### Lec 29:Analysis of signalized intersections

```Chapter 24. Analysis of Signalized Intersections
Chapter objectives: By the end of this chapter the student will:








Understand the conceptual framework for the HCM 2010 method,
including Critical lane group, v/s ratio, saturation flow rate, capacity of a
lane group, v/c ratio for lane group, approach and intersection v/c, LOS,
and effective green times and lost time
Have general ideas of the modules of the Highway Capacity Manual 2010:
Input data, Define movement groups, Compute lane group flow rate, Input
or compute phase duration, Compute capacity, and Compute performance
measures
Will be able to explain how Arrival Type is determined
Know how to enter input data into the Highway Capacity Software
(HCS2010)
Explain the terms of delay models (including Incremental Queue Analysis)
Know how to deal with initial queues
Understand how the permitted left turns are modeled by the HCM 2010
Understand how the left-turn adjustment factor for compound
(protected/permitted) phasing is modeled.
Chapter 24
1
24.1 Introduction
What’s new in HCM 2010 as compared with HCM 2000
1. The model has been set up to handle actuated signal
analysis directly.
2. The estimation of delay is now partially modeled
using Incremental Queue Analysis (IQA). IQA allows
a more detailed analysis of arriving and departing
vehicle distributions.
3. The definition of lane groups has been altered. Lane
groups are identified and separately analyzed as part
of the methodology.
“This text focuses on the analysis of pretimed signals because it is
more straight forward to present basic modeling theory for fixed
time signals.”
Chapter 24
2
HCM
2010
Analysis
Steps
Chapter 24
3
24.1 Conceptual framework for HCM 2010
Five fundamental concepts of the HCM 2010:
The critical lane group concept
 The v/s ratio as a measure of demand
 Capacity and saturation flow rate concepts
 Level-of-service (LOS) criteria and
concepts
 Effective green time and lost-time concepts

Chapter 24
4
24.2.1 The Critical-Lane Group Concept
Critical lane analysis (Section 17.3) vs. Critical lane group analysis
Critical lane analysis compares actual flow (v) with the
saturation flow rate (s) and capacity (c) in a single
lane. Critical lane group analysis compares actual flow
(v) with the saturation flow rate (s) and capacity (c) in
a group of lanes operating in equilibrium. In either
case, the ratio of v to c is the same (when traffic is
evenly distributed among the lanes in aChapter
lane 24group).
This applies to shared lanes, also.
Exclusive right- or leftturn lanes must be
separately analyzed
because they are
separate lane groups.
Lane utilization is
considered in computing
saturation flow rate.
5
24.2.2 The v/s ratio as a measure of demand & 24.2.3 Capacity and
saturation flow rate concepts
* The simple method in Chapter 21 (as
a comparison – Vc is adjusted by
converting into tvu (through vehicle
unit) & saturation flow is given):
Nt L
C des 
1
Vc
PHF ( v / c )( 3600 / h )
* In the HCM model, demand flow rates are not converted to tvu. It uses veh/hr
(though adjusted for PHF). A key part of the HCM 2010 model is a methodology
for estimating the saturation flow rate of any lane group based on known
prevailing traffic parameters.
si  s0 N  f i
i
We may not be able to compare directly lane groups because their
conditions are different. So HCM use the flow ratio, v/s, a dimensionless
value for comparison purposes. This
process
is called “normalization.”
Chapter
24
6
24.2.3 Capacity (continued)
 In the simple timing method in
Chapter 21, the capacity of the
intersection as a whole was
considered.
1
3600 
Vc 
  3600  Nt L
h
h
C 
TG
 HCM 2010 as well as HCM 2000 gives the capacity of
each lane group.
 Demand does not necessarily peak at all approaches
at the same time.
ci  si
gi
C
 Capacity may change for each approach during the
day. (like the effect of curb side parking, bus blocking,
etc.)
