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OCR for page 83
83
Chapter Eight
SPECIAL CAPACITY AND DELAY ISSUES
There are some conditions that have not been taken into
accost In the capacity and delay models discussed so far.
These conditions include:
the existence of ra~sed/staped median or a two-
way left-turn lane (TWLTL3 lane on the major
street where a two-stage gap acceptance process
was observed path some Livers
a flared minor street approach where right turn
sneakers are present
the existence of upstream signals where heavy
major street platoons were observed
Me existence of pedestrians which may affect Me
capacity and delay
Me enhance of a sham through and left turn lane
on the major sheet where left turn vehicles may
cause blockage of the through traffic
These conditions need special consideration when applying
the capacity end delay models. The purpose of this section
is to summarize some of the findings and
recommendations regarding these special situations.
TWO-STAGE GAP ACCEPTANCE
Among the sites studied for this project, there are a
significant number of sites where either a two-way left-turn
lane (TWLTL) lane or a raised median exist on the major
sew The existence of these facilities usually causes some
degree of special gap acceptance phenomena, such as a
t`vo-stage gap acceptance process. For example, Me
existence of a raised or striped median causes a significant
proportion of the minor street drivers to cross the first
major sheet approach, and then pause in the middle of the
road, to wait for another gap in the other approach. When
a TWLTE exists on tiLe major street, the minor street left
turn vehicle usually merges into the TWLTE first, Men
seeks a usable gap on the other approach while slowly
moving for some distance along Me TWLTL.
Empirical Data
The influence of a TWOS and liaised median on the minor
street capacity at TWSC intersections has been
investigated using the recommended cap acid mode} and
procedure. The methodology used is to compare Me
capacities calculated from the standard mode} and
procedure with the field capacities measured using
Equation 105. Figure 49 shows the mode} testing results
for sites with either a TWLTE or a raised median The
generalized cntical gap and follow-up times were used.
Two regression lines were also plowed in the figure. The
scattered data points suggest dial the capacity vanes
significantly under these circumstances. It is evident that
the theoreticalmode} underestimates the capacity for most
cases, which means the existence of a TWLTL or raised
median usually causes an Increase In the capacity for the
minor street. The magnitude of capacity underestimation
is slightly larger for sites with a TWLTE than for sites
with a raised median. In other words, the increase of
capacity due to a TWLTE is larger than due to a raised
median. Based on Me regression results, it was found the
ratios between the mode! capacity and field capacity for
sites pith a TWLTL is about 2.S, and Me ratio is about 2.l
for sites with a raised median. This means that with the
existence of a TWLTE or raised/striped median, the
capacities calculated using the standard procedure need to
be increased by multiplying 2.S or 2.1. However, this
result is based only on limited empirical data.
coo
- 800
600-
£
~ 400
200
O
1 1 1,-1
1 1,-1 1
~7
. ~ 1
. | Raised Med~- | o
. ~
. -a _1-
, ~ I
\ I
TWLTLN I
400 600 800
held Capacity, vehlhr
0 TWLTLN Raised Median
1 000
FIgure 49. Model Testing Results for Sites with TWLTL or Raised
Median (~nor LT CapaciW)
Analytical Theory
The basic capacitor model as well as Be capacity estimation
pr~e cannot tee applied directly to Opinions in which
two-stage gap acceptance exists. Briton, Wu (1995,
Working Paper 24) and WE Bnion (TRB 1996) studied
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84
the two-stage gap acceptance phenomena from a
theoretical point of view as part of this project. A set of
capacity models dealing with the two-stage gap acceptance
process was developed A summary of the mode! is
presented In this section.
At many unsignalized intersections there is a space In the
center of the major street available where several minor
street vehicles can be stored between the traffic flows of
the two directions of the major street, especially in the case
of multi-lane major traffic. This storage space within the
~ntersechon enables the minor street driver to pass each of
the major streams one at a time. This behavior can
contribute to increased capacity. A mode} that can
account for this type of behavior was developed by
Harders (1968~. His concept was used as a basis for the
following derivations. ''
_ _1-~ ~
However, some major
ampllilcatlons as well as a correction and an adjustment
for field conditions have been made to better correspond to
field conditions.
part 11
J
N7?
. ~ ~ in line
_ _ put
Ic spaces fc r passenger car
~ r
~,9
1~
~-
Figure 50. Illusl~abon of an ~tersechon wi~ Two~tage Gap
Acceptance Process
partl
These derivations are based on an intersection consisting
of two parts, as shown in Figure 50. ~ Figure 50, the
minor street through traffic (movement S) crosses the
major street In two phases. Between the partial
intersections ~ and ~ there is a storage space for k
vehicles. This area has to be passed by the left turner Tom
the major street (movement I) and the minor through
heroic (movement S). The minor left turner (movement 7)
also has to pass through this area. Movement 7 can be
treated like movement 8. Therefore, these derivations
concentrate on He minor through traffic (movement 8)
cross~ngbo~parts of He major street. The designation of
movements has been chosen in accordance with Chapter
10 of the 1994 HCM. It is assumed Hat the usual rules for
unsignalized intersections are applied by drivers at the
intersections. Thus movements 2 and 5 (major through
traffic) have priorly over all other movements. Movement
~ vehicles have to give way to priority movement 5,
whereas movement ~ has to give He right of way to each
of the movements shown In Figure 50. ~ these
derivations, movement 5 represents ad major traffic
streams at part II of the intersection. These, depending on
thelayoutof~e intersection, could include through traffic
(movement 5), led turners (movement 4) and right turners
(movement 6~.
Analytical Mode] for the Determination of the
Capacity
To determine the capacity of the whole intersection, a
constant queue on the minor approach (movement S) to
part ~ is assumed.
If wi is He probability for a queue of i vehicles queuing
in the storage space within the central reserve, then the
probabilities wi for all of the possible queue lengths i
must sunlupto ~ with O ~ i ~ k, as shown belong:
Kiwi=
f-O
where kis He number of spaces inure storage space within
the central reserve.
Considering central area of the intersection as a closed
storage system, which is limited by the input line and
output line, die capacity properties of He storage system
are restricted by the aspects of maximum input and
maximum output. Different states of the system are
distinguished below.
Starte 1: Part I of He intersection limits the input of
vehicles to He storage area. Under state 1, the number i
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85
of vehicles In the storage area is less than Me maximum
possible queue length k, i.e. i < k . Dur[ng this state a
minor street vehicle from movement ~ can enter the storage
space if the major streams (volume v, and vp provide
sufficient yaps. This case the capacity of Part ~ (possible
input from movement S) characterizes the capacity, i.e.:
car = eve + V2) (~o
where cove + v2) is the capacity of Part ~ in the case of no
obstruction by the subsequent Part 0, which is We capacity
of an isolated unsignalized cross intersection for through
minor traffic with major traffic volume v, ~ v2.
The probability for this state ~ is p, = ~ - we . Thus, the
contribution of state ~ to the capacity of Part ~ for
movement ~ is
c,~ =~1 -we +V2) (~D
Dunng state I, vehicles from movement ~ can also enter
Me storage space.
