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Course Logistics

• HW7 due today (9 total)• Midterm next Friday (Wednesday review)• Signalized Intersections (Chapter 7 of text)• Last material before midterm• Final set of topics: transportation planning

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Signalized Intersections

CEE 320Anne Goodchild

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Outline

1. Key Definitions 2. Baseline Assumptions3. Control Delay4. Signal Analysis

a. D/D/1b. Random Arrivalsc. LOS Calculationd. Optimization

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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 (a consistent period)

• Green Time (G)– The duration of the green indication of a given

movement at a signalized intersection

• Red Time (R)– The period in the signal cycle during which, for a given

phase or lane group, the signal is red

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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)– All red time– The period in the signal cycle during which all

approaches have a red indication

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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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Key Definitions (4)

• Effective Green Time (g)– Time effectively utilized for movement

– g = G + Y + AR – tL

• Effective Red Time (r)– Time during which a movement is effectively

not permitted to move.

– r = R + tL

– r = C – g

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Red Green Yellow Red

C

Y RG

AR

l1

headways typically

longer

saturation headway

l2

time

space

end of intersection

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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 (vehicles/hour) = 3600/headway (seconds per vehicle)

• Approach Capacity (c)– Saturation flow times the proportion of

effective green– c = s × g/C

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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

• 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

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Baseline Assumptions

• D/D/1 queuing• Approach arrivals < departure capacity

– (no queue exists at the beginning/end of a cycle)

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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)

– Average delay per vehicle (seconds per vehicle)

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D/D/1 Signal Analysis (Graphical)

ArrivalRate

DepartureRate

Time

Ve

hicl

es

Maximum delay

Maximum queue

Total vehicle delay per cycle

Red Red RedGreen Green Green

Queue dissipation

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D/D/1 Signal Analysis – Numerical

• Time to queue dissipation after the start of effective green

• Proportion of the cycle with a queue

• Proportion of vehicles stopped

0.1

10

rt

c

trPq

0

qs P

c

tr

gr

trP

00

c

t

c

t

gr

trPs

000

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D/D/1 Signal Analysis – Numerical

• Maximum number of vehicles in a queue

• Total delay per cycle

• Average vehicle delay per cycle

• Maximum delay of any vehicle (assume FIFO)

0.1

rQm

12

2rDt

12

1

12

22

c

r

c

rDt

rdm

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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

• Six levels of service (A through F)

• Based on control delay measure

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Control Delay

• Applies to both signalized and not signalized intersections

• Referred to as signal delay for a signalized intersection

• Total delay experienced by the driver as a result of the control

• Includes deceleration time, queue move-up time, stop time, and acceleration time

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Signal Analysis – Random Arrivals

• Webster’s Formula (1958) - empirical

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)

g = effective green time (sec)

c = cycle length (sec)

)/(52

3/1

2

2

65.012

' cgxc

x

xdd

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Signal Analysis – Random Arrivals

• Allsop’s Formula (1972) - empirical

d’ = avg. veh delay assuming random arrivalsd = avg. veh delay assuming uniform arrivals

(D/D/1)x = ratio of arrivals to departures (c/g)

x

xdd

1210

9'

2

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Signalized Intersection LOS

• Based on control delay per vehicle– How long you wait, on average, at the stop light

from Highway Capacity Manual 2000

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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).

321 ddPFdd

d = Average signal delay per vehicle in s/veh

PF = progression adjustment factor

d1, d2, d3 = as defined above

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Uniform Delay (d1)

Cg

X

Cg

Cd

,1min1

15.02

1

d1 = delay due to uniform arrivals (s/veh)

C = cycle length (seconds)

g = effective green time for lane group (seconds)

X = v/c ratio for lane group

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Incremental Delay (d2)

cT

kIXXXTd

811900 2

2

d2 = 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

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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…– d3 = 0

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Control Optimization

• Conflicting Operational Objectives

– minimize vehicle delay• Fuel consumption• Air quality

– minimize vehicle stops– minimize lost time– major vs. minor service (progression)– pedestrian service– reduce accidents/severity

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The “Art” of Signal Optimization

• Long Cycle Length– High capacity (reduced lost time)– High delay on movements that are not served– Less efficient if uneven demand

• Short Cycle Length– Reduced capacity (increased lost time)– Reduced delay for any given movement– More efficient if equal demand

• “snappy” operations

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Minimum Cycle Length

n

i cic

c

s

vX

XLC

1

min

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)

Xc = critical v/c ratio for the intersection

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Minimum cycle length

• set Xc = 1.0• critical v/c will be 1

– you can just squeeze all the vehicles through on that phase’s green time

• However, if you set Xc = 1 – there will be times when more arrivals than

your assumed v will show up and the cycle will fail

• Therefore, often values less than 1 are assumed for Xc (such as 0.90).

