About Traffic Flow Capacity Calculator (Greenshields Model)
The traffic flow capacity calculator analyses uninterrupted flow on a highway segment using the Greenshields model, the classic linear relationship between space-mean speed and density: v = vf*(1 - k/kj), where vf is the free-flow speed (at zero density) and kj is the jam density (where speed falls to zero).
From the fundamental traffic relation q = k*v, the flow is a parabola in density, q = vf*(k - k^2/kj). The maximum flow, or capacity, is q_max = vf*kj/4 and occurs at the optimum density k = kj/2 and optimum speed v = vf/2. The calculator reports the capacity and optimum conditions, the speed and flow at a density you choose, and — for a given sub-capacity flow — both the free-flow and congested density solutions, with a flow-density curve.
How It Works
- Enter the free-flow speed vf (speed as density approaches zero) and the jam density kj (density at which traffic stops).
- The calculator computes the capacity q_max = vf*kj/4 and the conditions at capacity: optimum density k = kj/2 and optimum speed v = vf/2.
- Enter a query density k; the speed follows from v = vf*(1 - k/kj) and the flow from q = k*v. Densities below kj/2 are on the free-flow branch and densities above kj/2 are on the congested branch.
- The flow-density chart plots the parabola q = vf*(k - k^2/kj) with the capacity point and the queried state highlighted; a given flow below capacity corresponds to two densities, one on each branch.
Worked Example
A freeway lane has a free-flow speed vf = 100 km/h and a jam density kj = 120 veh/km. The capacity is q_max = vf*kj/4 = 100*120/4 = 3000 veh/h, occurring at the optimum density k = kj/2 = 60 veh/km and optimum speed v = vf/2 = 50 km/h. At a lower density k = 30 veh/km the speed is v = 100*(1 - 30/120) = 75 km/h and the flow is q = 30*75 = 2250 veh/h on the free-flow branch. That same flow of 2250 veh/h also occurs on the congested branch at k = 90 veh/km, where the speed is v = 100*(1 - 90/120) = 25 km/h and q = 90*25 = 2250 veh/h.
Formulas
- Greenshields speed-density relationship
v = vf*(1 - k/kj)- Flow (fundamental relation)
q = k*v = vf*(k - k^2/kj)- Capacity and optimum conditions
q_max = vf*kj/4 at k_m = kj/2 , v_m = vf/2- Density roots for a given flow
k = (kj/2)*(1 -/+ sqrt(1 - q/q_max))
Standards & References
- Greenshields linear speed-density model (1935)
- Highway Capacity Manual (HCM) fundamental diagram of traffic flow
- Fundamental relation q = k*v (flow = density x speed)
Frequently Asked Questions
What is the Greenshields model?
The Greenshields model is the simplest model of uninterrupted traffic flow. It assumes the space-mean speed decreases linearly with density: v = vf*(1 - k/kj). Combined with the fundamental relation q = k*v, it produces a parabolic flow-density curve and a maximum flow (capacity) of vf*kj/4.
What is the capacity of a road in the Greenshields model?
Capacity is the maximum possible flow, q_max = vf*kj/4, which occurs at the optimum density k = kj/2 and the optimum speed v = vf/2. Below this density traffic is in free flow; above it, the road is congested and flow decreases even as density rises.
Why does one flow value correspond to two densities?
Because the flow-density curve is a downward parabola, every flow below capacity is reached at two densities: a lower one on the free-flow branch (higher speed) and a higher one on the congested branch (lower speed). Only the capacity flow corresponds to a single density, kj/2.
What is the difference between density and flow?
Density k is the number of vehicles per unit length of road (for example veh/km) at an instant, while flow q is the number of vehicles passing a point per unit time (for example veh/h). They are linked by speed through q = k*v, the fundamental relation of traffic flow.