State 2: For this state it is assumed that the storage area
is occupied, i.e., k vehicles are queuing in Me storage
space. In this state no minor vehicle from movement ~ or
vehicles from movement ~ can get into the storage area. If,
however, a sufficient gap for Me passage of one minor
street vehicle can tee accommodated at both parts ~ and I~
of the intersection simultaneously, then an additional
vehicle can get into the storage area. The capacity for vat
(possible input from movement S) during this stage is
C2 = eve + V2 + us) (~)
where cove + v2 + vs) is the capacity of an isolated cross
intersection for through traffic with major traffic volume
vat + V2 + vs.
Thus, the contribution of state 2 to the capacity of part ~
is
CI2 = Wit X ~Vt + V2 + V; (~19)
where wk is the probability that k vehicles are In the
storage space.
State ~ and state 2 are mutually exclusive. The capacity of
part ~ is Me total maximum input to the storage area. Here
the volume vat of movement I, in addition to the partial
capacities mentioned above, has to be included. Therefore,
Me total maximum input to the storage area is
Input = c`, + cry + v' = (1 - wit x c(v,, + v2)
+ WE X eve + V2 + Vs) + v~ (120)
State 3: To consider the output of Me storage area, this
analysis concentrates on part ~ of the intersection. For i
> 0, each possibility for a departure homage storage area
provided by the major stream of volume vs can be used.
The capacity (maximum output of Me storage area) of part
In this case is
C3 = revs) (121)
where C(V5) iS the capacity of part II In the case of no
obstruction by the upstream Part I, which is the capacity of
an isolated unsignalized cross intersection for through
minor traffic wad major traffic volume vs.
The probability for this state is: pa = ~ - Wo.
Thus the contribution of state 3 to Me capacity of part II
IS
Cars = (1 - wo) X revs) (122)
where we is the probability Mat O vehicles are In the
storage space.
No vehicles Dom movement ~ (volume vie can directly
pass through the storage area in this state (i.e., move
without being impeded by movement 5~.
State 4: For i = 0 (i.e. an empty storage area) no vehicle
can depart Me storage area even if the major stream of
volume vs would allow a departure. If, however, a
sufficient gap is provided In the major streams of bow
parts of the intersection simultaneously, a minor street
vehicle from movement ~ can pass the whole intersection
without teeing queued somewhere in the storage area. This
is the same situation as exists with single stage gap
acceptance. The possible output of the storage area from
movement ~ vehicles during this state is
C4 = ~V1 + V2 + VS) (1~)
Thus, the contribution of state 4 to Me capacity of part ~
IS
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86
C~4 = Wo X LEVI + V2 + VS) (124)
Also vehicles from movement ~ can pass through the
storage area in this state. The number of vehicles Tom
movement ~ which pass through the storage area In this
state is
Il.4. ~0 ~
ti2s'
Here, C~,4,,,~ does not mean Me cap acid for v,~but the
demand on the capacity. The traffic demand of vat should
be less than the capacil~,~ of the part ~ cove). i.e. vat is
subject to the restriction vat < cove). Obverse, the
intersection Is overloaded and as a result of this stationary
state, no solution can be denved.
States 3 and 4 are mutually exclusive. Therefore, the total
maximum output of We storage area is
0U~ut = Can ~ Ca,4 tCa,4,v,
= (1 - Why X C(VS) ~ We X COY ~ V2 ~ Van ~ Wo X VI(12o
= (1 - We) x C(VS) ~ Wo X twit ~ V2 ~ Van ~ Vt]
During times when the entire intersection is operating at
capacity, due to reasons of continuity, Me maximum input
and output of the storage area must be equal.
Thus, since Me input equals the output (see Equations 120
and 126):
(} - Wk) X OVA + V2) + W' X ~V1 + V2 + VS) + V!
(} - Wo) X C(VS) + Wo X [c(v1 + V2 + Vs) + VI]
The total capacity CT for minor Trough traffic (movement
8) when considering the whole intersection is identical to
both sides of this equation minus vl.
The probabilities wO and wit can be denved as:
y- 1
Y - 1
y~1 _ yl
w =
~yam _ 1
where
(128)
(129)
~v1 + V2) - act + V2 + Vs) (130)
revs) - V1 - dVl + V2 + Vs)
Note Mat wO is Me probability (proportion of the analysis
time period) of state 4 under capacity conditions
(continuous queuing) on the minor approach. This
situation is identical to when there is no median storage
and single-stage gap acceptance occurs. Thus, the
remailing time proportion contnbudon to capacity, I-wO,
is due to the existence of a median.
ti27)
The remaining proof and assumption details are presented
in Working Paper 24. Based on these assumptions, the
computational steps which are necessary to estimate the
capacity of an unsignalized intersection where the minor
movements have to cross the major street in two stages are
presented In Table 46. Alternatively, graphical methods
are presented In the following sections.
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Representative terms from entire chapter:
major street
87
Table 46. Summary of Steps in the Capacity Calculation atIntersechons win Tw~Stage Gap Acceptance Process
.
vl= volume of priory sheet leRt~uning traffic et pert I l
v2= volume of major street through1raffic comings the left at part I L
v'= volume ofthe sum of all major sheet flows coming In the right at part IL |
Here the volumes of all prior movements et pert II have to be included. These are: major right (6, except if this movement is guided along a biangular
island Ted from We ~roughbaffic), m~yor~rough (S), major left (4); numbed of movements according to HCM 1994, Chapter 10.
c(v1 EVE)= capacity at pelt I
c(v,)= capacity at part II
c(v'+v2+v5)= capacity at a cross intrusion for minor through traffic with a major sheet traffic volume of v,+v2+v5
Note: all capacity terms apply for movement 8 . They are to be calculated by any useful capacity formula,(e.~, Siegloch~s model or Hardcr~s model)
where
cull + V2) - c(vl + V2 + V5) (131)
C(VS) - Al - C(V1 + V2 + Vs)
Ct=~{kX[
88
Realistic values for the to and tf values can be obtained
from Table 47. The given cndcal gaps to and follow-up
times tf are of realistic magnitude compared with the
measurement results obtained In this project (NCHRP 3-
46, 1995, wowing paper 16~. 1h Table 47, the cndcal gap
and the foDow-up time for the case without central reserve
tic = 0) are larger than for the two-stage pnont~r case.
Table 47. Typical Values for Critical Gap and Follow-Up Time for
Two Stage Priority Situations win Multi-Lane Major Suet
0~1
of
0&
7.0
3.8
Based on Equation IQ5, assuming that aD of Me If values
are nearly identical,
c(v~+v2+v') C(V~+V2)XC(V5) (136)
cO cO cO
where }/tf is We mid capacity when Were is no
conflicting traffic.
This relationship for cO makes it possible to standardize all
of We capacity terms by cO. If cO is defined In units of
veh/s, We other capacity terms must have this unit (the unit
veh/h could also be used for ad of the capacity terms). It
is also useful to standardize CT In Equations 132 and 133:
CT = - (13n
o
(which has to be obtained using Equations 132 and 133)
can then be expressed as a function of cove + v2~/cO and
[ckVs) -vim ~ /cO. The results ofthese derivations can then
be shown graphically as In Figure 5 I.
