page_title: Traffic flow
section_title: Cumulative vehicle count curves (N-curves)
hierarchical_section_title: Traffic flow / Cumulative vehicle count curves (N-curves)
caption_attribution_description: English: Arrival, virtual arrival and departure curves
context_page_description: In mathematics and transportation engineering, traffic flow is the study of interactions between travellers and infrastructure, with the aim of understanding and developing an optimal transport network with efficient movement of traffic and minimal traffic congestion problems.
context_section_description: A cumulative vehicle count curve, the N-curve, shows the cumulative number of vehicles that pass a certain location x by time t, measured from the passage of some reference vehicle. This curve can be plotted if the arrival times are known for individual vehicles approaching a location x, and the departure times are also known as they leave location x. Obtaining these arrival and departure times could involve data collection: for example, one could set two point sensors at locations X₁ and X₂, and count the number of vehicles that pass this segment while also recording the time each vehicle arrives at X₁ and departs from X₂. The resulting plot is a pair of cumulative curves where the vertical axis (N) represents the cumulative number of vehicles that pass the two points: X₁ and X₂, and the horizontal axis (t) represents the elapsed time from X₁ and X₂.
  If vehicles experience no delay as they travel from X₁ to X₂, then the arrivals of vehicles at location X₁ is represented by curve N₁ and the arrivals of the vehicles at location X₂ is represented by N₂ in figure 8. More commonly, curve N₁ is known as the arrival curve of vehicles at location X₁ and curve N₂ is known as the arrival curve of vehicles at location X₂. Using a one-lane signalized approach to an intersection as an example, where X₁ is the location of the stop bar at the approach and X₂ is an arbitrary line on the receiving lane just across of the intersection, when the traffic signal is green, vehicles can travel through both points with no delay and the time it takes to travel that distance is equal to the free-flow travel time. Graphically, this is shown as the two separate curves in figure 8.
  However, when the traffic signal is red, vehicles arrive at the stop bar (X₁) and are delayed by the red light before crossing X₂ some time after the signal turns green. As a result, a queue builds at the stop bar as more vehicles are arriving at the intersection while the traffic signal is still red. Therefore, for as long as vehicles arriving at the intersection are still hindered by the queue, the curve N₂ no longer represents the vehicles’ arrival at location X₂; it now represents the vehicles’ virtual arrival at location X₂, or in other words, it represents the vehicles' arrival at X₂ if they did not experience any delay. The vehicles' arrival at location X₂, taking into account the delay from the traffic signal, is now represented by the curve N′₂ in figure 9.
  However, the concept of the virtual arrival curve is flawed. This curve does not correctly show the queue length resulting from the interruption in traffic (i.e. red signal). It assumes that all vehicles are still reaching the stop bar before being delayed by the red light. In other words, the virtual arrival curve portrays the stacking of vehicles vertically at the stop bar. When the traffic signal turns green, these vehicles are served in a first-in-first-out (FIFO) order. For a multi-lane approach, however, the service order is not necessarily FIFO. Nonetheless, the interpretation is still useful because of the concern with average total delay instead of total delays for individual vehicles.
caption: Figure 9. Arrival, virtual arrival, and departure curves
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