1.2: Rails vs Roads

We are now warmed up and ready to use divide-and-conquerreasoning for more substantial estimates. Our next estimate, concerning traffic, comes to mind whenever I drive the congested roads to JFK Airport in New York City. The route goes on the Van Wyck Expressway, which was planned by Robert Moses. As Moses’s biographer Robert Caro describes [6, pp. 904ff], when Moses was in charge of building the expressway, the traffic planners recommended that, in order to handle the expected large volume of traffic, the road include a train line to the then-new airport. Alternatively, if building the train track would be too expensive, they recommended that the city, when acquiring the land for the road, still take an extra 50 feet of width and reserve it as a median strip for a train line one day. Moses also rejected the cheaper proposal. Alas, only weeks after its opening, not long after World War Two, the rail-free highway had reached peak capacity.

Let’s use our divide-and-conquer tool to compare, for rush-hour commuting, the carrying capacities ofrail and road. The capacity is the rate at which passengers are transported; it is passengers per time. First we’ll estimate the capacity of one lane of highway. We can use the 2-second-following rule taught in many driving courses. You are taught to leave 2 seconds of travel time between you and the car in front. When drivers follow this rule, a single lane of highway carries one car every 2 seconds. To find the carrying capacity, we also need the occupancy of each car. Even at rush hour, at least in the United States, each car carries roughly one person. (Taxis often have two people including the driver, but only one person is being transported to the destination.) Thus, the capacity is one person every 2 seconds. As an hourly rate, the capacity is 1800 people per hour:

\[\frac{1 \textrm{person}}{2 \cancel{s}} \times \frac{3600 \cancel{s}}{1hr} = \frac{1800 \textrm{people}}{hr}\]

The diagonal strike-through lines help us to spot which units cancel and to check that we end up with just the units that we want (people per hour).

This rate, 1800 people per hour, is approximate, because the 2-second following rule is not a law of nature. The average gap might be 4 seconds late at night, 1 second during the day, and may vary from day to day or from highway to highway. But a 2-second gap is a reasonable compromise estimate. Replacing the complex distribution of following times with one time is an application of lumping—the tool discussed in Chapter 6. Organizing complexity almost always reduces detail. If we studied all highways at all times of day, the data, were we so unfortunate as to obtain them, would bury any insight.

How does the capacity of a single lane of highway compare with the capacity of a train line?

For the other half of the comparison, we’ll estimate the rush-hour capacity of a train line in an advanced train system, say the French or German system. As when we estimated the volume of a dollar bill (Section 1.1), we divide the estimate into manageable pieces: how often a train runs on the track, how many cars are in each train, and how many passengers are in each car. Here are my armchair estimates for these quantities, kept slightly conservative to avoid overestimating the train-line’s capacity. A single train car, when full at rush hour, may carry 150 people. A rush-hour train may consist of 20 cars. And, on a busy train route, a train may run every 10 minutes or six times per hour. Therefore, the train line’s capacity is 18 000 people per hour:

\[\frac{150 \textrm{people}}{\cancel{\textrm{car}}} \times \frac{2 \cancel{\textrm{cars}}}{\cancel{\textrm{train}}} \times \frac{6 \cancel{\textrm{trains}}}{\textrm{hr}} = \frac{18000 \textrm{people}}{\textrm{hr}}\]

This capacity is ten times the capacity of a single fast-flowing highway lane. And this estimate is probably on the low side; Robert Caro [6, p. 901] gives an estimate of 40000 to 50000 people per hour. Using our lower rate, one train track in each direction could replace two highways even if each highway had five lanes in each direction.