1~ (a
Figure Slat Total Capacity (0T = cllco)
Figure Slb. Total Capacitor (Delco)
Graphs of this type may be used as sufficient
approximations under circumstances that differ Tom the
conditions used in capacity models based on gap
acceptance theory. For example:
Capacities ctv1 +v2) and cove) maybe computed
from other theories than gap acceptance, or they
could be measured in the field.
gap acceptance theory, the critical gaps tc are
different for each part of the ~ntersechom
The only necessary condition for the application of these
graphs is that the follow-tip times If are of nearly identical
magnitude.
Graphs for Practical Application
Alternatively, the its for the peony given in this section
are illustrated In Figures 52(a) and 52(b) for k = 1 and 2
respectively. Here We capacities ckv, + v2) and cove) can
be introduced independent of We type of formula Tom
89
which they have been de~minei Another advantage of
these graphs is that they can tee applied for any value of v,
(major left sum).
1
0.e .
0.. .
0.7
~ 0.e
~- 0.6
~ 0~
D.S
T
t
t
t
..~..
...~..
!..
~~t~\ V.~-~~
1 'I.\ ~ At;;
'',"2 Vt\",~;t'~;'~';:
...,.,..~..~..~.~..~.;.i~
...~....~..~;...~.~
. ~ or. so s ~_ ~
' 1 1 1 1 1
0 0.1 02 0 ~0~ 0.5 0 ~0.? 0.8 0.9 1
~v,tv~l"
0.9
0.e
0.7
0.e
0.6
0
0~
02
0.1
Figure 52a. Capacities (CT) for Movement 8 in Relation to
Standardized Values of Capacities and vat. - 1)
1
0.9
0.e
O.7
0.e
0.e
0.4
0.e
02
O.
o
I..
at
t'
..~..
.... , . , , .. , ... , .. , . , ., ... , .. ,
0 0.1 02 0.3 0.4 0.6 0.. 0.7 0.. 0.0 1
~V10~"
0..
0.e
0.7
0.s
0.5
0.4
0.3
02
0.1
Figure 52b. Capacities (CT) for Movement 8 in Relation to
Standardized Values of Capacities and v,. - )
For example, for a two-stage priority intersection witch the
leaf icvol~es v~=IOO veh/br, v2-600 veh/br and vs=400
veh/hr, assume that possible storage spaces within Me
central reserve - 2, see Figure 52~. The capacities for
movement ~ crossing the Intersection separately can be
calculated Tom Siegioch's formula 03quation 135~. The
corresponding values of to and If can be obtained from
Table 47. The parameters for entering the application
graph (Figure 52) can be calculated as follows:
Pa I: Aim = 1 - ~0
t'
(138)
1 ~1~),'_ 3.8`
= a
3.8
3COO -- 2 ) = 0.119 (veh/sec)
Part HI: c(vs) = 1 -I
If
~ ( - )(6-3-8)
= 3g~r 3600 2 = 0.167 (veh/sec)
wit
(140)
c0 = tf = 31g = 0.263 (veh/sec) (140)
100
v =
1 3600
=
0.028 (v~h/sec) (141)
resulting in the parameters for use in Figure 52:
.
cave v2) = 0.119 = 045
cO 0.263
(142)
C(V53 - V1 _ 0.167 - 0 028 = 0.53 (143)
With these two parameters, We relative capacity for
movement 8 is shown below:
CT = - = 0.36
cO
(Few 52(b)). (144)
Therefore, the absolute capacitor for movement 8 is:
CT= CT X CO
= 0.36x 0.263
= 0.095 (veh/sec)
= 342 (veh/hr)
(14~
If gap acceptance theory is applied (Equation 135) to
Do
estimate the basic capacity terms c(vl +v2) and c(vs) and
if Me tc and If values are known, the capacity for movement
8 can also be shown by the graphs depending directly on v2
and vs. However, one graph would have to be developed
for each possible v1 value. An example of this type of
graph, using tc and If values Mom Table 47 (night
columns), is shown In Figure 53.
1 600
1 400
1 200
800
600
400
200
O
W-",.~>',:.,,''\, 1
W~ N ' \.1
.. ~ V ~ Hi. ~ Hi- ~ .`,'''''i''.''t
-`'\>'`\i'`\~` \ A, W. It'
~200 400 600 800 1000 12001400 16OO
: .
.,....Ij...
1
1
2 1
· 1
1
. I
_ i
v2 [veh/h]
Figure 53. Capacities car for Movement 8 Depending on Traffic
Volumes V2 and VS win v~=0
The theory described here can also be used to determine
the capacity of the minor left turner (movement 7) under
two-stage priority conditions. K there is no separate lane
Norris movements the central storage area, the so-called
mixed lane formula (Equation 10-9 of the HEM, 1994)
should be used to calculate ~e total capacity for
movements 7 and g . Delay estimates for the two-stage
priority situation can also be performed using the delay
formulae In chapter seven.
Limitations and Conclusions of the Theory
The theory which led to Equation 132 should be treated
with caret The concept wo~dbe true if capacities c(v1 + v2
+ vs) and c(vs) were completely estimated. However,
during a specific maximum time, only k vehicles can enter
one major stream gap both in part ~ and part II of Me
intersection due to the restricted storage space within the
median reserve ofthe two-stage situation. This restriction
applies especially for c(v~ + v2) and c(vs) (states ~ and 3
above). This restriction does not apply for c(v, + v2 + vs.)
since during state 2 and state 4 for the number of minor
stream vehicles (>k) departing during one large gap (being
provided simultaneously in major streams ~ and 2 as well
as 5) is not~mited Each of the conventional formulae for
the capacitor c(v) (e.g. Siegloch's model ~ Equation 135)
is based on the assumption that, during large major stream
gaps, a greater number of minor stream vehicles can be
accommodated. This is not true In the two-stage gap-
acceptance situation (state ~ and 3) since here the number
of minor vehicles perlarge gap is limited to k in both parts
of the intersection.
To account for this limited validity of Equation 132,
different approaches have been tested. The derivation of an
analytical formula that accounts for these effects seems to
be impossible. Only a partial solution to address realistic
conditions was possible 03rilon, Wu, Lemke, 1995).
Therefore, some approximations were necessary: a
correction term, ~ (Equation 134), was obtained by
calibrating the results of the analytical formula against
simulation results.
The two-stage priority situation, as it exists at many
unsignali~d intersections within multi-lane major skeets,
provides larger capacities than intersections without
central resene areas. Capacity estimation procedures for
this situation have not been available until now. The above
model (Table 46) provides an analytical solution for this
problem. ~ addidon, simulation studies lead to a
correction of the theoretical results. Based on these
denvations, a set of graphs was evaluated which enable a
simple estimation of the capacity at an unsignalized
intersection under two-stage priority. These graphs are
ready to be used In practice, nevertheless, an emp~ncal
confirmation of this model approach is desirable in filture
research Meanwhile, since tills is a commonly encountered
phenomenon at TWSC intersections with multilane
highways, He theory presented here is recommended for
use at unsignalized intersections.