 Capacity is provided to movements to satisfy
movement demands. (Note: the critical capacity ratio v/c
(for the intersection as a whole) is still calculated in
HCM 2010 just like HCM 2000).
Chapter 24
7
The v/c ratio  “degree of saturation”
Three issues:
(1) Capacity is practically always estimated (because it is difficult to
measure.)
(2) In existing cases demand is often measured by “departure flows”
although it should be “arrival flows.”
(3) For future cases, predicted arrival volumes are given (by a planning
model) instead of actually counted volumes.
Case 1 & 2: v/c > 1.0 resulted in a HCM analysis for an existing signalized
intersection.
If demand is measured by a departure flow (assuming it was correct),
this cannot be accepted because max value v/c = 1.0. If arrival flows are
measured, v/c > 1.0 may occur – this becomes obvious because queue
forms). Capacity must have been underestimated if queue is not formed
despite the fact v/c > 1.0 results. Capacity underestimation is possible
because HCM models are nationalChapter
average
24 models.
8
Case 3: v/c > 1.0 resulted in an analysis for a planned signalized
intersection.
In a planning case, both demand and capacity are estimates. But, it may
indicate that the forecast demand flow exceeds the estimated capacity of the
lane group, and a problem will likely occur. Demand is an arrival flow for a
predicted case because those values come from a planning model.
Computation of a v/c ratio (degree of saturation) for a given lane
group (this model does not change among different HCM versions:
Xi 
vi
ci
vi

si
gi

vi si
Flow ratio/Green ratio
gi C
C
Chapter 24
9
Computation of a v/c ratio for an intersection as a whole (p.576):
The critical v/c ratio for the intersection  defined as the sum of the critical
lane group flows divided by the sum of the lane group capacities available to
serve them (compare this one with the Simple Method in Ch 20).
Xc 

Xc 
v
ci

 s ci

g ci 

C 
 v s 

 v s 

g ci

 v s 
CL
CL
ci

 v s 
C
ci
CL
C
C
C
ci
ci
X c min 
 v s 
C max
ci
C max  L
If the Xc > 1.0, then the physical design, phase plan, and cycle length specified
do not provide sufficient capacity for the anticipated or existing critical lane
group flows.  Do something to increase capacity: (1) longer cycle lengths
(less number of cycles, less lost time), (2) better phase plans (improved LT
treatment), and (3) add critical lane group or groups (meaning change approach
layouts  increase capacity)
Chapter 24
10
Computation of a v/c ratio for an intersection as a whole
 If the critical v/c ratio is less than 1.00, the cycle length, phase
plan, and physical design provided are sufficient to handle the
demand and flows specified.
 But, having a critical v/c ratio under 1.00 does not assure that
every critical lane group has v/c ratios under 1.00. When the
critical v/c ratio is less than 1.00, but one or more lane groups have
v/c rations greater than 1.00, the green time has been misallocated.
Chapter 24
11
24.2.4 Level of service concepts and criteria
 All the HCM delay models assume random arrivals. Hence, the delay
model produce delays for approaches with random arrivals. Urban signals are
coordinated; hence, many do not have random arrivals. This is corrected by
the “quality of progression” factor called “Arrival Type” factor. See Table
24.3 and 24.4. There are 6 arrival types: 1 = poor coordination, 6 =
exceptionally good coordination.
 For uninterrupted facilities, like freeways, v/c has a direct connection with
the performance of the facility. So, if v/c = 1.0, the facility is at the capacity.
 For signalized intersections (interrupted facilities), this is not necessarily
true – especially when delay is used as the MOE.
 You may get LOS=F even if v/c is well below 1.0. For instance LT
vehicles may have a long stopped delay even if its v/c is low.
