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APPENDIX C
ESTIMATION OF DESIGN LENGTHS OF ATL COMPONENTS
Introduction
A key design element of ATLs is the appropriate length of the ATL's
upstream and downstream components. Although it may be hypothesized that a
longer ATL promotes higher ATL use, extensive field observations of ATLs tend
to contradict this theory. In fact, some of the ATL sites used to develop the
operational models in the "ATL Volume Estimation" section in Chapter 3 had
high ATL utilization and short downstream lengths. Instead, the primary
motivator for using the ATL appears to be a defensive one: avoiding a cycle
failure when traffic in the adjoining CTL is moderately to highly congested.
Based on the above premise, the required ATL upstream length is predicated
on the provision of adequate storage for and access to the ATL from the
neighboring CTL. The downstream length, on the other hand, is predicated on
servicing the queued vehicles in the ATL so that they can accelerate to the
approach free flow speed and smoothly merge before reaching the end of the
downstream taper. Gap availability and acceptance in the CTL for ATL vehicles
operating under relatively high-speed, uninterrupted conditions must also be
considered. Therefore, the recommended minimum downstream length is the
greater of the lengths determined from these two operating conditions.
Note that the lengths determined from this method represent minimum
design requirements for ATLs. Poor downstream sight distance, lack of proper
signage (or existence of overhead lane signs), presence of downstream
driveways, and significant right-turn-on-red (RTOR) flow from cross-street
traffic may all necessitate adjustments to the minimum length to accommodate
those effects. Finally, the minimum ATL lengths developed in this section are
predicated on the assumption that an ATL will in fact be built. This is a strong
assumption, but it is one that relies on the engineer's judgment on the practical
need for such a lane. Because one of the major outputs of these guidelines is the
predicted ATL through flow rate under various conditions, it is incumbent on
the practitioner to decide whether the estimated ATL volume indeed warrants
the additional lane, especially if the anticipated flow is only one to three vehicles
per cycle on average. However, if the decision is to proceed with an ATL
installation, then the procedure described in the next section can be followed.
ATL Length Estimation Procedure
This procedure is built around the ATL flow rate estimation models
described in the "ATL Volume Estimation" section in Chapter 3. Since there are
separate models for one-CTL and 2-CTL cases, the same reasoning applies to the
ATL length estimation process. The procedure is implemented in two Microsoft
Excel spreadsheets that estimate minimum ATL length and provide other
important performance measures as outputs. The starting point of the analysis
has no ATL presence. For the one-CTL case, the procedure considers an
approach with a single shared through-right continuous lane, while the 2-CTL
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case assumes an exclusive through-movement lane and a shared through-right
continuous lane. In all cases, left turns are assumed to operate from an exclusive
lane or pocket and therefore are not part of the analysis.
An outline of the procedure as it relates to the ATL upstream length
determination is explained in the following steps:
1. Identify whether the one-CTL or 2-CTL case applies.
2. Supply the data required for ATL flow rate estimation including:
a. Total approach through and right-turn flow rates,
b. Cycle length and effective green time for the subject approach,
and
c. Saturation flow rate for both through and right-turn movements.
3. Estimate the ATL flow rate based on the one-CTL or 2-CTL model in the
"ATL Volume Estimation" section in Chapter 3.
4. Calculate the ATL through flow rate assuming equal lane volume-to-
adjusted saturation flow rate (v/s) based on the HCM 2010 shared or
exclusive lane-group volume distribution.
5. Take the predicted ATL flow rate as the lower estimate from steps 3 and
4.
6. Calculate the ATL and CTL volumes, capacity, control delay, and back of
queue using the HCM 2010 signalized intersection procedures. For
shared ATLs, include the right-turn flow rate in the lane flow
computations.
7. Estimate the 95th percentile queues in both the ATL and CTL (for one
CTL lane in the case of two CTLs) using HCM procedures.
8. Select a storage length based on the greater of the 95th percentile queues
in the ATL and CTL. Queue storage or access distance is calculated based
on an estimate of average vehicle spacing in a stopped queue.
