CEE 320
Winter 2006
Signalized Intersections
CEE 320
Steve Muench
Outline
1.
2.
3.
4.
Key Definitions
Baseline Assumptions
Control Delay
Signal Analysis
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Winter 2006
a.
b.
c.
d.
D/D/1
Random Arrivals
LOS Calculation
Optimization
Key Definitions (1)
• Cycle Length (C)
– The total time for a signal to complete a cycle
• Phase
– The part of the signal cycle allocated to any combination
of traffic movements receiving the ROW simultaneously
during one or more intervals
• Green Time (G)
– The duration of the green indication of a given
movement at a signalized intersection
• Red Time (R)
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– The period in the signal cycle during which, for a given
phase or lane group, the signal is red
Key Definitions (2)
• Change Interval (Y)
– Yellow time
– The period in the signal cycle during which, for a given
phase or lane group, the signal is yellow
• Clearance Interval (AR)
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– All red time
– The period in the signal cycle during which all
approaches have a red indication
Key Definitions (3)
• Start-up Lost Time (l1)
– Time used by the first few vehicles in a queue while reacting
to the initiation of the green phase and accelerating.
2 seconds is typical.
• Clearance Lost Time (l2)
– Time between signal phases during which an intersection is
not used by traffic. 2 seconds is typical.
• Lost Time (tL)
– Time when an intersection is not effectively used by any
approach. 4 seconds is typical.
– tL = l1 + l2
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Winter 2006
• Total Lost Time (L)
– Total lost time per cycle during which the intersection is not
used by any movement.
Key Definitions (4)
• Effective Green Time (g)
– Time actually available for movement
– g = G + Y + AR – tL
• Extension of Effective Green Time (e)
– The amount of the change and clearance interval at the
end of a phase that is usable for movement of vehicles
• Effective Red Time (r)
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– Time during which a movement is effectively not
permitted to move.
– r = R + tL
– r=C–g
Key Definitions (5)
• Saturation Flow Rate (s)
– Maximum flow that could pass through an intersection if
100% green time was allocated to that movement.
– s = 3600/h
• Approach Capacity (c)
– Saturation flow times the proportion of effective green
– c = s × g/C
• Peak Hour Factor (PHF)
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– The hourly volume during the maximum-volume hour of
the day divided by the peak 15-minute flow rate within
the peak hour; a measure of traffic demand fluctuation
within the peak hour.
Key Definitions (6)
• Flow Ratio
– The ratio of actual flow rate (v) to saturation flow rate (s)
for a lane group at an intersection
• Lane Group
– A set of lanes established at an intersection approach
for separate analysis
• Critical Lane Group
– The lane group that has the highest flow ratio (v/s) for a
given signal phase
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Winter 2006
• Critical Volume-to-Capacity Ratio (Xc)
– The proportion of available intersection capacity used
by vehicles in critical lane groups
– In terms of v/c and NOT v/s
from Highway Capacity Manual 2000
Baseline Assumptions
• D/D/1 queuing
• Approach arrivals < departure capacity
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Winter 2006
– (no queue exists at the beginning/end of a
cycle)
Quantifying Control Delay
•
Two approaches
–
Deterministic (uniform) arrivals (Use D/D/1)
–
Probabilistic (random) arrivals (Use empirical equations)
• Total delay can be expressed as
– Total delay in an hour (vehicle-hours, person-hours)
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Winter 2006
– Average delay per vehicle (seconds per vehicle)
D/D/1 Signal Analysis (Graphical)
Vehicles
Departure
Rate
Arrival
Rate
Queue dissipation
Total vehicle delay per cycle
Maximum delay
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Winter 2006
Maximum queue
Time
Red
Green
Red
Green
Red
Green
D/D/1 Signal Analysis – Numerical



  1.0
• Time to queue dissipation after the start of effective green
r
t0 
1   
• Proportion of the cycle with a queue
r  t0
Pq 
c
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Winter 2006
• Proportion of vehicles stopped
 r  t0  r  t0
Ps 

