Streetcar Mystery

Copyright ©2006 by Paul Niquette. All rights reserved.

Every car bound for the Brooklyn Bridge 
follows a Prospect Park car by 20 minutes.

Measured in minutes, 'headway' is the interval between successive buses or streetcars or trains departing from a given station.  The Streetcar Mystery puzzle told us that back in 1890, the streetcars on the DeKalb “ran four times per hour.” Putting time in the numerator where it belongs, we conclude that the line must have been operating with “15-minute headways.”

Our courier in the puzzle could have shown up at a given station just in time to catch a streetcar -- or slightly later, just missing it.  Missing a streetcar would mean waiting the full headway time for the next one.  The Streetcar Mystery puzzle said that the courier "boarded the streetcars completely at random."  Sophisticated solvers would estimate that he waited an average of 7.5 minutes for the next streetcar to come along -- half the headway.

Two alternating services operated on the DeKalb tracks.  For patrons with destinations on either of the two branch lines, headways would have been 30 minutes.  With waiting times that long, of course, regular transit patrons and commuters make sure to schedule their arrivals at the platform according to the published time-table.

Inasmuch as all his destinations were along the DeKalb line, the courier did not care which service he used and so he had an average wait of 7.5 minutes.  If he had ever bothered to consult the published time-table, the courier would have noticed that the schedule for the two services were staggered.  Inter-service headways alternated between 10 minutes and 20 minutes.  The narrative in the Streetcar Mystery puzzle neglected to mention that fact, thereby tempting solvers into making an inappropriate assumption.

Why have staggered headways?

Sophisticated solvers may enjoy speculating about the purpose for staggered headways.  Here are a couple of possibilities...

More patrons on the DeKalb line are headed for Prospect Park than for the Brooklyn Bridge.  Both services will carry passengers who, like the courier in the puzzle, are going to disembark before Flatbush.  A longer headway will encourage those "locals" to ride the more lightly loaded cars.

Sophisticated commuters would be recognizable as the ones who routinely arrive at their platforms just seconds before the Brooklyn Bridge-bound car departs, cutting their waiting time to a minimum, secure in the knowledge that the next Prospect Park car will be coming along in only 10 minutes.
Streetcars running between Prospect Park and Brooklyn Bridge share the tracks on Flatbush.  These cars must operate in synchrony with converging and diverging DeKalb cars. Longer inter-service headways on the DeKalb line provide "slots" for cars operating on the Flatbush line.
Sophisticated solvers will observe that each "line" really operates as a loop (think of a conveyer belt) with movements in opposite directions to and from each end-point maintaining the system's operating headway independent of respective trip-times and with the stagger necessarily held constant.
The reciprocal of headway takes the form of a flow-rate (trains per hour, buses per hour, streetcars per hour). Nota bene, we have warned against the practice of putting time in the denominator (see Train Speed or Short Cut?).  Nevertheless, putting headway in the denominator invites a useful analogy to electrical current, which is everywhere the same throughout a circuit.  That comparison is especially appropriate for "fixed guideway" transit systems operating in closed loops.  Thus, headway is the same no matter where in the loop you measure it.

For more insights about headway, see Headway Versus Leeway.


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