Operations Research · 1960 · 209 citations · 1 references
Traffic TheoryEngineeringTraffic FlowTraffic PredictionIntegral Differential EquationTraffic ModelNetwork AnalysisStochastic NetworksTransport PhenomenaProbability TheoryComputer ScienceTraffic EngineeringDistribution FunctionTraffic-flow ProblemTraffic SimulationTransportation EngineeringBoltzmann Transport Equation
The study extends a Boltzmann‑type integral differential equation approach to traffic flow, building on prior work. It aims to analyze how a single car influences others by defining reduced n‑car distribution functions for clusters moving at the same velocity. The authors incorporate passing into the Boltzmann equation and derive an evolution equation for the reduced n‑car distribution function. They find that at high vehicle density a collective flow emerges, and the model describes the transition from free to condensed traffic.
The approach to the traffic-flow problem based on an integral differential equation of the Boltzmann type which has been considered by one of us (I P ) in a recent paper is further developed. The possibility of passing is explicitly introduced into the equation for the velocity distribution function. As in the previous paper, it is shown that at sufficiently high concentration a collective flow process must take place. In order to study more specifically the effects of one car on another, we define reduced n-car distribution functions giving the probability of finding a cluster of n cars all having the same velocity. We derive an equation for the evolution of this distribution function. Study of it yields some information as to the way traffic changes from relatively free flow to completely hindered, “condensed” flow.
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<i>Molecular Theory of Gases and Liquids</i>
Joseph O. Hirschfelder, C. F. Curtiss, R. Byron Bird · Physics Today · 1955 · 11.4K citations