The Driver's "Rule of Thumb"

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

Prof. David Bernstein
James Madison University

Computer Science Department

bernstdh@jmu.edu

Motivation

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

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.)

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

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

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

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

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

Using the Same Assumptions as Above

Congestion (cont.)

What About Travel Times?

Congestion (cont.)

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?

Question: How fast can you hit a speed bump while driving and live?