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Optimum Cycle Length Estimation

n

i ci

opt

s

v

LC

1

1

55.1

Copt = estimated optimum cycle length (seconds) to minimize vehicle delay

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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Green Time Estimation

iii X

C

s

vg

g = effective green time for phase, i (seconds)

(v/s)i = flow ratio for lane group, i (seconds)

C = cycle length (seconds)

Xi = v/c ratio for lane group i

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Pedestrian Crossing Time

ft. 10for 7.22.3

E

E

ped

pp W

W

N

S

LG

ft. 10for 27.02.3 Epedp

p WNS

LG

Gp = minimum green time required for pedestrians (seconds)

L = crosswalk length (ft)

Sp = average pedestrian speed (ft/s) – assumed 4 ft/s

WE = effective crosswalk width (ft)

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

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Examples

Signalized Intersections

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ExampleAt an intersection, saturation headways of 3 seconds are observed.

What is the saturation flow rate?

s=3600/3=1200 vehicles/hour

NB

SB

EB

WB

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ExampleIf GNB= 20 seconds, YNB= 3 seconds, RNB= 18 seconds, and AR= 2 seconds. What are the effective red and green times? The cycle time, and the lane-group capacity?

g=20+3+2+4=29 seconds

r=18+4=22 seconds

C=22+29=51 secondsc=1200*29/51=682 vehicles/hour

NB

SB

EB

WB

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D/D/1 Signal Analysis (Graphical)

ArrivalRate

DepartureRate

Time

Ve

hicl

es

Maximum delay

Maximum queue

Total vehicle delay per cycle

Red Red RedGreen Green Green

Queue dissipation

Assumes g>C

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Determine total vehicle delay over 3 cycles if arrival rate is 500 veh/hr

• Arrival rate: 500/3600=.139 veh/sec• Departure rate: 1200/3600=.333

vehicles/second• Traffic intensity = .139/.333 = .417

• Check capacity exceed arrivals:– .333x29=9.65 vehicles can get through on

green– .139x51=7.09 vehicles arrive in cycle

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Total vehicle delay

• One cycle – 0.139*(22)2/2(1-.417)=57.5 vehicle seconds

• Three cycles = 173 vehicle seconds

12

2rDt

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ExampleAn intersection operates using a simple 3-phase design as pictured.

NB

SB

EB

WB

Phase Lane group

Saturation Flows

1 SB 3400 veh/hr

2 NB 3400 veh/hr

3 EB 1400 veh/hr

WB 1400 veh/hr

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Example

SB

NB

EB

WB

30

150

50

30400

1001000

200

30020

What is the sum of the flow ratios for the critical lane groups (Yc)? 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?

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Key Definitions

• Flow Ratio: The ratio of actual flow rate (v) to saturation flow rate (s) for a lane group

• Critical Lane Group: The lane group that has the highest flow ratio (v/s) for a given signal phase

• Volume to Capacity ratio (X): v/c or C/g

• Critical Volume-to-Capacity Ratio (Xc): The proportion of available intersection capacity used by vehicles in critical lane groups

• Sum of the Flow Ratios for the Critical Lane Groups (Yc): sum of flow ratios for critical lane groups

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Example

• PHASE 1: SB T/left/right= (400+150+30)/3400 = 0.171

• PHASE 2: NB T/left/right = (1000+100+50)/3400 = 0.338

• PHASE 3: – EB T/right = (200+20)/1400 = 0.157– WB T/right = (300+30)/1400 = 0.236 limiting since v/s

is highest

• Yc=0.171 + 0.338 + 0.236 = 0.745

• Total lost time = 3(2+2) = 12 seconds

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ExampleCalculate an optimal cycle length using Webster’s formula, and a minimum cycle length.

n

ici

opt

sv

LC

1

1

55.1

Copt = 1.5(12 seconds) + 5/(1-0.745) = 90.2 seconds

n

i cic

c

s

vX

XLC

1

min

Cmin = (12 seconds) (0.9)/ (0.9-0.745) = 69.7 seconds

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Minimum cycle length

• set Xc = 1.0• critical v/c will be 1

– you can just squeeze all the vehicles through on that phase’s green time

• However, if you set Xc = 1 – there will be times when more arrivals than

your assumed v will show up and the cycle will fail

• Therefore, often values less than 1 are assumed for Xc (such as 0.90).