 Pq
 r  g 
c
 r  t0  t0 t0
Ps 


 r  g  c c
D/D/1 Signal Analysis – Numerical

• Maximum number of vehicles in a queue
Qm  r
• Total delay per cycle
r 2
Dt 
21   
• Average vehicle delay per cycle
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Winter 2006
r 2
1
r2
Dt 


21    c 2c1   
• Maximum delay of any vehicle (assume FIFO)
dm  r


  1.0
Signal Analysis – Random Arrivals
•
Webster’s Formula (1958) - empirical
d' d 
1/ 3
x
 c 
 0.65 2  x 25( g / c )
2 1  x 
 
2
d’ = avg. veh. delay assuming random arrivals
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Winter 2006
d = avg. veh. delay assuming uniform arrivals (D/D/1)
x = ratio of arrivals to departures (c/g)
g = effective green time (sec)
c = cycle length (sec)
Signal Analysis – Random Arrivals
•
Allsop’s Formula (1972) - empirical
9 
x2 
d '  d 

10 
2 1  x  
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Winter 2006
d’ = avg. veh delay assuming random arrivals
d = avg. veh delay assuming uniform arrivals
(D/D/1)
x = ratio of arrivals to departures (c/g)
Definition – Level of Service (LOS)
• Chief measure of “quality of service”
– Describes operational conditions within a traffic
stream
– Does not include safety
– Different measures for different facilities
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Winter 2006
• Six levels of service (A through F)
Signalized Intersection LOS
• Based on control delay per vehicle
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Winter 2006
– How long you wait, on average, at the stop light
from Highway Capacity Manual 2000
Typical Approach
• Split control delay into three parts
– Part 1: Delay calculated assuming uniform arrivals (d1).
This is essentially a D/D/1 analysis.
– Part 2: Delay due to random arrivals (d2)
– Part 3: Delay due to initial queue at start of analysis time
period (d3). Often assumed zero.
d  d1 PF   d 2  d3
d = Average signal delay per vehicle in s/veh
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PF = progression adjustment factor
d1, d2, d3 = as defined above
Uniform Delay (d1)
 g
0.5C 1  
C

d1 
g

1  min 1, X  
C

d1 = delay due to uniform arrivals (s/veh)
C = cycle length (seconds)
g = effective green time for lane group (seconds)
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Winter 2006
X = v/c ratio for lane group
Incremental Delay (d2)

d 2  900T  X  1 

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Winter 2006
d2 =
8kIX 
 X  1 

cT 
2
delay due to random arrivals (s/veh)
T =
duration of analysis period (hours). If the analysis is based on the
peak 15-min. flow then T = 0.25 hrs.
k =
delay adjustment factor that is dependent on signal controller mode.
For pretimed intersections k = 0.5. For more efficient intersections k
< 0.5.
I =
upstream filtering/metering adjustment factor. Adjusts for the effect of
an upstream signal on the randomness of the arrival pattern. I = 1.0
for completely random. I < 1.0 for reduced variance.
c =
lane group capacity (veh/hr)
X =
v/c ratio for lane group
Initial Queue Delay (d3)
• Applied in cases where X > 1.0 for the
analysis period
– Vehicles arriving during the analysis period
will experience an additional delay because
there is already an existing queue
• When no initial queue…
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– d3 = 0
Control Optimization
•
Conflicting Operational Objectives
–
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minimize vehicle delay
– minimize vehicle stops
– minimize lost time
– major vs. minor service (progression)
– pedestrian service
– reduce accidents/severity
– reduce fuel consumption
– Air pollution
The “Art” of Signal Optimization
•
Long Cycle Length
–
–
–
•
High capacity (reduced lost time)
High delay on movements that are not served
Pedestrian movements? Number of Phases?
Short Cycle Length
–
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Winter 2006
Reduced capacity (increased lost time)
– Reduced delay for any given movement
Minimum Cycle Length
C min
L Xc

