page_title: Traffic flow
section_title: Tandem queues
hierarchical_section_title: Traffic flow / Cumulative vehicle count curves (N-curves) / Applications / Tandem queues
caption_attribution_description: English: N-curve with two Bottlenecks
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: As introduced in the section above, the N-curve is an applicable model to estimate traffic delay during time by setting arrival and departure cumulative counting curve. Since the curve can represent various traffic characteristics and roadway conditions, the delay and queue situations under these conditions will be able to be recognized and modeled using N-curves. Tandem queues occur when multiple bottlenecks exist between the arrival and departure locations. Figure 14 shows a qualitative layout of a tandem-queue roadway segment with a certain initial arrival. The bottlenecks along the stream have their own capacity, 'μᵢ [veh/time], and the departure is defined at the downstream end of the entire segment.
  To determine the ultimate departure, D(t), it can be an available method to research on the individual departures, Dᵢ(t). As shown in the Figure 15, if the free-flow travel-time is neglected, the departure of BNᵢ−1 will be the virtual arrival of BNᵢ, which can also be presented as Dᵢ−1(t) = Aᵢ(t). Thus, the N-curve of a roadway with two bottlenecks (minimum number of BNs along a tandem-queue roadway) can be developed as Figure 15 with μ₁ < μ₂. In this case, D₂(t) will be the ultimate departure of this 2-BN tandem-queue roadway.
  Regarding of a tandem-queue roadway having 3 BNs with μ₁ < μ₂, if μ₁ < μ₂ < μ₃, similarly as the 2-BN case, D₃(t) will be the ultimate departure of this 3-BN tandem-queue roadway. If, however, μ₁ < μ₃ < μ₂, D₂(t) will then still be the ultimate departure of the 3-BN tandem-queue roadway. Thus, it can be summarized that, the departure of the bottleneck with the minimum capacity will be the ultimate departure of the entire system, regardless of the other capacities and the number of bottlenecks. Figure 16 shows a general case with n BNs.
  The N-curve model describing above represents a significant characteristic of the tandem-queue systems, which is that the ultimate departure only depends on the bottleneck with the minimum capacity. In a practical perspective, when the resources (economy, effort, etc.) of the investment on tandem-queue systems are limited, the investment can mainly focus on the bottleneck with the worst condition.
caption: Figure 15. N-Curve of Tandem Queues with Two BNs
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