page_title: Traffic flow
section_title: X-models
hierarchical_section_title: Traffic flow / Car-following models / Introduction to three representations of traffic flow / X-models
caption_attribution_description: English: X-models in car-following model
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: Considering the triangular flow-density fundamental diagram, we can get  and , and correspondingly the car-following model can be described via  model:
   
  where  is the headway between vehicles, and the bumper-to-bumper spacing at which vehicles are jammed can be derived as . The solution of  can be graphically shown in Figure 28 and Figure 29.
  When the constant spacing is applied, the initial data is linear, and the car-following model can be simplified into:
   
  Then, if we partition the time-space plane into grids of , and we interpret the origin (0,0) as , the general car-following model becomes:
   
  The solution can be interpreted intuitively in the time-space diagram of Figure 30: the trajectory vehicle n is the lower envelope between (i) the trajectory of leading vehicle shifting along the characteristics of slope , and (ii) its own trajectory under free-flow conditions.
  However, the general car-following model assumes an infinite vehicle acceleration, which is impractical. To compensate for this drawback, we can incorporate a vehicle kinematics model into the car-following model. The vehicle kinematics can be expressed as a linear acceleration model:
   
  in which  is the acceleration coefficient,  is the desirable speed.
  Define  as the resulting displacement at time  for vehicle  starting at a speed of  at time 0, the general-car following model with acceleration bounds would be:
caption: Figure 29.
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