CE

E 3

20W

inte

r 20

06

Signalized Intersections

CEE 320Steve Muench

CE

E 3

20W

inte

r 20

06

Outline

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

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

CE

E 3

20W

inte

r 20

06

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)– The period in the signal cycle during which, for a given

phase or lane group, the signal is red

CE

E 3

20W

inte

r 20

06

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

CE

E 3

20W

inte

r 20

06

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

• Total Lost Time (L)– Total lost time per cycle during which the intersection is not

used by any movement.

CE

E 3

20W

inte

r 20

06

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)– Time during which a movement is effectively not

permitted to move. – r = R + tL

– r = C – g

CE

E 3

20W

inte

r 20

06

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)– 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.

CE

E 3

20W

inte

r 20

06

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

from

Hig

hway

Cap

acity

Man

ual 2

000

CE

E 3

20W

inte

r 20

06

Baseline Assumptions

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

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

CE

E 3

20W

inte

r 20

06

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)

CE

E 3

20W

inte

r 20

06

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

CE

E 3

20W

inte

r 20

06

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

µλρ = 0.1<ρ

( )ρρ−

=10

rt

c

trPq

0+=

( )( ) qs P

c

tr

gr

trP =+=

++= 00

λλ ( )

( ) c

t

c

t

gr

trPs ρλ

µλλ 000 ==

++=

CE

E 3

20W

inte

r 20

06

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)

µλρ = 0.1<ρ

rQm λ=

( )ρλ

−=

12

2rDt

( ) ( )ρλρλ

−=×

−=

12

1

12

22

c

r

c

rDt

rdm =

CE

E 3

20W

inte

r 20

06

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 (λc/µg)g = effective green time (sec)

c = cycle length (sec)

( ))/(52

3/1

2

2

65.012

' cgxc

x

xdd +

−

−+=

λλ

CE

E 3

20W

inte

r 20

06

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 (λc/µg)

( )

−

+=x

xdd

1210

9'

2

λ

CE

E 3

20W

inte

r 20

06

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)

CE

E 3

20W

inte

r 20

06

Signalized Intersection LOS

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

from Highway Capacity Manual 2000

CE

E 3

20W

inte

r 20

06

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.

( ) 321 ddPFdd ++=

d = Average signal delay per vehicle in s/veh

PF = progression adjustment factor

d1, d2, d3 = as defined above

CE

E 3

20W

inte

r 20

06

Uniform Delay (d1)

( )

−

−

=

Cg

X

Cg

Cd

,1min1

15.0

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

CE

E 3

20W

inte

r 20

06

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

CE

E 3

20W

inte

r 20

06

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

CE

E 3

20W

inte

r 20

06

Control Optimization

• Conflicting Operational Objectives– minimize vehicle delay– minimize vehicle stops– minimize lost time– major vs. minor service (progression)– pedestrian service– reduce accidents/severity– reduce fuel consumption– Air pollution

CE

E 3

20W

inte

r 20

06

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– Reduced capacity (increased lost time)– Reduced delay for any given movement

CE

E 3

20W

inte

r 20

06

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

CE

E 3

20W

inte

r 20

06

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)

CE

E 3

20W

inte

r 20

06

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

CE

E 3

20W

inte

r 20

06

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) – often assumed 4 ft/s

WE = effective crosswalk width (ft)

3.2 = pedestrian startup time (seconds)

Nped = number of pedestrians crossing during an interval

CE

E 3

20W

inte

r 20

06

Effective Width (WE)

from Highway Capacity Manual 2000

CE

E 3

20W

inte

r 20

06

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

CE

E 3

20W

inte

r 20

06

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? 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?

CE

E 3

20W

inte

r 20

06

ExampleCalculate an optimal signal timing (rounded up to the nearest 5 seconds) using Webster’s formula.

( )( )∑

=

−

+= n

ici

opt

sv

LC

1

1

55.1

CE

E 3

20W

inte

r 20

06

ExampleDetermine the green times allocation using v/c equalization. Assume the extension of effective green time = 2 seconds and startup lost time = 2 seconds.

=

iii X

C

s

vg

LC

Csv

X

n

i ic −

×

=∑

=1

CE

E 3

20W

inte

r 20

06

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).

∑∑

=

ii

iii

A v

vdd

∑∑

=

kk

kkk

I v

vdd

CE

E 3

20W

inte

r 20

06

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

CE

E 3

20W

inte

r 20

06

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

CE

E 3

20W

inte

r 20

06

Intersection Control Type

from Highway Capacity Manual 2000

FYI – NOT TESTABLE

CE

E 3

20W

inte

r 20

06

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.