Greenshields Speed
Calculate the speed of a traffic stream by the linear Greenshields model, v = v_f·(1 − k ÷ k_j), from the free-flow speed v_f, the current density k and the jam density k_j (vehicles/km). The result, in the unit of v_f, shows speed falls linearly with density: on an empty road (k = 0), vehicles travel at free-flow speed; as density rises, speed decreases, reaching zero at total jam (k = k_j). It is the most classic macroscopic traffic flow model, the basis of the parabolic flow-density relationship. Enter the free-flow speed, the current density and the jam density.
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Greenshields model (speed-density)
In 1935 Bruce Greenshields proposed the first and best known macroscopic traffic relationship, built on a simple hypothesis: speed falls linearly with density. The model is v = v_f·(1 − k ÷ k_j), where v_f is the free-flow speed (the speed when the road is practically empty, limited only by geometry and by the posted limit), k is the current density and k_j is the jam density, at which vehicles are at a standstill. The reading is direct: on an empty road (k = 0), v = v_f (maximum speed); as more vehicles enter and density climbs, speed drops proportionally; and at full gridlock (k = k_j), speed reaches zero. The far-reaching consequence shows up when this straight line is combined with the fundamental equation q = k · v: substituting, flow q becomes a parabolic function of density — it starts at zero (empty road, nobody passing), grows to a maximum (the capacity, which occurs at the critical density k_j/2 and speed v_f/2), and falls back to zero at gridlock (many vehicles, all of them stopped). That parabola is the famous fundamental diagram of traffic, and it explains why adding vehicles to a congested road reduces total flow. Simple as it is (real relationships are messier, with abrupt drops at the transition), the Greenshields model captures the essence of the phenomenon and remains the gateway to traffic flow theory. Enter the free-flow speed, the current density and the jam density.
Related Tools
Space Mean Speed
Calculate the space mean speed of two vehicles by the harmonic mean, v_s = 2 ÷ (1/v₁ + 1/v₂), from the individual speeds v₁ and v₂. The result, in the same unit as the speeds, is the harmonic mean — not the arithmetic — which is the correct way to compute the mean speed of a traffic stream when observing a road section (average over space). Space mean speed is always less than or equal to the time mean speed (the arithmetic mean observed at a point), because it gives more weight to slow vehicles, which spend more time in the section. It is the speed used in the fundamental equation q = k·v. Enter the two speeds.
Average Vehicle Spacing
Calculate the average vehicle spacing, s = 1000 ÷ k, dividing 1000 metres by the traffic density k (vehicles/km). The result, in metres, is the average distance between the fronts of two consecutive vehicles in a traffic stream. Spacing is the inverse of density: congested roads have high density and small spacing; free-flowing roads have low density and large spacing. It is the spatial analogue of headway (which is temporal) and relates to speed by s = v·h. The smallest spacing, at jam density, equals the vehicle length plus the minimum gap. Enter the traffic density.
Saturation Flow
Calculate the saturation flow of a signalized approach, S = S₀ × N, multiplying the base saturation flow per lane S₀ (vehicles/h per lane, typically ~1800–1900) by the number of lanes N. The result, in vehicles/h, is the maximum rate of vehicles that can cross the stop line if the signal stayed green continuously and a queue existed — the queue discharge rate during green. It is a central parameter in signal design and intersection capacity, adjusted by lane width, grade, turning and parking factors. Enter the base saturation flow per lane and the number of lanes.
Average Headway
Calculate the average headway (time interval between successive vehicles), h = 3600 ÷ q, dividing 3600 seconds by the flow rate q (vehicles/h). The result, in seconds, is the average time between two consecutive vehicles passing a point. Headway is the inverse of flow: the higher the traffic volume, the shorter the intervals. It is a central concept of traffic flow theory, used in signal design, capacity analysis and car-following models. The smallest safe headway defines the maximum capacity of a lane. Enter the flow rate.
Equivalent Flow (PCE)
Calculate the equivalent flow in passenger car equivalents (PCE), q = Q_cars + Q_heavy × E, adding the car flow to the heavy-vehicle flow multiplied by the equivalence factor E (how many passenger cars each truck or bus equals in road occupancy — typically 1.5 to 3.0). The result, in PCE/h, converts a mixed traffic stream into an equivalent homogeneous one, allowing volumes to be compared and the capacity of roads with different traffic compositions to be computed. Heavy vehicles occupy more space and accelerate more slowly, especially on grades. Enter the car flow, the heavy-vehicle flow and the equivalence factor.
Peak Flow Rate (PHF)
Calculate the peak flow rate of a roadway, q = V ÷ PHF, dividing the hourly volume V (vehicles/h) by the peak hour factor PHF (between 0 and 1, the ratio of the hour's volume to four times the busiest 15-minute volume). The result, in vehicles/h, is the equivalent flow rate of the busiest 15-minute period — always greater than or equal to the hourly volume, since traffic does not arrive uniformly. It is the design flow used in capacity and level-of-service analysis by the HCM, since sizing by the hourly average would underestimate the peaks. Enter the hourly volume and the peak hour factor.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.