Do You Know Your Relative Driving Speed?

Abstract

Drivers on a busy highway or freeway typically select a driving speed based not only on the posted speed limit, but also on the velocities of nearby vehicles. Many drivers attempt to stay at or near the flow of traffic, while others prefer to go a bit faster or slower. Traffic safety engineers have stated that the safest speed to travel on a busy freeway is at the 85th percentile of traffic speeds. A natural question arises: How can individuals gauge their speed percentiles from observing traffic in the vicinity? In this note, I utilize a simple idealized model for traffic flow: Assume that each vehicle travels at a constant speed and that the locations of vehicles and their speeds are described by what is called a marked Poisson process. This means that vehicles are randomly spaced along the highway, and that their speeds are independent of their locations and of all other vehicles' speeds and locations. Assume also that the distri bution of traffic speeds (the marks) has density function f(x), defined for speeds x > 0, continuous and positive on its support and not changing over time. That is, the number of vehicles per mile of road traveling between speeds x and x + Ax is ap proximately proportional to f(x) Ax, when Ax is small. For example, if f(x) is the uniform density on some interval [a, b], then we expect an equal number of vehicles traveling at each speed between a and b. The na?ve estimate of one's percentile rank in the distribution of speeds on the high way is simply the observed proportion of vehicles passed out of the total number that one passes or is passed by. In particular, if the number of vehicles passing is equal to the number of vehicles being passed, then a driver might conclude that (s)he is driving at the median speed. Clevenson, Schilling, Watkins, and Watkins [1] recently showed that this is not the case. In actuality, when the number of vehicles passing equals the number being passed, the driver is, surprisingly, traveling at the mean speed rather than the median. More generally, the driver's speed percentile cannot be obtained merely by counting the vehicles passing and being passed. This article explores the relationship between the na?ve estimate, based on counting vehicles passing or being passed, and the actual speed percentile rank of a driver under the assumed model. We show that a driver traveling at a relatively slow or relatively fast speed will, (perhaps subconsciously) using the na?ve estimate, judge his or her speed to be in a more extreme percentile than is actually the case. For instance, a person driving at the 85th percentile may perceive that (s)he is driving in an even higher speed percentile. To begin, suppose that a particular vehicle V is traveling at speed s. Then the ac tual speed percentile of this vehicle's speed under the assumed model is p = F(s) = fo /( *) dx, while the driver's observed speed percentile is equal to the proportion of passed vehicles out of all other vehicles seen (either passing V or passed by V). Since vehicles at speed x will be encountered at a rate proportional to both the number of