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ExampleDetermine the green times allocation (assume C=95 seconds)

iii X

C

s

vg

LC

Csv

X

n

i ic

1

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• DETERMINE Xc

• Xc = 0.745(95)/(95 – 12) = 0.853

• CALCULATE EFFECTIVE GREEN TIMES• gSB = 0.171(95/0.853) = 19.04 seconds• gNB = 0.338(95/0.853) = 37.64 seconds• gEBWB = 0.236 (95/0.853) = 26.28 seconds

• CHECK• 19.04 + 37.64 + 26.28 + 12 = 94.96 = 95

seconds

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ExampleWhat 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).

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Signalized Intersection LOS

• Based on control delay per vehicle– How long you wait, on average, at the stop light

from Highway Capacity Manual 2000

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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).

321 ddPFdd

d = Average signal delay per vehicle in s/veh

PF = progression adjustment factor

d1, d2, d3 = as defined above

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Uniform Delay (d1)

Cg

X

Cg

Cd

,1min1

15.02

1

d1 = delay due to uniform arrivals (s/veh)

C = cycle length (seconds)

g = effective green time for lane group (seconds)

X = v/c ratio for lane group

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Incremental Delay (d2)

cT

kIXXXTd

811900 2

2

d2 = 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

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Determine the delay for each lane group

SB lane group• c = s (g/C) = 3200(19.04/95) = 641.35 vehicles

• d1 = (0.5)(95)(1 – 19.04/95)2/(1 – 0.853(19.04/95)) = 36.63 seconds

• d2 = 900(0.25)((0.853-1) + sqrt((0.853 – 1)2 + 8(0.5)(1.0)(0.853)/((641.35)(0.25))) = 13.52 seconds

• d3 = 0 (assumed)

• d = 36.63 + 13.52 + 0 = 50.15 seconds

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NB lane group• c = s (g/C) = 3200(37.64/95) = 1267.87 vehicles

• d1 = (0.5)(95)(1 – 37.64/95)2/(1 – 0.853(37.64/95)) = 26.16 seconds

• d2 = 900(0.25)((0.853-1) + sqrt((0.853 – 1)2 + 8(0.5)(1.0)(0.853)/((1267.87)(0.25))) = 7.41 seconds

• d3 = 0 (assumed)

• d = 26.16 + 7.41 + 0 = 33.57 seconds

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EB lane group• c = s (g/C) = 1400(26.28/95) = 387.28 vehicles

• d1 = (0.5)(95)(1 – 26.28/95)2/(1 – 0.853(26.28/95)) = 32.53 seconds

• d2 = 900(0.25)((0.853-1) + sqrt((0.853 – 1)2 + 8(0.5)(1.0)(0.853)/((387.28)(0.25))) = 20.57 seconds

• d3 = 0 (assumed)

• d = 32.53 + 20.57 + 0 = 53.10 seconds

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WB lane group• c = s (g/C) = 1400(26.28/95) = 387.28 vehicles

• d1 = (0.5)(95)(1 – 26.28/95)2/(1 – 0.853(26.28/95)) = 32.53 seconds

• d2 = 900(0.25)((0.853-1) + sqrt((0.853 – 1)2 + 8(0.5)(1.0)(0.853)/((387.28)(0.25))) = 20.57 seconds

• d3 = 0 (assumed)

• d = 32.53 + 20.57 + 0 = 53.10 secon

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• Find the weighted average (by flow) of delay for the four lane groups

• dI = ((50.15)(580) + 33.57(1150) + 53.10(220) + 53.10(330))/(580 + 1150 + 220 + 330) = 42.50 seconds

• From Table 7.4 this equates to LOS D (not very good)

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ExampleIs 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.

ft 10for 27.02.3 Epedp

p WNS

LG

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• EB/WB

• Gp = 3.2 + 24/4 + 0.27(5) = 10.55 seconds

• NB/SB

• Gp = 3.2 + 48/4 + 0.27(10) = 17.90 seconds

• Shortest green time was 19 seconds, so OK

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Intersection Control Type

from Highway Capacity Manual 2000

FYI – NOT TESTABLE