n
v
X c   
i 1  s  ci
Cmin = estimated minimum cycle length (seconds)
L = total lost time per cycle (seconds), 4 seconds per
phase is typical
(v/s)ci = flow ratio for critical lane group, i (seconds)
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Winter 2006
Xc = critical v/c ratio for the intersection
Optimum Cycle Length Estimation
C opt
1.5L   5

n
v
1  
i 1  s  ci
Copt = estimated optimum cycle length (seconds) to
minimize vehicle delay
L = total lost time per cycle (seconds), 4 seconds per
phase is typical
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Winter 2006
(v/s)ci = flow ratio for critical lane group, i (seconds)
Green Time Estimation
 v   C 
gi     
 s i  X i 
g = effective green time for phase, i (seconds)
(v/s)i = flow ratio for lane group, i (seconds)
C = cycle length (seconds)
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Winter 2006
Xi = v/c ratio for lane group i
Pedestrian Crossing Time
N ped
L 
G p  3.2 
  2.7
Sp 
WE

 for WE  10 ft.

L
G p  3.2 
 0.27 N ped  for WE  10 ft.
Sp
Gp = minimum green time required for pedestrians (seconds)
L = crosswalk length (ft)
Sp = average pedestrian speed (ft/s) – often assumed 4 ft/s
WE = effective crosswalk width (ft)
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3.2 = pedestrian startup time (seconds)
Nped = number of pedestrians crossing during an interval
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Effective Width (WE)
from Highway Capacity Manual 2000
Example
An intersection operates using a
simple 3-phase design as
pictured.
SB
WB
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EB
Phase
Lane
group
Saturation Flows
1
SB
3400 veh/hr
2
NB
3400 veh/hr
3
EB
1400 veh/hr
WB
1400 veh/hr
NB
Example
What is the sum of the flow ratios for the critical lane groups?
What is the total lost time for a signal cycle assuming 2 seconds of
clearance lost time and 2 seconds of startup lost time per phase?
SB
150
30
400
EB
30
200
300
20
1000
50
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100
NB
WB
Example
Calculate an optimal signal timing (rounded up to the nearest 5
seconds) using Webster’s formula.
Copt 
1.5L   5
n
1   v s ci
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i 1
Example
Determine the green times allocation using v/c equalization.
Assume the extension of effective green time = 2 seconds and
startup lost time = 2 seconds.
v
  C

i 1  s  i
Xc 
CL
n
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Winter 2006
 v   C 
gi     
 s i  X i 
Example
What is the intersection Level of Service (LOS)? Assume in all
cases that PF = 1.0, k = 0.5 (pretimed intersection), I = 1.0 (no
upstream signal effects).
d v

v
i i
dA
i
i
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Winter 2006
i
d v

v
k
dI
k
k
k
k
Example
Is this signal adequate for pedestrians? A pedestrian count showed
5 pedestrians crossing the EB and WB lanes on each side of the
intersection and 10 pedestrians crossing the NB and SB
crosswalks on each side of the intersection. Lanes are 12 ft. wide.
The effective crosswalk widths are all 10 ft.
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L
G p  3.2 
 0.27 N ped  for WE  10 ft
Sp
FYI – NOT TESTABLE
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Winter 2006
Signal Installation: “Warrants”
•
Manual of Uniform Traffic Control
Devices (MUTCD)
•
Apply these rules to determine if a
signal is “warranted” at an
intersection
•
If warrants are met, doesn’t mean
signals or control is mandatory
•
8 major warrants
•
Multiple warrants usually required
for recommending control
http://mutcd.fhwa.dot.gov/
FYI – NOT TESTABLE
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Winter 2006
Intersection Control Type
from Highway Capacity Manual 2000
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Primary References
•
Mannering, F.L.; Kilareski, W.P. and Washburn, S.S. (2003). Principles
of Highway Engineering and Traffic Analysis, Third Edition (Draft).
Chapter 7
•
Transportation Research Board. (2000). Highway Capacity Manual.
National Research Council, Washington, D.C.