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.

An approach is generalized for enhancing a standard input-output diagram to represent graphically the time and distance that vehicles spend in a queue upstream of a bottleneck. The approach requires the construction of a curve depicting the cumulative number of vehicles to have reached the back of the queue as a function of time. The original technique, described in a previous paper, is reviewed for bottlenecks with constant capacity and for those where capacity changes once. The approach is then generalized to allow multiple changes in bottleneck capacity, and the original assumption of a triangular flow-density ( q-k) relation is relaxed to one that is piecewise-linear concave. Although it is consistent with the kinematic wave theory of traffic flow, the proposed approach is simpler to apply to complex problems because it avoids the laborious construction of a time-space diagram. It allows the estimation of several measures required in the evaluation of the impacts of bottlenecks, including the (accurate) number of vehicles in queue and the physical extents of queues at any time and the total time spent by vehicles in different traffic states.