EARED MINOR STREET APPROACH
The geometric elements near the stop line on the stop-
controDed approaches of many intersections may result In
a higher capacity than the shared lane capacity formula or
an estimation of field capacity may predict. This is
because, at such approaches, two vehicles may occupy or
depart from the stop line simultaneously as a result of He
large curb radius, tapered curb, or a parking prohibition.
The magnitude of this effect win depend In part upon He
Ming movement volumes and the resultant probability of
two vehicles at the stop line, and upon He storage length
available to feed the second position at the stop line.
Figure 54 shows a situation where the curb line provides
space for two vehicles to proceed one beside the over
forward to the stop line. In this case, we could define the
storage as key.
k spaces
/ T
V - ,.
or
V7 Vp
Figure 54. Illustration of an Intersection with Flared Minor Sheet
Approach
91
800
700
.~
400
300
E 2X
I]J 100
Go. .
~ ''.
;,cG.-~ I
. . . . . .
7
-
0 100 200 300 400 500 600 700 800
True Capac ty, veh/hr
Figure SS. Capacity at~tersections win Flared Minor Street
Approaches
Clearly, He estimated capacity using this method is less
than the true capacity at Intersections with flared minor
street approaches, i.e. the capacity at intersections where
a flared minor street approach exists is increased to some
extent Compaq to when Here is no flared approach. This
effect challenges the definition of a continuous queue
required to observe capacity in the field: does one exist
whenever there are queued vehicles behind those at the
stop line, or only when both "sewers" at the stop line are
occupied? It was decided that the original definition should
be maintained (i.e., at least one vehicle always present at
the stop line) and that the ability for rift turn vehicles to
"sneak' was a geometric factor which should be explicitly
recognized in any new procedure. The issue then is what
level of accuracy can be provided based on empirical data
and new theoretical models.
A comparison between observed and field estimated Troutbeck (1995, Phase ~ report, NCHRP 3~6) has
capacity for an example site exhibiting this "right turn reasoned that this is really a short-lane effect. If the
sneaker" elect is shown In Figure 55. number of cars queued In the short lane is one (that is,
Here are two vehicles at the stop line and there is a line of
vehicles behind them), then an estimate can be made of the
increase in capacity by calculating the number of gaps that
are acceptable: that is, the expected number of gaps per
hour that are greater than the critical gap. ~ each one of
these gaps allows a vehicle to use the short lane, this would
be aneshm~ofthe ex~a~ncrease In capacity (absorption
capacity3. However, there may be some of these gaps
which are not fully used by the short-lane vehicles, and this
may be accounted for by a factor related to the availability
of the short lane to arriving vehicles.
Each data point In Figure 55 represents a 15-minute
interval. The depart flow rate was recorded during each
15-minute interval when a continuous queue existed on the
minor sweet approach This observed departure flow rate
is regarded as the true capacity of He minor street. The
minor street approach capacity was also estimated by
dinding 3600 by the sum of the service time and move-up
time (see chapter four). This method is only valid for a
single-lane minor street approach without a flared position.
The majority of the data points shown are below He line
on which estimated capacity is equal to He true capacity.
92
At most TWSC Intersections that operate at below
capacity, the average queue on the cntical approach is
usually low, say less than 4. For intersections with an
average queue of less than 4, the second position at the
stop line may be available upon arrival, or soon thereafter,
so Headmost of these gaps In the major stream traffic could
be used. This likelihood would increase at a given site
with an increase in the length of the "short lane". Given
these observations, for most under-capacity cases, a
storage length sufficient to accommodate two vehicles may
be sufficient to enable the approach to operate and be
analyzed as two separate lanes.
Briton et al. CHOP 3-46, 1995, Working Paper 24)
suggest that~his effect may be modeled using reasoning
similar to the two-stage gap acceptance mode! for minor
street left or through vehicles when median storage is
available. Their initial thinking is outlined below.
Harders (1968) derived He following equation to calculate
shared lane capacity for the minor street approach:
~ V7 Ve Vg
_ = + +_ (~4
Cm C7 C8 Cg
where cm is the shared lane capacity, c', c8 and, c9are He
capacity ofeachindiv~dualmovement 7, 8, and 9 (led,
through, and right turn); and v', v8, and v, are He traffic
volumes of each movement 7, 8, and 9.
This equation is also used In the HEM. It is valid for each
arbitrary traffic system where haBic flows of different
capacities are using the same service facilitr.
The usual geometric design of an Signalized ~ntersechon,
however, provides space for more than one vehicle waiting
at the sup line side by side, e.g., vehicles performing more
than one movement can use He stop line position at He
same time. If k is defined as He number of spaces for
passenger cars belonging to one movement that can queue
at the stop line without obstructing the access to the stop
line for other movements, it is clear that with k > 0, the
capacity of the minor street approach is increased
compared with the sharedIane oondidom Wig He increase
of k, the total capacitor approaches He case that each
movement has its own individual lane of infinite length.
No special solution for He estimation of capacity in He
case for k >0 could be obtained from the literature.
Therefore, a new solution is needed Consider a T
intersection where only two minor street movements exist
(movement 7 and 9~. For movement 9, if less than k
vehicles are queuing In movement 7, each right tum
movement can proceed to the stop line. Doing this time,
movement 9 can depart throve stop line with its capacity
car During over times when there is a longer queue Dan k
vehicles in movement 7, no turn vehicle outside the storage
area of the flare can depart. Thus during these times, He
capacity for movement 9 is zero. Therefore, the capacity
for movement 9 is
C9 = P(n,~k).c, (147)
Similarly, He capacity for movement 7 is given by
C7 = P(n,~k).c7 (148)
This Is not a solution that is exact in a mathematical sense
of queuing theory; rather it is a pragmatic approximation.
To solve Equations 147 and 148, a set of equations for He
queue lengths of bow of He movements should be applied.
The result of an M/M/1 queuing mode! was chosen as a
usefill approximation for the queue length distribution and
is given by Equation 149.
p(n~c)=l-px+~ (~149)
where n is He queue length and p is the degree of
saturation.
The solution is still not correct as it neglects the
interdependencies between both movements, especially
between He queue length distributions. The foldowing
issues would skill need to be resolved before a ~eoredcally
robust model can be developed:
· overcome He problem of dependence between
movements
· include movement 8 for the case of a 4-leg
intersection
Until a better model is available, He following simplified
93
solution teased on an extension of the above discussion can
be used (refer to Figure 56~:
r ~: I
~ Of
SEPARATE ~1
UNSHARED
Gil ~ 11~1
INFRARED
i
1
mar
at_
KACTUAL
QUEUEk
Figure 56. Capacitor Approximation atIdtersections with Flared
Minor Street Approach
Assuming the shared lane case is the worst case,
calculate the total approach capacity clod, then
the delay In vehicle hours. This is equivalent to
Me average queue on the approach for the shared
lane case.
.
.
.
Assuming the separate lane case is the best case,
calculate the total approach capacity c,`~ =
c,+c8+cg, Men the delay in vehicle hours. I-his is
equivalent to the average queue on the approach
for the separate lane case.