 HCM 1994 delay model focuses on the first 15-min interval. So, even if it
is over-saturated (v/c > 1.0), we get a relatively smaller delay. HCM 2010 has
3 study approaches: Single analysis period for 15 min and 1 hour, and
multiple 15-min analysis periods. Chapter 24
12
The 2010 HCM uses “total control delay” consisting of three terms
Total control delay per
vehicle = time in
queue delay +
accelerationdeceleration delay
New to HCM 2010:
Any lane group operating
at a v/c ratio greater than
1.00 is also labeled as
LOS F.
Because delay is difficult to measure in the field and because it cannot be measured for
future situations, delay is estimated using analytic models. The delay models are
discussed in section 24.3.7.
Chapter 24
13
24.2.5 Effective green times and lost times
A
B
G
y
l1
e
ar
R
l2
R
C
tL
g
R
D
r
g
r
A. Actual signal indications
B. Actual use of green and yellow; e is extended green, i.e. part
of the yellow used as green
C. Lost times l1 and l2 are added and placed at the beginning of
the green for modeling purposes
D. Effective green and effective red
l2  Y  e
Y  y  ar
l1 = 2 sec/phase
e = 2 sec/phase
Default by HCM2010
t L  l1  l 2
n
L 
Chapter 24
t
i 1
Li
14
Effective green times and the application of the lost times:
 HCM delay models use “effective green time” and “effective red time.”
 HCM 2010 models assume that all lost times happen at the beginning of the
phase.
g i  G i  y i  ar i  l1  l 2
g i  G i  l1  e
ri  C  g i
Watch out where tL takes
place, especially when an
overlap phase exists. That’s
where you must add y and
ar in the phase section of
the HCS input module.
Chapter 24
15
24.3 The Basic Model
24.3.1 Model structure
The HCM 2010
signalized
intersection
analysis consists
of 6 modules.
Chapter 24
16
Chapter 24
17
24.3.2 Analysis time periods



The peak 15 minutes within the
analysis hour (no oversaturation exists, no v/c > 1.0.
Use PHF.)
The full 60-min analysis hour
Sequential 15-min periods for
an analysis period of one hour
or greater (Most
comprehensive. PHF = 1.0 is
used.
vp 
V
PHF
Chapter 24
18
24.3.3 Input
Input Module: Many parameters are considered. See Table 24-2 in the text).
Geometric, traffic, and signalization conditions are considered  Some of them
are self-explanatory. See pages 582 – 585 for details and default values.
 Area type: CBD intersections have lower saturation flow rates (in
general). Saturation flow rates for CBD is about 10% less than for nonCBD.
 Parking conditions and parking activity: Parking activity within 250 ft
of the stop line is considered. Parking activities interfere traffic flow
 Conflicting pedestrian flow (for RT vehicles): Pedestrian flow between
1700 to 2100 ped/hr completely blocks right-turn vehicles. HCM 2010
considers bicycles as well. Also, check pedestrian min green times
 Local bus volume: Buses must stop to be considered in this parameter. If
they pass through the intersection, not stopping for passengers, they are
considered as heavy vehicles.
 Arrival type: The single mostChapter
important
factor influencing delay
24
predictions.
19
More discussion on Arrival type (Table 24.3, p.583 text):
1
Dense platoon, containing over 80% (P) of the lane group
volume, arriving at the start of the red phase  very poor
progression
2
About 40 to 80% arriving at the start of the red phase 
unfavorable progression
3
Main platoon contains less than 40% of the lane group volume
 random arrival
4
40 to 80% arriving throughout the green time  favorable
progression
5
Over 80% arriving at the start of the green phase  highly
favorable progression
6
Exceptional progression, with minimal or negligible side-street
entries.
AT  3
P
g C
Chapter 24
P = Proportion of
vehicles arriving
on green.
20
More discussion on Arrival type:
Need to compute a platoon ratio, Rp:
Rp = 1.00, when the proportion of vehicles arriving
on green is equal to the g/C ratio.
P = Proportion of
vehicles arriving
on green.
Rp 
P
gi C
Table 24.4 (They
accidentally (?)
forgot two
columns for
platoon ratio.
