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Signalized Intersections. CEE 320 Steve Muench. Outline. Key Definitions Baseline Assumptions Control Delay Signal Analysis 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

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signalized intersections

Signalized Intersections

CEE 320Steve Muench

  • Key Definitions
  • Baseline Assumptions
  • Control Delay
  • Signal Analysis
    • D/D/1
    • Random Arrivals
    • LOS Calculation
    • Optimization
key definitions 1
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
key definitions 2
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
key definitions 3
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.
key definitions 4
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
key definitions 5
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.
key definitions 6
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
baseline assumptions
Baseline Assumptions
  • D/D/1 queuing
  • Approach arrivals < departure capacity
    • (no queue exists at the beginning/end of a cycle)
quantifying control delay
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)
d d 1 signal analysis graphical
D/D/1 Signal Analysis (Graphical)






Queue dissipation

Total vehicle delay per cycle

Maximum delay

Maximum queue








d d 1 signal analysis numerical
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
d d 1 signal analysis numerical14
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)
signal analysis random arrivals
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)

signal analysis random arrivals16
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)

definition level of service los
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)
signalized intersection los
Signalized Intersection LOS
  • Based on control delay per vehicle
    • How long you wait, on average, at the stop light

from Highway Capacity Manual 2000

typical approach
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.
initial queue delay d 3
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
control optimization
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
the art of signal optimization
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
effective width w e
Effective Width (WE)

from Highway Capacity Manual 2000



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




















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?


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


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


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).


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.

signal installation warrants


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

intersection control type


Intersection Control Type

from Highway Capacity Manual 2000

primary references
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.