blocking effect by
left turns In the shared lane. On the other hand, if the value of the Dee queue increases, the blocking
effect is reduced because left tarrying vehicles are able to wait in the free queue before turning left.
The larger the Dee queue, the smaller the blocking effect.
These two phenomena can be clearly seen in Figure C-8 when the volume is between 400 vph and
720 vph and the value of the free queues is between 0 and 1.2. It is clear that the phase times will
depend on the combined effects of EL} and the value of the free queue. In Figure C-8, it is also easy
to observe that the effect of the free queue is reduced when the approach volume becomes large.
Art this example, when the volume exceeds 720 vph and the free queue exceeds 1.2, the phase times
are always equal to the maximum chase time (84 seconds in this case). This indicates that the effect
. ~
ofthe Dee queue is too small to prevent the phase time from reaching its maximum.
The computational structure of the proposed analytical model has been modified to incorporate the
free queue value. While it would be difficult to verify the model satisfactorily, either by simulation,
or in the field, it is suggested that the analysis is robust and the value of the model will be enhanced
if the free queue is included.
Appendix C: Page 31
APPENDIX C REFERENCES
I. Ak~elik, R. Analysis of Vehicle-Actuated Signal Operations. Australian Road Research
Board. Working Paper WD TE 93/007, ~ 993.
Ak~celik R Estimation of Green Times and Cycle Time for Vehicle-Actuated Signals. Paper
No. 94-0446, 73rd Annual Meeting of Transportation Research Board, Washington, January
1994.
Fin, F.B. Estimation of Average Phase Durations for FuD-Actuated Signals. Transportation
Research Record S8l, TRB, National Research Council, Washington, DC, 1982, pp. 65-72.
4.
kin, F.B. Predictive Models of Traff~c-Actuated Cycle Splits. Transportation Research 16B
(5), 1982, pp.65-72.
5. Cowan, RJ. Usefill Headway Models. Transportation Research 9 (6), 1975, pp. 371-375.
6.
Ak~celik, R. and Chung, E. Calibration of the Bunched Exponential Distribution of Arrival
Headways. Road and Transport Research 3 Ail, 1994, pp.42-59.
7. Briton, W. Recent Developments in Calculation Methods for Unsignalized Intersections in
West Germany. Intersections Without Traffic Signals, Ed. W. Briton, 1988.
Appendix C: Page 32