Chapter 24
21
24.3.4 Movement Groups, Lane Groups, and
1. Conversion of hourly demand volumes to peak 15-min flow rates
needs to be done first.
2. Establish analysis lane groups (6 types)
v
V
PHF
These two
rules are new
in HCM2010.
Chapter 24
3. Determination of total lane group demand flow rates, vgi
22
Figure 24.5 Common Movement and Lange
Groups on a Signalized Intersection Approach
New in
HCM2010
Chapter 24
23
24.3.5 Estimating the Saturation Flow Rate
for Each Lane Group
The saturation flow rate module is the most important part of HCM2010.
The prevailing total saturation flow rate for each lane group is estimated.
s  s 0 Nf w f HV f g f p f bb f a f LU f RT f LT f Rpb f Lpb
• fw = 0.96 Lane width < 10ft
• fw = 1.00 10 ft ≤ Lane width ≤ 12.9 ft
• fw = 1.04 Lane width ≥ 12.9 ft
Chapter 24
24
1
 G 
fg  1 

 200 
f HV 
1  PHV  E HV  1  EHV = 2.00
 18 N m 
P  0 .9  

 3, 600 
fp 
N
 1  P
N
 18 N m 
N  0 . 10  

 3 , 600 
fp 
 0 . 05
N
Limitations:
• 0 ≤ Nm ≤ 180; if Nm > 180, use 180 mvts/h
• fp(min) = 0.05
• fp(no parking) = 1.00
Chapter 24
25
 14 . 4 N B 
B  1 .0  

 3, 600 
f bb 
N
 1  B
N
f bb
 14 . 4 N B 
N 

 3 , 600 

 0 . 05
N
Limitations:
• 0 ≤ NB ≤ 250; if NB > 250, use 250 b/h
• fbb(min) = 0.05
• CBD location: fa = 0.90
• Other location: fa = 1.00
f LU 
vg
v g1 N
See Table 24.7 for default values (next page).
Chapter 24
26
Chapter 24
27
• From an exclusive RT lane (fRT = 0.85)
• From a shared lane
• From a single-lane approach
Adjustment for Left Turns (discussed in section
24.5 of the text):
• Case 1: Exclusive LT lane with protected phasing (fLT
= 0.95)
• Case 2: Exclusive LT lane with permitted phasing
• Case 3: Exclusive LT lane with compound phasing
• Case 4: Shared lane with protected phasing
• Case 5: Shared lane with permitted phasing
• Case 6: Shared lane with compound phasing
Chapter 24
28
Adjustment for Pedestrian and Bicycle Interference with Turning
Vehicles (There are seven steps; see pages 589-591):
Chapter 24
29
Pedestrian and Bike Interference Adjustment (continued):
gped = walk +
clearance interval
• 2,000 = 3,600/1.8 sec a ped occupies the ped conflict
area.
• 10,000 = 3,600/0.36 sec a ped occupies the ped conflict
area walking parallel. 0.4 = 40% occupied.
Chapter 24
30
Pedestrian and Bike Interference Adjustment (continued):
• 2,700 = 3,600/1.33 sec a bike
occupies the bike conflict area.
• 0.02 = 2% occupied
Because bikes follow
the same rule as cars
Chapter 24
31
Pedestrian and Bike Interference Adjustment (continued):
4
Joint probability
(Venn diagram)
Chapter 24
32
Pedestrian and Bike Interference Adjustment (continued):
Meaning “After turning
from an exclusive RT
lane, there is only one
lane in the receiving
side”
40% less
impact
Chapter 24
33
Pedestrian and Bike Interference Adjustment (continued):
Chapter 24
34
24.3.6 Determine Lane Group Capacities
and v/c Ratios
1. The v/s ratio for each lane group is computed.
2. Relative v/s ratios are used to identify the criticallane group in the phase plan; the sum of critical lane
group v/s ratios is computed.