Derive a maximum length in vehicles ken of the
flared area above which the traffic flows
approximately like it would be on two separate
lanes. This could be assumed to be equivalent to
the average queue In the second bullet point above
plus one vehicle rounded up to an Integer number
of vehicles (altematively, a more conservative
approach would be to calculate and use the 95th
percentile queues from the 1994 HCM instead of
the average queues in Me steps above).
Using the queue storage Ken, at the site,
interpolate between the shared lane formula
capacity (with 1~0) and the sum of separate lane
capacities (wish k detelInined above) - using either
a linear Interpolation (conservative) or an
interpolation that follows the shape of the curve
given in Equation 149 (more lenient Man the
linear assumption). Linear interpolation can be
shown to yield Me following capacity:
Cedus.1 2 (Cscpe~te- Cohered) k8Ct~ / ~ + Cow (~50)
2 CS,P,,TC KaIl31 / \~ + Coed ~ 1~ I/ kit)
Because k will usually be an integer of low positive value
less than 10 vehicles, the ratio kCb,~/k~ will only be
accurate to one decimal place, or 10 percent of the
difference between We extreme capacities. Therefore it is
recommenced that linear interpolation be used The critical
parameter to determine in this procedure is k,,,=.
100
Table 48. kaput Parameters for Experiment to Assess the Effect of Signals on TWSC Capacitor
, ,., , A.,., , .. , , ............. ........ ., ,., .,.,.~. .. ,. , ,., ... ,.,, ...........
Cycle tsec) | 90 90 Ill
Offset~sec) | 0 0 45
Sahllation Flow Rate (vphgpl) ~ 1800 1800
Gr~n/Cycle ~ 0.4 0.5 0.6 0.7 0.4 0.5 0.6 0.7
Volume/Capacity 1 0 0.25 0.5 0.75 0.9 1.0 0 0.25 0.5 0.75 0.9 1.0
(Proportion of Green Discharging at Sanction
Headway)
| Percent Flow within Platoons | 80% 80%
| Distance from TWSC (~) | Incremerds of 500 up to limit of signal spacing
.
Spacing Between Sigr~alstmi) | 0.25 0.5 1.0
Progression Speed (mph) | 30 30
"a" Dispersion Factor | 0.15 0.15
' a" Dispersion Factor | 0.87 | 0.87 It
The TWSC intersection was assumed to be a four legged
intersection with each turning movement having an
exclusive lane (therefore, knowledge of volumes was not
required for capacity calculation). The major sheet was
ass~edto have two lanes In each direction, troth left turn
pockets. No impedance effects were taken into account.
Capacity calculations were performed for the minor left
(HCM movement #7), minor through All, minor right (9),
and major left approaching from the right (43.
The above combinations of inputs produced a batch of
over 500 separate runs. Numerous output statistics are
product for each run including the proportion of We time
period within each flow regime, the weighted capacity of
each movement, and the percent and absolute difference
from capacity assuming random arrivals.
Table 49 provides an aggregation of the percent increase
In capacity for each movement compared to random
arrivals sort by green time as columns, and signal
spacing and distance to signal 2 as rows. For the same
experiment, Table 50 provides an aggregation of the
vehicular increase In capacity for the minor through
movement compared to random sort by signal spacing,
distance to signal 2 as columns, and green time, total
approach vogue, and offset as rows. Inspection of either
tables confirms expected trends:
closer signal spacing has a greater effect
closer distance to a signal has a greater effect
the results are very sensitive to a change in Me
Offset between the signals
capacity decreases wig increasing opposing
volumes
Me Rest percent change In capacity from random was
84 percent for the minor left ~ movement, 59 percent
for the minor through, 23 percent for the minor right turn,
and 10 percent for Me major left turn The greatest
vehicular change In capacity from random for the minor
through movement was 200 vph. It is interesting to note in
this case that chang~ng~che offset by one-half Cycle reduced
the change to 106 vph. lThe capacity increases become
quite modest when the TWSC intersection is 2000 feet or
more Tom either signal (less than 10 percent), and with
most input combinations (less than 20 percent).
The results of this fictitious experiment hint at potentially
significant differences of greater magnitude end beyond the
distance downstream found using the bunched distribution
model. However, they have not been confirmed by field
observations. They may be a result of unrealistic input
101
parameters such as only20 percent of flow teeing outside
platoons. This, combined with a high proportion of time
for this flow regime would make the non platooned periods
very productive in terms of providing capacitor. An
important observation to be made from this experiment is
that most of the chosen buts are important to account for
(cycle length, green time, saturation flow, distance, speed,
platooned volume, non-platooned volume). Therefore, a
realistic procedure should account for Hem Felicity,
probably using computer software, rather than producing
tables of average percent increases in capacity to apply by
hand
Fable 49. Percentage Idcrcasc in Capacity As Function of Upstream Signals as Comb to Random Major Street Arrivals
~ ~ 2e2~ ~5~2-2~ ~ 5~ ~ 5~T ~J~5 ~ ~ ' ~
MinLT 68 78 84
MinTH 49 S6 S9
SOD MinllT 22 23 22
MajLT 9 10 9
0.2S MinLT 64 7S 84
MinTH 46 S4 60
1000 MinRT 13 13 12
MALT S S S
MinLT 39 42 40
MinTH 28 31 29
SOO MinRT 22 23 22
MajLT 9 9 9
MinLT 33 34 3S
MinTH 24 24 25
1000 MinRT 14 13 12
MajLT 6 S S
MinLT 31 33 36
MinTH 22 24 26
lSOO MinRT 7 6 S
O.S MajLT 3 2 2
MinLT 3S 41 46
MinTH 26 30 34
2000 MinRT 2 2 2
MajLT 1 1 1
MinLT 42 S2 S9
MinTH 32 39 4S
2SOO MinRT 1 1 O
MajLT O O O
MinLT 31 34 34
MiIITH 23 2S 2S
SOO MinRT 22 23 22
MajLT 9 9 9
MinLT 20 19
MinTH 14 14 14
1000 MinRT 13 13 12
MajLT S S S
MinLT 10 8
MinTH 7 4 6
1 lSOO MinRT 7 4 S
MajLT 3 0 2
MinLT 4 3 3
MinTH 3 2 2
2000 MinRT 2 2 2
MajLT 1 1 1
MinLT 3 1 1
MinTH 2 1 O
2SOO MinRT 1 1 O
MaiLT ~ O O O
77
SS
18
7
81
S9
9
30
22
17
7
28
20
34
25
4
49
36
o
66
50
o
27
20
18
7
14
10
9
6
4
4
1
2
1
1
o
1
1
1
o
77
SS
21
9
76
SS
12
5
38
28
21
9
33
23
12
5
34
24
2
43
32
2
5S
42
o
o
31
23
21
9
18
13
12
S
8
2
3
2
2
1
o
o
Notes: (1) D2 - Di~to Uh;beam Signal on~e I~ Side of ~e Minor S~ Appn~acb, (ft); (2) Green Time - Green Time for ~e Major Sbeet ofthe Upsbeam SiF-l':
see, (3) Ihe Numbers Shown in~e Table are Capac~ Idcreases in Perce~age Comparetto Random Major S1reet Arrivals.
