1. Signalized Intersections
CEE 320 Steve Muench

2. Outline Key Definitions Baseline Assumptions Control Delay
Signal Analysis
D/D/1
Random Arrivals
LOS Calculation
Optimization

3. Key Definitions (1) Cycle Length (C) Phase Green Time (G) Red Time (R)
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

4. Key Definitions (2) Change Interval (Y) Clearance Interval (AR)
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

5. Key Definitions (3) Start-up Lost Time (l1) Clearance Lost Time (l2)
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.

6. 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

7. Key Definitions (5) Saturation Flow Rate (s) Approach Capacity (c)
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.

8. Key Definitions (6) Flow Ratio Lane Group Critical Lane Group
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

9. from Highway Capacity Manual 2000

10. Baseline Assumptions D/D/1 queuing
Approach arrivals < departure capacity
(no queue exists at the beginning/end of a cycle)

11. 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)

12. D/D/1 Signal Analysis (Graphical)
Departure
Rate
Arrival
Rate
Vehicles
Queue dissipation
Total vehicle delay per cycle
Maximum delay
Maximum queue
Time
Red
Green
Red
Green
Red
Green

13. 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

14. 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)

15. 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 (lc/mg)
g = effective green time (sec)
c = cycle length (sec)

16. Signal Analysis – Random Arrivals
Allsop’s Formula (1972) - empirical
d’ = avg. veh delay assuming random arrivals
d = avg. veh delay assuming uniform arrivals (D/D/1)
x = ratio of arrivals to departures (lc/mg)

17. 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)

18. Signalized Intersection LOS
Based on control delay per vehicle
How long you wait, on average, at the stop light
from Highway Capacity Manual 2000

19. 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 =
Average signal delay per vehicle in s/veh PF
progression adjustment factor
d1, d2, d3
as defined above

20. Uniform Delay (d1) 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

21. Incremental Delay (d2) 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 = 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
When intersection v/c approaches 1.0 then an actuated controller will tend to behave like a pretimed one

22. 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

23. 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

24. 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

25. Minimum Cycle Length 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
If you truly want to minimize cycle length then set Xc = 1.0, which means that your critical v/c will be 1 and you can just squeeze all the vehicles through on that phase’s green time. However, due to the stochastic nature of arrivals, if you set Xc = 1 then there will be times when more arrivals than your assumed v will show up and the cycle will fail (not all vehicles will be let through on a particular green). Therefore, often values less than 1 are assumed for Xc (such as 0.90).

26. Optimum Cycle Length Estimation
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)
This is only one estimation
Values between 0.75Copt and 1.5 Copt give similar delay times

27. Green Time Estimation 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

28. Pedestrian Crossing TimeGp =
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
Assumes 15th percentile walking speed of pedestrians is 4 ft/s

29. Effective Width (WE)
from Highway Capacity Manual 2000

30. Example
An intersection operates using a simple 3-phase design as pictured. SB WB EB
Phase
Lane group
Saturation Flows 1 SB
3400 veh/hr 2 NB 3 EB
1400 veh/hr WB NB

31. 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 NB EB WB 30
150 50
400
100
1000
200
300 20
PHASE 1
SB T/left/right= ( )/3400 = 0.171
PHASE 2
NB T/left/right = ( )/3400 = 0.338
PHASE 3
EB T/right = (200+20)/1400 = 0.157
WB T/right = (300+30)/1400 =  limiting since v/s is highest
Yc = = 0.745
Total lost time = 3(2+2) = 12 seconds

32. Example
Calculate an optimal signal timing (rounded up to the nearest 5 seconds) using Webster’s formula.
Copt = 1.5(12 seconds) + 5/( ) = 90.2 seconds = 95 seconds (rounded up to nearest 5 seconds)

33. 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.
DETERMINE Xc
Xc = 0.745(95)/(95 – 12) = 0.853
CALCULATE EFFECTIVE GREEN TIMES
gSB = (95/0.853) = seconds
gNB = 0.338(95/0.853) = seconds
gEBWB = (95/0.853) = seconds
CHECK
= = 95 seconds
ACTUAL GREEN TIMES
In this case, they are the same as g
G = g+e-l1 = g = g

34. 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).
Determine the delay for each lane group
SB lane group
c = s (g/C) = 3200(19.04/95) = vehicles
d1 = (0.5)(95)(1 – 19.04/95)/(1 – 0.853(19.04/95)) = seconds
d2 = 900(0.25)(( ) + sqrt((0.853 – 1)2 + 8(0.5)(1.0)(0.853)/((641.35)(0.25))) = seconds
d3 = 0 (assumed)
d = = 59.33
NB lane group
c = s (g/C) = 3200(37.64/95) = vehicles
d1 = (0.5)(95)(1 – 37.64/95)/(1 – 0.853(37.64/95)) = seconds
d2 = 900(0.25)(( ) + sqrt((0.853 – 1)2 + 8(0.5)(1.0)(0.853)/(( )(0.25))) = 7.41 seconds
d = = 54.91
EB lane group
c = s (g/C) = 1400(26.28/95) = vehicles
d1 = (0.5)(95)(1 – 26.28/95)/(1 – 0.853(26.28/95)) = seconds
d2 = 900(0.25)(( ) + sqrt((0.853 – 1)2 + 8(0.5)(1.0)(0.853)/((387.28)(0.25))) = seconds
d = = 65.54
WB lane group
Find the weighted average of delay for the four lane groups
dI = ((59.33)(580) (1150) (220) (330))/( ) = seconds
From Table 7.4 this equates to LOS E (not very good)

35. 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.
EB/WB
Gp = / (5) = seconds
NB/SB
Gp = / (10) = seconds OK

36. Signal Installation: “Warrants”
FYI – NOT TESTABLE
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
Warrant 1, Eight-Hour Vehicular Volume. Warrant 2, Four-Hour Vehicular Volume. Warrant 3, Peak Hour. Warrant 4, Pedestrian Volume. Warrant 5, School Crossing. Warrant 6, Coordinated Signal System. Warrant 7, Crash Experience. Warrant 8, Roadway Network.

37. Intersection Control Type
FYI – NOT TESTABLE
Intersection Control Type
from Highway Capacity Manual 2000

38. 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.