3. Lane group capacities are computed (Eq. 24-2).
4. Lane group v/c ratios are computed (Eq. 24-3)
5. The critical v/c ratio for the intersection is computed
(Eq. 24-5).
ci  si
gi
C
Eq. 24-2
Xi 
vi
ci
vi

si
Eq. 24-3
gi

vi si
gi C
C
Xc 
 v s 
i
 g
ci
ci
C




C L
 v s ci 
i
C
i
Chapter 24
Eq. 24-5
35
Step 1 and 2 of 24.3.6
These values are
flow ratios (v/s).
Finding critical lane
groups is similar to
the simple method;
the only difference is
that HCM uses v/s to
find critical lane
groups.
Chapter 24
36
24.3.7 Estimating Delay and Level of Service
d = d1 + d2 + d3
Where, d = average control delay per vehicle, s/veh
d1=average uniform delay per vehicle,
In HCM2010,
d2 = average incremental delay per vehicle,
AT is part of the
d3= additional delay per vehicle due to a
uniform delay,
preexisting queue
d1, computation
(see slide #39).
d  d 1 PF  d 2  d 3
2
In HCM2000, arrival type
factor was multiplied to d1
as shown on the right.
g 
0 . 5 C  1 
C 

d1 
g 
1   min 1, X  
C 


d 2  900 T  ( X  1) 

Chapter 24 d 3 
1800 Q b 1  u t
cT
X
 1
2
 8 kIX  


cT

 
37
Incremental Queue Accumulation (IQA)
(Note: what’s in the textbook is for pre-timed signals)
By HCM2000
By HCM2010
Difference?
Chapter 24
38
Incremental Queue Accumulation (IQA)
The effect of progression is built in into the methodology in IQA.
First, we need to find out the P value, which is the portion of
platoon arriving during the green.
If you don’t have field data for it, you may estimate it by equation
24-26.
 AT 
P 
 g C 
 3 
AT  3
P
g /C
P = Proportion of
vehicles arriving on
green.
Chapter 24
39
Incremental Queue Accumulation (IQA) Steps
Step 1: Determine the arrival rate
during the effective red, Vr
Vr 
1  P  * V
*C
r
V = average arrival flow rate, veh/hr: The numerator gives the
number of vehicles arriving during the effective red in a cycle.
Step 2: Determine the queue at
the end of the red time, q2
 vs 
q 2  q1  
 * t
 3600 
q2  0
During red time, r,
v = Vr
s=0
q1 = 0 for a single interval analysis
Δt = Elapsed time, during red = r
Chapter 24
40
Incremental Queue Accumulation (IQA) Steps
Step 3: Determine the uniform delay during the effective red time
 q1  q 2 
d r  t * 

2


q1 = 0 for a single interval analysis
Δt = r
Step 4: Determine the arrival rate
during the effective green time
Vg 
Chapter 24
V *P
g
C

V * P *C
g
41
Incremental Queue Accumulation (IQA) Steps
Step 5: Determine Δt2, the time needed to dissipate the queue
Δt2
 Vg
q 2  
 3600
t2 

 s 
 * t2  
 * t2

 3600 

3600 * q 2
s  Vg
Step 6: Determine the uniform delay
during the effective green
dg
 q2  q3 
 t2 * 

2


q3 = 0 for Δt2 ≤ g, undersaturated flow
Chapter 24
42
Incremental Queue Accumulation (IQA) Steps
Step 6: Determine the uniform delay
during the effective green (continued)
 V g * g   V r * r   ( s * g ) 
q 3  q1  

3600


Step 7: Determine the uniform delay,
d1 s/veh
d1 
na 
d
r
 dg 
q 2  n a 
Vg * g
3600
na = the number of vehicles arriving on green
Chapter 24
43
The incremental delay term, d2

d 2  900 T  ( X  1) 

X
 1
2
 8 kIX  


 cT  
Chapter 24
Eq. 24-35
44
Upstream Filtering or Metering
Degree of Saturation at Upstream Intersection, Xu
I
0.4
0.5
0.6
0.7
0.8
0.9
≥ 1.0
0.922 0.858 0.769 0.650 0.500 0.314 0.090
 An I-value of 1.0 is used for an isolated intersection (i.e., one that is 1
mile or more from the nearest upstream signalized intersection). This
value is based on a random number of vehicles arriving per cycle so that
the variance in arrival equals the mean.)