1
=
~ )!
c ~
o
o
pi
on
2
,
,
Cal
a
x
a
of
lo
~ 00
~ ¢~
0 0 is Cal ~ ~ ~ ~ 00 00 0 0 ~ lo- a, 00 ~ 0 0 0 0
T ~141 Ma AT MITT FI:T 51~ 1
Lit ~t
~ ~°
f :~:
~ ~ ~ ~ ~ ~ ~
~ ~ 1 ~ m" -~1~N
oo Coo ., mo oo ~V) o`m <~m -~-
~ ~ ~ ~ ~ so ~ so ~ ~ Cal ~ ~o ., Cr so ~4 ~
14t Lo
~ ~ ~ ~ ~ ~ ~
1~ ~ -~1~t
1~143171417151~ 13;71]
1~1441414141414141~41~
t ~t~L
L 12
1 1115 _ ~ = _ ~
o
3
TIC
L
U.
g
-
1 . -
l .-..-..
1 :::::::::
·:.: :.:
·:-:-:-:
1 :::::
1 .......
1 .~
1 ..~.
1 ::C;:
1 :0
1 .:.:-,:.
1 .........
-: -:
_~
~^ ^^ 1
i'
! Olga I ~ 11 1~ tt
to 134~
:~E
Lt1~t'
I
Ah
:~-
:~1:
-~E
-~E
q:~-
ti ~1- 1m 1'
104
Microscopic Simulation
The spreadsheet mode! discussed above is a simple
macroscopic deterministic model. However, the interplay
between the venous input parameters is Rely to be best
captured by use of a microscopic simulation model. To
evaluate tibe behavior of a permit leR turn model, Velan
and Van Aerde (1996) have used microscopic simulation
to determine the effects of various degrees of platooned
flow from an upstream signal on the major left tum. The
degree of platoon~ng is expressed In terms of the
green/cycle ratio through which the same opposing flow is
filtered.
The results of one experiment is shown in Figure 62. For
example, with an opposing flow of 900 veh/br, the
capacibr increased from approximately 400 vet shout
platooning (g/c=~.O) to approximately 800 veh~r with
strong platooning (glc=0.5) of the opposing flow. Since in
this test, 100 percent Lithe opposing flow was assumed to
be platooned, the unplatooned periods yield most of the
capacity as We green/cycle parameter is reduced. In
reality, cap acid increases of this magnitude are less likely.
On most signalized arsenals, there are numerous
intermediate driveways, as well as mining movements at
the upstream signal, and laggards dropping out of the
platoons. They ad contribute to some non-zero proportion
of unplatooned flow.
1800 , :
1500
s
~1200 .
c,
~ 900~
g 600 .
Q 300-
1 1 1
1 1 1
~1 1 ~ 1
, ~
_ ~ .
. ~
. ~ l
stew
~ 1 1
l l l l l l
~1
. . . .
o
0 300 600 900 1200 1500 1800 2100
Opposing Flow, veh/hr
. .
1 1 l
_ gIc=0.3_ gic=o.s 1 gic=0.7_ 9/C=0.~_~ g/c=1.0
Figure 62. impact of me Level of Platooning in We Opposing Plow
on the Minor Street Capacity
The above discussion indicates that it is Important In many
urban situations to include the effect of upstream signals
Intersections when determining the capacil~,r of adjacent
TWSC Intersections. This was confined by a user survey
during Phase ~ of Me project where users aIrnost
n~mmously requested Mat a new pro - =e acknowledge
the effect of upstream signals. Therefore, it is
recommended Mat a procedure be incorporated directly
Into the bow of the HCM chapter for unsignalized
intersections, rather Han the current Appendix I.
THE EFFECT OF PEDESTRIANS ON CAPACITY
AND DELAY
Capacity and delay of vehicular tragic at unsignali~d
intersections may also be affected by pedestrians. Where
a cross walk is provided, pedestrians In some states have
the highest priority over any other traffic movements.
Since most of the sites collected during the project are
either located In suburban area or rural areas, very low
pedestrian volumes were observed. Therefore, the effect of
pedestrians could not be obtained using the collected data.
At many IWSC intersections with low pedestrian
volumes, whatever methodology is Implemented is not
expected to result in a large decrease in capacity.
However, as the level of pedestrians increases to levels
common in downtown situations, some decrease in
capacity should be accounted for.
Conversely, the capacity of pedestrian movements In
situations requiring gap acceptance is not addressed In the
current HCM. Many of the required input parameters
which are needed such as pedestrian bunching, critical
gaps, and crossing speeds should probably be under the
preview of tile Pedestrian chapter of the HCM. So should
simpler gap acceptance processes such as bock
crossings be covered in the Pedestrian chapter. It is
possible ~at, given such information, the Unsignalized
intersection chapter could expand upon the midblock
crossing methodology to account for the more complex
~nte~ctions with vanoush,r ranked boning movements at an
unsignalized intersection. Therefore, the cap acid of
pedestrian movements are not addressed here, although it
is recognized as an important issue. Such a procedure
would provide guidance for signal warrants, and the need
for median refuges for pedestrians to cross in two stages.
In the absence of empirical data, a critical gap could be
derived based on perception, safety margin, and walldng
times.
Further studies will be necessary to accost for the effect
of pedestrians on capacity and delay of vehicular traffic at
unsignalized intersections. In the interim, two theoretical
procedures were developed to account for pedestnans. The
first procedure takes a simple view of the time that
105
cross~ngpe~ians "block" lower ranked movements. In
the second procedure, pedestrians crossing each leg of Me
intersection are assigned a "rally,', as for vehicular tuning
movements, and~eated in much the same way as vehicles,
but with specific pedestrian charactenstics. A theoretical
adjustment to vehicular capacity is made by including
conflicting, higher pr~onty pedestrian streams into the
conflicting volume, and including their impedance due to
their higher pnority. These procedures indicate a range of
possibilities for Including pedestrians into a TWSC
capacity model. The Highway Capacity Committee will
need to consider at what level of detail pedestrians should
be incorporated, in light of pedestrians' actual behavior.
This will probably require validation TV emp~ncal
observations.
Blockage Time Method
Pedestrian impedance is not Me same as vehicle impedance
since the pedestrians. can all cross He road at the same
time. This means that they do not consume intersection
capacity unless there are so many pedestrians that they
form a congruous stream. Perhaps, therefore, they should
not be included as an Impedance In the sense of vehicular
impedance.
Impedance is essentially a way to account for the time that
the higher ranked traffic streams wig be using the
intersection. For the case of pedestrians, the additional
time that they will use the intersection (and cars cannot)
win be a constant time based on walking speed. Consider
a major street left mining vehicle exiting the Intersection
onto a minor street leg which pedestrians need to cross.
The few issue is to establish the average time between the
left Owners being able to depart (ignoring pedestrians for
now3. This is a "block" time and TV ndicate the times
that the rank ~ major sheet through and right vehicles are
using the intersection (The concept of a block time is also
used for impedance calculations in the HCM).
The block time is:
e v/C_1
h= (153)
Vp
where vp is the opposing traffic to major left turn
movement and to is the critical gap for major left turn
movement.
