33.2.1. Relationship between Speed and Density

Greenshield assumed a linear speed-density relationship as illustrated in figure 33:1 to derive the model. The equation for this relationship is shown below. v f v =v .k (33.1) f − k [j] where v is the mean speed at density k, v is the free speed and k is the jam density. This equation ( 33.1) is often referred to as the Greenshields’ model. It indicates that when density becomes zero, speed approaches free flow speed (ie. v f when k 0).

Greenshield's model proposes a simple linear equation relating speed (v) and density (k) of traffic. The important parameters are:

1. Mean Speed (v): The average speed of vehicles at a certain density k.
2. Free Speed (v_f): The maximum speed when there are no vehicles on the road (density = 0).
3. Jam Density (k_j): The highest density at which vehicles can still move (maximum congestion). At a density of zero, vehicles can travel at free speed, but as density increases, speed decreases linearly until they reach jam density where movement becomes very slow.

Imagine a highway with no cars. Drivers can go as fast as they want - this is free speed. As more cars join the highway, everyone has to slow down because there isn’t enough space for everyone to travel at high speeds. In the extreme case where every lane is completely full, cars are at jam density and can barely move. This is similar to what happens in a traffic jam, where the relationship between the number of cars (density) and how fast they can go (speed) is linear until they can't go any faster.

Once the relation between speed and flow is established, the relation with flow can be derived. This relation between flow and density is parabolic in shape and is shown in figure 33:3. Also, we know that q =k.v (33.2)

After understanding how speed and density relate, we can derive the relationship involving flow (q). The flow is calculated by the product of density (k) and speed (v). This relationship demonstrates that as density increases, the flow initially increases, reaches a maximum, and then decreases, forming a parabolic curve.

Imagine pouring water into a funnel. Initially, as you pour in slowly (low density of water), the water flows out smoothly (increasing flow). If you pour too much too quickly (high density), the funnel can't handle it anymore, and the flow decreases (decreasing flow). This shows that there is an optimal 'density' for maximum flow.

To find the density at maximum flow, differentiate equation 33.3 with respect to k and equate it to zero. ie., dq/dk = 0 v f v .2k = 0 f − k j k j k = 2 Denoting the density corresponding to maximum flow as k_0, k j k = (33.5) 0 2

To find the density where traffic flow is maximized, we apply a method used in calculus called differentiation. We take our earlier equation (related to flow) and differentiate it to find where it peaks. This maximum flow density, denoted as k_0, occurs at half of the jam density (k_j), showing that there is a specific point before reaching maximum congestion where traffic can flow best.

Consider a highway merging from multiple lanes into two lanes. If the merging happens too slowly, vehicles start to back up and flow decreases. The best density of cars allows for a smooth transition without bottlenecks, just like a carefully timed relay race, where the smooth changes between runners allows maximum performance.

Substituting equation 33.5 in equation 33.3, k v k^2 j f j q_max = v . . f 2 − k^2 j [ ] Thus the maximum flow is one fourth the product of free flow and jam density.

To calculate the maximum flow, we substitute k_0 (the density at maximum flow) into our previously derived flow equation. It turns out that the maximum flow (q_max) is a quarter of the product of free speed (v_f) and jam density (k_j). This gives us a precise mathematical relationship that helps predict how many vehicles can pass a point in a given time under optimal conditions.

Think of a lemonade stand at a fair. If more customers come than you can serve (jam density), even if they want lemonade (flow), your ability to serve them (speed) suffers. At the optimum point, you can serve the most customers effectively without running out of inventory, ensuring maximum satisfaction.

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