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The Driver's "Rule of Thumb"
Understanding the Implications


Prof. David Bernstein
James Madison University

Computer Science Department
bernstdh@jmu.edu

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Motivation
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  • Recall:
    • When you learned to drive you were told to leave one car length between your car and the car in front of you for every 10mph of velocity
  • A Question:
    • What are some of the implications of this rule
Definitions and Notation
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  • Speed/Velocity:
    • The common speed/velocity of the vehicles in mi/hr (denoted by \(v\))
  • Density:
    • The number of vehicles on a piece of a road in veh/mi (denoted by \(k\))
  • Flow:
    • The number of vehicles that pass a point on a road in veh/hr (denoted by \(q\))
Definitions and Notation (cont.)
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  • Spacing:
    • The distance between vehicles at (at a point in time) in mi/veh (denoted by \(s\))
  • Headway:
    • The amount of time between vehicles (i.e., the time between when two vehicles pass a point on the road) in hr/veh (denoted by \(h\))
Some Basic Calculations
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  • Assumptions:
    • A car is 15ft long (i.e., 15/5280 = 0.002840909mi)
    • All cars are traveling at the same speed
    • All cars are obeying the "rule of thumb"
  • An Example:
    • \(v = 30\text{mi/hr}\)
    • \(s = 0.002840909\text{mi/veh} \cdot (30/10) = 0.008522727 \text{mi/veh}\)
    • \(k = 1/s = 117.33\text{veh/mi}\)
    • \(h = s/v = 0.008522727 / 30 = .0002840909\text{hr/veh}\)
Headway vs. Velocity
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  • Expanding:
    • \(h = \frac{s}{v} = \frac{0.002840909 \cdot (v/10)}{v} = 0.0002840909\)
  • The Interpretation:
    • The headway doesn't change with the speed (i.e., the increase in spacing is offset by the increase in speed)
Density vs. Velocity
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  • Expanding:
    • \(k = \frac{1}{s} = \frac{1}{0.002840909 \cdot (v/10)} = \frac{1}{0.0002840909} \cdot \frac{1}{v} = \frac{3520.0}{v}\)
  • The Interpretation:
    • As the velocity increases the density decreases
Reinterpretations by an Outside Observer
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  • Velocity vs. Headway:
    • The headway has no impact on velocity
  • Velocity vs. Density:
    • An increase in density results in a decrease in velocity (which is commonly called congestion)
    • For the parameters above: \(v = \frac{3520.0}{k}\)
Congestion
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Using the Same Assumptions as Above

images/VelocityVsDensity_RuleOfThumb.png
Congestion (cont.)
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What About Travel Times?

images/TravelTimeVsDensity_RuleOfThumb.png
Congestion (cont.)
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  • Bureau of Public Roads (1950s):
    • \(t = e \cdot \left(1.0 + 0.15 \cdot \left(\frac{n}{Z}\right)^4 \right)\)
    • where \(Z\) is the capacity and \(n\) is the "number" of vehicles on the road
  • Other Models:
    • Mosher (1963)
    • Davidson (1966)
    • Akcelik (1978, 1980, 1981)
Nerd Humor?
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Question: How fast can you hit a speed bump while driving and live?

Answer: Courtesy of xckd

There's Always More to Learn
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