Then, during this block time only, the probability that no
pedestrians will arrive should be established. The
probability that zero pedestrians wiD arrive during tb, is
~ a t o f t h e h e a d w a y b e t w e e n p e d e s ~ i ~ s b e i n g g r e a t e r ~ a n
Ah:
-vend (154)
This is then the proportion of lime a pedestrian is expected
to tee present when a left turner is present. If the walk time
is t,,~,, then the product pO x tw should be subtracted from the
time available to all the lower ranks (~an rank I) each
time a left turn occurs. The Impedance is then:
Po tw
(155)
to
This impedance should be used as a factor that is
multiplied with the other impedance factors. The vehicle
impedance factors would be calculated the same way as In
the current HEM procedure.
Pedestrians Treated as Distinct Movements
While recognizing some peculanties associated with
pedestrians, this method takes a more uniform approach to
both vehicular and pedestrian movements at an
intersection. Unless otherwise noted, the terms
"pedestrian" or "volume of pedestrians" in the following
discussion are defined as the number of groups of
pedestrians accepting the same gap in the opposing tragic
stream Defining the average size of groups for venous
pedestrian flows is a separate task not addressed here.
.
Firstly, the venous pedestrian movements must be ranked
between (or equal to) the appropriate vehicular
movements. Once Cat has been done, they can be included
in the cat HEM procedure in a manner quite consistent
wig how vehicles are treated:
They are part of the cornicing volume, since they
define "begin gap" or "end gap" events. This will
usually be a small elect for normal pedestrian
volumes. Pedestnansafebrisimpliedby the act
of assigning them a rank or priority.
Theyimpede lower ranked movements. However,
tile duration of the impedance due each pedestrian
crossing should only be for blocking one travel
lane, since vehicles performing a given honing
106
movement tend to turn in front of or behind
pedestnans once their target lane is clear. This
rule win prevent an overestimate of Impedance
due to pedestnans crossing wide streets.
With reference to the HEM chapter, pedestrian
impedance should also be adjusted for the
codependence effect between impedances of
higlaer ranked movements as is done for vehicles.
It is useful to define which leg of the intersection is being
crossed. Let subscnpt:
S denote crossing of We subject minor street
approach
O denote crossing of the opposite minor street
approach
~ denote crossing of We major street to the leg of
the subject minor sweet approach
R denote crossing of the major street to the right
of the subject minor street approach
Ranking Pedestrian Movements. A decision also needs to
be made regarding the appropriate ranldng of pedestnans
between the vehicular movements. This may be a policy
issue which vanes by jurisdiction. For example, both
AASHTO and He MUTCD imply Hat pedestrians must
use acceptable gaps In major street (rank 1) traffic Stearns
and that pedestrians have pnonty over all minor sweet
traffic at a TWSC intersection. Refer to Figure 63.
Cross Intersection
Rank 1: 2, 3, 5, 6
1,l,
~ R. S. O
9, 12
a) ~8 11
4 7 10
Figure 63. Rink of Movements
T-lotersection
t tl 5
_ _ _ _ ! ~ Id_
2 ~
a i,' I'
Rank 1: 2, 3, s
WAS
a) 3: 7
In the HCM, major lens 1,4 and minor rights 9,12 ad have
rank 2. An issue for major street crossing pedestrian types
~ and R is whether they have higher or equal rank Han
major leR turns 1 and 4. If so, then they belong to a new
rank"1.5". This is probably an unsafe assumption. If not,
then they need a new rank "2.5". However, they still rank
higher than minor rights 9,12 which means that these
rights new a lower rank This would suggest the following
rank sequence:
· rank 1; (1,4); (L,R); (9,12); rank 3; rank 4
where ranks 1,3 and 4 are as in the HCM
For pedestrians O and S crossing the minor legs, they rank
higher than all minor approach movements. They may also
rank higher than the major left turns 1,4 and major right
turns 3, 6 while crossing the exit legs of the minor street
approaches, although pedestrians often yield to them. If
this yielding to major lefts is assumed (a safer
assumudon). then the entire ranking seau~ce for all
pedestrians becomes quite simple for a cross intersection:
.
rack 1; (1,4); (all pedestrians L,R,O,S); (9,12);
rank 3; rank 4
for a T intersection the ranking sequence is:
· rank 1; 4; (all pedestrians L, R. S); 9; rank 3
In general, the ring can be summanzed as: major sweet
vehicles, pedestrians; miller street vehicles
This ranking, together wad the recommendation that
pedestrian groups crossing any leg only be assumed to
block the leg for a subject vehicle movement for the time
required to cross one lane is conservative in terms of die
magnitude of impedance it will produce for vehicles. This
is appropriate given the lack of emp~ncal calibration or
validation data.
Conflicting volume. Following He sequence of the HEM
Methodology section in Chapter 10, it is questionable
whether or how to include passenger car equivalents as a
function of approach grade for pedestrians in Table 10-!
of He HCM. This would be difficult since pedestrian
group sizes vary as a function of pedestrian flow. Once
group size was [mown, a pedestrian group on a flat grade
could be assigned a weight based on crossing time for one
travel lane. To get p.c.e.'s for non-zero grades, crossing
dine could vary based on variation in walking speeds on
different grades.
107
The diagrams and conflicting tragic equations ~ Figure
10-3 of the HEM would be modified to include the
pedestrian group volumes VL, VR, VS, and VO (as shown
In Figure 63), where V is the number of pedes~ians/group
size.
For planning applications, the number of pedestrians could
be defined as it is In the chapter on signalized
Intersections:
low= 50/hr
moderate= 200~r
high= 400/hr
As is done wad major sweet nght turning vehicles which
In some cases have a weight of 0.5, each of We four
pedestrian movements could be included in conflicting
traffic equations, with weights of I.0 for higher ranked
pedestnan movements and weights of O for lower ranked
movements, except for a proportion of lower ranked
pedestrians in "non-compliance" with their ranking rules.
In practice, pedestrian volumes would usually comprise
only a small proportion of total conflicting volume.
Pedestrian Critical Gap and Crossing Time. Although a
critical gap for pedestrians is not needed if calculation of
pedestrian capacity is not a goal of Me Chapter Ten
procedure, a discussion of it is worthwhile to raise some
interesting issues. Conceptually the critical gap could be
defined as:
[c fr + t~ + fir (~156)
where
.
