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This paper describes a simple approach for modifying an
input-output (or queueing) diagram to measure the time and distance spent
by vehicles in a queue in a much simpler and self-serving manner than
using
a time-space diagram. The graphical technique requires construction of a
curve depicting the cumulative number of vehicles to have reached the back
of the queue as a function of time, but as shown herein, the technique can
be easily automated with a spreadsheet. Application of the technique is
shown for the simple case of a constant departure rate from a bottleneck,
and for the slightly more general case of a bottleneck capacity which
changes once, which is demonstrated to be applicable to the study of an
undersaturated traffic signal. In the course of describing the usefulness
of this technique for estimating several measures, including the maximum
length of a physical queue and the time at which this maximum occurs, the
paper clarifies the difference between "delay" at a bottleneck and the
"time spent in queue," which appear to have been confused in some of the
literature.
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