page_title: Traffic flow
section_title: The bottleneck model
hierarchical_section_title: Traffic flow / Cumulative vehicle count curves (N-curves) / Applications / The bottleneck model
caption_attribution_description: English: Roadway section experiencing a bottleneck.
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: One application of the N-curve is the bottleneck model, where the cumulative vehicle count is known at a point before the bottleneck (i.e. this is location X₁). However, the cumulative vehicle count is not known at a point after the bottleneck (i.e. this is location X₂), but rather only the capacity of the bottleneck, or the discharge rate, μ, is known. The bottleneck model can be applied to real-world bottleneck situations such as those resulting from a roadway design problem or a traffic incident.
  Take a roadway section where a bottleneck exists such as in figure 12. At some location X₁ before the bottleneck, the arrivals of vehicles follow a regular N-curve. If the bottleneck is absent, the departure rate of vehicles at location X₂ is essentially the same as the arrival rate at X₁ at some later time (i.e. at time TTFF – free-flow travel time). However, due to the bottleneck, the system at location X₂ is now only able to have a departure rate of μ. When graphing this scenario, essentially we have the same situation as in figure 9, where the arrival curve of vehicles is N₁, the departure curve of vehicles absent the bottleneck is N₂, and the limited departure curve of vehicles given the bottleneck is N′₂. The discharge rate μ is the slope of curve N′₂, and all the same traffic flow characteristics as in figure 11 can be determined from this diagram. The maximum delay and maximum queue length can be found at a point M in figure 13 where the slope of N₂ is the same as the slope of N′₂; i.e. when the virtual arrival rate is equal to the discharge / departure rate μ.
  The N-curve in the bottleneck model may also be used to calculate the benefits in removing the bottleneck, whether in terms of a capacity improvement or removing an incident to the side of the roadway.
caption: Figure 12. Roadway section experiencing a bottleneck
exif: {}