.
range of 4 to 6 seconds. It is probably a function of group
size. The 4 second value is used by Me MUTCD as a
minimum "walk" indication. The safety margin is
salmon, but perhaps in the order of 2 seconds. Although
these first two terms is not needed unless a calculation of
pedes~iancapaci~is Arm, the Bird term wig n=l, Me
time one travel lane is occupied, is needed to calculate the
impedance due to pe~rians. Wing speed is a function
of age and flow and has been documented to be between
2.5 to 6.0 feet~second. The MUTCD recommends 4
feet/second, except for elderly or child pedestrians where
3 feet/second may tee more appropnate. The HCM chapter
on pedestrian capacity provides speed flow profiles for
pedestrians and uses a crossing speed of 4.5 feet/second.
tr is the perception and reaction time
t, is the safety margin
tw= w . n / v + (N-~)tf is the walk time for group
w is the travel lane wide
n is the number of lanes crossed at a time (this
could be a two stage process if a median is
present). This is related to which lanes are being
used by the vehicle defining the "end gap" event.
v is the waking speed
N is the number of rows In pedestrian group
if is the follow-up time for consecutive rows of
pedestrians
Perception reaction time could be assumed to be in the
Using the above equation, and typical parameter values,
crossing a 5 lane road may require 21 seconds, whereas a
3 lane crossing may require 15 seconds. Whether these are
strictly critical gaps is debatable. The decision process as
to how a pedestrian decides to cross a road is unclear. On
a multilane road the decision may be quite complex,
involving decisions about which lanes opposing tragic is
in, and whether to make a two-stage crossing (note: two-
stage gap acceptance by pedestrians could be analyzed In
a similar way to the method presented earlier for
automobiles). The blockage time for crossing one 12 feet
urge Ravel lane will be in the range of 2.5 to 4 seconds for
one row of pedestrians.
Follow up time for pedestrian groups may be assumed to
be He time between connive rows of pedestrians wit}lin
a group, in the order of 2 seconds per row.
The relatively simple expression for wale time to in
Equation 156 was adapted from He ITE Traffic
Engineering Handbook (Pline, 1992). It should be noted
that adaptation of other expressions may be more
appropriate at high volume pedestrian crossings with two-
way platoons of pedestnans, such as have obtained Tom
shockwave analysis at signalized crosswalks by Viricler, et
al. (1995~. However, at high pedestrian volumes, He
pedestrian signal warrant may in any case be met.
Impedance byPedesf1ians. Generally, pedestrian groups
would impede He lower ranked minor street vehicles while
crossing one travel lane. The equivalent vol,~me/capaci~
ratio of pedestrians required to calculate impedance would
be:
Y/C = Np-~600 t~57y
108
Then, the impedance factor
1= 1 - V/C
(158)
Table 51 shows approximate impedance factors that may
be appropriate for planning applications. The assumptions
would need to be verified with empirical data. As with
vehicular impedances, the statistical dependence between
queues of different streams could be accounted for using
the HEM equation 10-6 and Figure 10-6.
Field observations have shown Mat such a blockage effect
is usually very small, because the major sweet usually
provides enough space for the blocked rank ~ vehicle to
sneak around. Models could be developed Dom a
theoretical point of view when Me major street width does
not allow a through vehicle to sneak. At a minimum,
incorporating this effect requires Me following
information:
.
Table 51. Sensi~vi~ Test For Iinpedance Factor Due To Pedestrians
(s=4fps;w=128)
l
..~:::::.~ .~ ~ . .~ : -~::::~. ~.:.:~ .~ ~.
........................................ . ~. ~..... . . Pa - r
Low 50 1 1 ~ 0.96
Moderate 200 1.2 1 0.86
High 400 1.S 1 0.77
This section has raised the need to adequately account for
the effect of pedestnans on capacity at TWSC
intersections. It is apparent that there are many issues
regarding bow methodology, and parameter values that are
inadequately addressed In the literature. However, it is
imperative to account for the effect of pedestrians in high
pedestrian use areas, at least at a planning level of
analysis, as well as Dom a policy perspective.
DELAY TO MAJOR STREET THROUGH
VEHICLES
TraDic engineers are also interested In knowing He effect
of a shared lane on the major sheet approach where left
tum vehicles may block rank ~ through or right mining
vehicles. If no exclusive left turn pocket is provided on the
major street, a delayed left turn vehicle may block the rank
~ vehicles behind it and cause them some delay. The
current HEM procedure does not account for this effect
and assumes no delay would be experienced by major
street rank ~ vehicles. This effect not only delays rank ~
vehicles. While the delayed rank ~ vehicles are
discharging from the queue formed behind a major left
turning vehicle, they impede lower ranked movements
with which t hey conflict. However, He current 1994 HEM
does provide an impedance for major street left turn
vehicles In a shared lane. This section proposes Cat it also
be used to estimate delay to rank ~ vehicles.
the proportion of rank ~ vehicles being blocked
(the entire shared lane would be I/n)
the average delay to He major sheet left Ming
vehicles which are blocking Trough vehicles
In the simplest procedure, the proportion of major rank ~
vehicles not being blocked (i.e. In a queue free state) is
given byp*0 in equation 10-10 of the 1994 HCM (p*0j
should be substituted for the major left turn factor poj in
equation 10-3 of the 1994 HCM when calculating He
capacity of lower ranked movements which conflict).
Therefore, the proportion of rank ~ vehicles being blocked
is I-p*O~ . Note that on a multilane road, only the major
street volumes in the lane which may be blocked should be
user! in the calculation as Vi, and Viz. On multilane roads
if it's assumed that blocked rank ~ vehicles do not bypass
the blockage by moving across into other through lanes (a
reasonable assumption under conditions of high major
skeet flows) Men Vi, = Vim.
The average delay to rank ~ vehicles on this approach is
given by:
d,,,,,, I =
(1 p 0) d~49 ( N )
V,l + Y,3
(1 - p Of) x dame Kit
No 1 (159)
N=1
Because ofdle unique characteristics associated with each
site, the decision on whether or not to account for this
effect should be left to He analyst. Geometnc design
features such as an adjacent exclusive right turn lane, a
large curb radius, or a wide shared left and through travel
lane may enable rank 1 vehicles to bypass He blockage
caused by major left tuning vehicles. Also, convicting
traffic vogues in such adjacent bypass lanes must provide
sufficient gaps to accept bypassing vehicles.
It is interesting to note that when investigating factors
affecting He critical gap for He major sweet left tom (see
Wowing Paper 16, NC~P 346, 1995), it was found that
109
Me critical gap for the left turn was affected by the
designation of an exclusive versus shared lane as follows:
tc = 3.92 + 0.59 x.Exclusive (1603
where Exclusive = 0 or ~ wad a shared or exclusive left
turn lane, respectively. This effect may be as a result of
the "pressure" that a major street left turner feels while
blocking through traffic.
CONCLUSIONS
This section has introduced both emp~ncal evidence and
suggest theoretical models to adjust the basic capacity or
delay equations to account for some common occurrences
at TWSC intersect `~ two-stage gap acceptance; flared
minor street approaches; effects of upstream signals;
effects of pedestrians; and delay to major street rank ~
vehicles. Singly or In combination, these effects can cause
significant adjustments to the basic capacity and delay
models. The user survey conducted earlier in this study
indicated that methods to account for all these effects are
desirable. If one of these effects were to be included In Me
recommended computational procedure, then it may be
Important to include all of them, if the conditions are
present at a given site, since they may be either additive
(two-stage, flared approaches and signals increase
capacity), negative (pedestrians and shared major left ~
through lanes decrease capacity) or canceling In Weir
overall effect. Some effects may require significant
judgement or data to be provided by the analyst. For
example, the proportion of platooned and non-platooned
flows. Other physical inputs such as flared, median, or
shared lane geometries may be easier to obtain or judge.
110