The determination of the requisite downstream length requires a further set
of input parameters, some of which may be defaulted as shown in parentheses,
namely:
· Approach free flow speed or speed limit,
· Average acceleration rate from a stop on the ATL (10 feet/second2),
· Intersection width measured from the stop line to the far curb (40 feet),
· Minimum acceptable headway in CTL traffic stream (6 seconds), and
· Driver reaction time (1 second).
The downstream length estimation based on storage of vehicles at the
desired spacing in the downstream length (DSL1) proceeds as follows.
Estimate the average uniform, random, and oversaturation back of queue
(BOQ) for ATL through traffic only (Q1 + Q2 in HCM terminology). This
approach incorporates two opposing and simplifying assumptions. The first is
that the required length will be based on the average BOQ as opposed to the 95th
percentile value as was done in the upstream case. This is offset by another
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assumption where all through-movement vehicles in the ATL are assumed to be
contiguous in the queue and not separated intermittently by righ t-turning
vehicles in a shared lane, which would result in a larger separation between
through-movement vehicles. This procedure assumes that the effects of the two
assumptions will balance.
The downstream storage criterio n is based on providing sufficient spacing
between ATL vehicles at the free flow speed or speed limit. Since v ehicles
accelerate from the stop line position, the downstream distance me asured from
the far curb can be shown to be:
Where:
V = free flow speed or speed limit (in feet/second),
A = acceleration rate from stop line (in feet/second2),
L = spacing between vehicles at stop (in feet),
T = driver reaction time (in seconds), an d
INTW = intersection width measure d from the stop bar to the far
curb (in feet).
The second criterion for est imating required downstream length is based on
gap availability and acceptance under uninterrupted flow conditions, especially
on high-speed approaches. The concept is that, after tr aveling a reaction distance
past the intersection, an ATL driver must find an acceptable merge gap in the
neighboring CTL within the confines of the downstream ATL length. Using
assumptions on the headway distribution in the CTL and a minimum acceptable
merge headway value, the distance me asured from the far curb can be shown to
be:
Where:
NUM = the number of rejected gaps in the CTL. This could be
either the mean value of rejected gaps or a pre-specified
percentile number of rejected gaps, as explained below.
Gr = expected or average size of a rejected headway in the CTL
(in seconds).
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This model used to calculate DSL2 is based upon a gap acceptance procedure
with the following assumptions:
· Drivers begin searching for gaps as soon as they pass the stop bar,
· Drivers have reached the operating speed of the arterial,
· Drivers are homogeneous with regard to a critical headway or gap (tc),
and
· Traffic in the adjacent CTL follows an exponential headway distribution.
The following steps describe the model development:
Step 1. Determine the number of rejected gaps encountered until an acceptable
gap is found. Let p be the probability of rejecting a gap in the CTL, tc be the size
of the critical headway, and h be the time headway between vehicles in the CTL.
Then
where is the flow rate in the CTL (in vehicles per hour).
Then the probability of rejecting exactly i gaps is pi (1 p) and the expected
number of rejected gaps is:
An alternative approach to using Nr is to design the downstream length to
accommodate the 95th number of rejected gaps, as opposed to the mean value. In
this case, we would like to determine the number of rejected gaps that would
only be exceeded at most (1 ) percent of the time. In other words, find I such
that the number of rejected gaps X is such that
or conversely
which can be then expressed as
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Solving for I gives the condition for the percentile rejected gap:
For example, if the probability of a rejected gap p = 0.50 and a 95th percentile
confidence level on the number of rejected gaps is desired, then
This compares with a mean number of rejected gaps of
In the remaining steps, the user may choose to apply either the percentile or
mean value of rejected gaps.
Step 2. Determine the expected size of a rejected gap, E(t|t < tc):
where
using integration by parts, and after simplifying gives:
Since
Step 3. Calculate the expected waiting time for an acceptable gap, which is equal
to the product of the number of rejected gaps and the expected size of a rejected
gap:
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Optionally, if one selected the percentile gap approach, then the waiting time for
the (alpha) percentile rejected gap would be
Step 4. Calculate the distance traveled before an acceptable gap is found:
or in the case of the percentile gap:
where V is the operating speed in feet per second.
Incorporating the reaction time T, the total distance traveled (in feet) is given by
or in the case of the percentile gap,
The computational engine described in Appendix B provides both a mean and
percentile option for computing the design value of DSL2.
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