 An I-value of less than 1.0 is used for non-isolated intersections. This
reflects the way that upstream signals decrease the variance in the number
of arrivals per cycle at the subject intersection. As a result, the amount of
delay due to random arrivals is reduced.
Chapter 24
45
Aggregating Delay (p.597)
dv
i
dA 
Average delays are weighted by
the number of vehicles
experiencing delays.
i
i
v
d v
i
A
dI 
A
v
A
A
A
Computing total control delay,
dI, per vehicle for the
intersection as a whole (This
computation is not
recommended according to the
HCM2010, p.597)
Chapter 24
46
24.3.8 Interpreting the results of signalized
intersection analysis





v/c ratio for every lane group
Critical v/c ratio (Xc) for the intersection as a whole
Delays and levels of service for each lane group
Delays and levels of service for each approach
Delays and levels of service for the overall intersection
(not recommended by HCM2010.)
The following scenarios are possible:
 Scenario 1: Xc ≤ 1.00, all Xi ≤ 1.00. No capacity deficiency
 Scenario 2: Xc ≤ 1.00, some Xi > 1.00. As long as Xc ≤ 1.00, the
current conditions can handle; reallocate green times
 Scenario 3: Xc > 1.00, some or all Xi > 1.00. Need to change timing
and phasing and if necessary physical layout changes
Chapter 24
47
24.4 A “Simple” Sample Problem
Chapter 24
48
21.5 Complexities

24.5.1 Left-turn adjustment factor, fLT, for
permitted left turns

24.5.3 Using analysis parameters to adjust
signal timing
Chapter 24
49
24.5.1 LT Permitted Left Turns from a shared lane
Interaction between LT vehicles and opposing vehicles
No gaps are available for LTs when the standing queue is released right
after the signal turns green.
If a LT vehicle arrives during this time, it must wait, blocking the leftmost lane, until the opposing queue has cleared.
After the opposing queue has cleared the intersection, LTs may be made
through gaps in the unsaturated opposing flow.
LTs have no impact on the subject approach until the first LT vehicle
arrives (for a shared LT lane).
gq = avg. amount of green time required for the opposing standing queue
to clear the intersection, sec.
gf = avg. amount of green time before the arrival of the 1st LT vehicle, sec
(gf = 0.0 sec for an exclusive LT lane)
gu = avg. amount of green time after the arrival of the 1st LT vehicle that is
not blocked by the clearance of the opposing standing queue, sec
Chapter 24
50
Figure 24.10 Portion of the Green Phase Illustrated
g u  g  g q if g q  g
g u  g  g f if g
f
f
 gq
Opposing queued vehicles clearing Green not blocked by the opposing
clearing queue, usable by LT vehicles
The first LT vehicle must wait, i.e., TH is blocked.
Opp. Queued vehicle clearing
First LT veh has not arrived yet
Green not blocked by the opposing
clearing queue, usable by LT vehicles
Chapter 24
51
Fig. 24.11Queue Accumulation Polygon for a Shared Lane
with Permitted Left Turns
gq
gdiff
Int. 3
Int. 0
Sth = saturation flow rate for
through veh.
