Traffic flow

In mathematics and transportation engineering, traffic flow is the study of interactions between travellers (including pedestrians, cyclists, drivers, and their vehicles) and infrastructure (including highways, signage, and traffic control devices), with the aim of understanding and developing an optimal transport network with efficient movement of traffic and minimal traffic congestion problems.The minimum highway capacity C m i n {displaystyle C_{min}} , at which the F→S phase transition can still be induced at a highway bottleneck as stated in Kerner’s theory, has no sense for other traffic flow theories and models. In mathematics and transportation engineering, traffic flow is the study of interactions between travellers (including pedestrians, cyclists, drivers, and their vehicles) and infrastructure (including highways, signage, and traffic control devices), with the aim of understanding and developing an optimal transport network with efficient movement of traffic and minimal traffic congestion problems. Attempts to produce a mathematical theory of traffic flow date back to the 1920s, when Frank Knight first produced an analysis of traffic equilibrium, which was refined into Wardrop's first and second principles of equilibrium in 1952. Nonetheless, even with the advent of significant computer processing power, to date there has been no satisfactory general theory that can be consistently applied to real flow conditions. Current traffic models use a mixture of empirical and theoretical techniques. These models are then developed into traffic forecasts, to take account of proposed local or major changes, such as increased vehicle use, changes in land use or changes in mode of transport (with people moving from bus to train or car, for example), and to identify areas of congestion where the network needs to be adjusted. Traffic behaves in a complex and nonlinear way, depending on the interactions of a large number of vehicles. Due to the individual reactions of human drivers, vehicles do not interact simply following the laws of mechanics, but rather display cluster formation and shock wave propagation, both forward and backward, depending on vehicle density. Some mathematical models of traffic flow use a vertical queue assumption, in which the vehicles along a congested link do not spill back along the length of the link. In a free-flowing network, traffic flow theory refers to the traffic stream variables of speed, flow, and concentration. These relationships are mainly concerned with uninterrupted traffic flow, primarily found on freeways or expressways. Flow conditions are considered 'free' when less than 12 vehicles per mile are on a road. 'Stable' is sometimes described as 12–30 vehicles per mile per lane. As the density reaches the maximum mass flow rate (or flux) and exceeds the optimum density (above 30 vehicles per mile), traffic flow becomes unstable, and even a minor incident can result in persistent stop-and-go driving conditions. A 'breakdown' condition occurs when traffic becomes unstable and exceeds 67 vehicles per mile. 'Jam density' refers to extreme traffic density when traffic flow stops completely, usually in the range of 185–250 vehicles per mile per lane.