Traffic Flow Calculator

Traffic stream analysis rests on a single identity: flow equals density times space mean speed, q = k x u. Flow is vehicles per hour past a fixed point, density is vehicles per mile of roadway, and speed is the average over that section. Density is harder to measure directly but is the more informative quantity, because it describes how crowded the road actually is rather than how many vehicles happen to be passing. Related descriptors follow immediately: average headway is 3600 divided by flow in seconds per vehicle, and average spacing is 5280 divided by density in feet. The Greenshields model closes the system by assuming speed falls linearly with density, u = uf x (1 - k/kj), where uf is free-flow speed and kj is jam density, typically 185 to 250 vehicles per mile per lane. Substituting into q = k x u produces a parabola whose maximum flow, uf x kj / 4, occurs at half the jam density and half the free-flow speed. Highway Capacity Manual practice grades freeway operation by density per lane, with Level of Service A up to 11 passenger cars per mile per lane, B to 18, C to 26, D to 35, E to 45, and F beyond that where flow breaks down. Greenshields is a useful screening and teaching model but a coarse one: measured speed-density data are markedly non-linear near capacity, and observed freeway capacity of roughly 2,400 passenger cars per hour per lane usually exceeds the parabolic estimate.

Quantitative study of traffic streams begins with Bruce D. Greenshields, who presented A Study of Traffic Capacity to the Highway Research Board in 1935. Working on two-lane roads in Ohio, he used a hand-cranked camera to photograph traffic at fixed intervals, then measured vehicle positions between successive frames to extract speed and spacing. Plotting speed against density, he fitted the straight line that still carries his name, and in doing so established that capacity is a property of the speed-density relationship rather than a fixed count of vehicles. The Bureau of Public Roads and the Highway Research Board published the first Highway Capacity Manual in 1950, converting this research into design practice. The 1965 second edition introduced the Level of Service concept, grading operating quality on an A to F scale keyed initially to volume-to-capacity ratio and operating speed. Later editions shifted the freeway service measure to density, which discriminates far better near capacity, where speed stays almost constant while crowding rises sharply. Subsequent theory broadened the field considerably, including the car-following models developed at the General Motors research laboratories in the late 1950s and the kinematic wave theory published by Lighthill and Whitham in 1955 and independently by Richards in 1956, which treats a traffic stream as a compressible fluid carrying shock waves.