Ssl3 = saturation flow rate for
shared-lane interval 3
Int. 1
Int. 4
Int. 2
Chapter 24
52
Modeling permitted left-turns (cont)
Summary for Permitted Left Turns from a shared lane (general concept)
• For multilane
opposing
approaches
f LT 
f LT 
gdiff = gq - gf
• For single lane
opposing
approaches
1.00 >= F1 >= 0.0
f LT 
f LT 
g
f
(1 . 0 ) 
g
g
f
g

g
gf
gu
g
(1 . 0 ) 
g
g diff
g

g diff
g
0 . 0  
gu
g
( F1 )
( F1 )
g
gf
g diff
 F2  
 F2  
gu
g
gu
g
( F1 )
( F1 )
F2 = 0 when opposing
approach is multilane. Why?
Chapter 24
53
Basic structure of the permitted LT model (cont): multilane
opposing approaches
f LT 
g
f
g

gu
g
( F1 )
F1 
1
1  PL  E L 1  1 
But the minimum is (sn = sneakers):
f sn 
2 1  PL 
g
Observations show at least one vehicle can turn
during the clearance interval and may be two if the
second vehicle is a LT vehicle. 2 = 2 second
headway, 1 = minimum 1 LT sneaker, and PL is the
proportion (probability, that is) of LT vehicles in the
left lane.
Chapter 24
54
Basic structure of the permitted LT model (cont): single-lane
opposing approaches
When the opposing flow is in a single-lane approach, a LT
vehicle on that approach creates a gap in the opposing flow
through which a subject LT vehicle may move. We need to
consider this available gap (2nd term below), which does not
exist if there are multiple lanes in the opposing approach.
f LT 
gf
(1 . 0 ) 
g
F2 
Proportion of LT
vehicles in the
opposing singlelane approach,
decimal
gq  g
g
f
( F2 ) 
1
1  PL  E L 2  1 
g
( F1 )
Proportion of through and RT
vehicles in the opposing singlelane approach, decimal.
n = No. of opposing vehicles in the period gq
– gf, about (gq – gf)/2. n can be zero. 2 is 2
sec/veh headway and n is for joint
PLTo
Chapter 21probability. The numerator is the probability
55
that one or more vehicles are LT vehicles.
1  PTHo
n
EL2 
gu
24.5.3 Altering Signal Timings Based on v/s Ratio
After v/s ratios are computed, we may need to make adjustments – either
reallocation of green time, modifying cycle length, or modifying the intersection
layout. For the first two cases, v/s ratios can be used to reduce the amount of
trial-and-error computations. (Assume the phase design does not change.)
First, we solve Xc for C:
C 
LX
Xc 
Xc 
c
 v / s 
ci
v

ci
g ci 

 s ci

C



 v / s 

ci
 g ci 


C



 v / s ci
C
CL
When Xc = 1.0, it is like C equation for simple signal timing (eq. 20-13).
Suppose sum(v/s) = 0.9, and we desire to achieve Xc = 0.95. What
would be the cycle length to serve this (assume max C = 90 sec)?
C 
9 ( 0 . 95 )
0 . 95  0 . 90
 171 sec  C max  90 sec
Xc = 0.95 cannot be achieved in this case. C = 171 sec is too long.
Chapter 24
56
Modifying signal timing based on v/s ratios (cont)
C needs to be contained within the common cycle lengths (171 sec in
the previous example is too long). Typically C = 120 sec is the
maximum cycle length accepted. Hence,
Xc 
 v / s 
C
i
CL
 0 . 90
120
120  9
 0 . 973
Where, Σgi = C – L.
With sum(v/s) = 0.90 and C = 120 sec, Xc = 0.973 is the minimum that can be
achieved (although Xc = 0.95 cannot be achieved). Once C is determined, we
can compute new effective green times, then new actual greens for the critical
lane groups and for the next trial-and-error analysis.
Xi 
vi
ci
vi

si
gi

(v / s ) i
gi C
 (v / s ) i
C
gi
C
Chapter 24
g i   v / s i
C 
g
i
C
Xi
L
G i  g i  t L 57
 Yi
```