33.4. Other Macroscopic Stream Models

In Greenshield’s model, linear relationship between speed and density was assumed. But in field we can hardly find such a relationship between speed and density. Therefore, the validity of Greenshields’ model was questioned and many other models came up. Prominent among them are Greenberg’s logarithmic model, Underwood’s exponential model, Pipe’s generalized model, and multiregime models. These are briefly discussed below.

Greenshield’s model simplifies the relationship between speed and density to a straight line, but real-world data often shows a different pattern. This led researchers to develop alternative models that better capture the actual traffic behavior. The models mentioned—Greenberg’s, Underwood’s, Pipe’s, and multiregime models—propose different ways to understand how traffic flow works by taking into account varying conditions and different types of relationships between speed and density.

Imagine trying to understand how a river flows using only one type of measurement. If you only measured water flow as a straight line and ignored curves, your understanding would be limited. Just like the river has different flows and speeds at various points, traffic behaves differently under various conditions, prompting the development of multiple models.

Greenberg assumed a logarithmic relation between speed and density. He proposed,

$v = v_0 \cdot \ln \left( \frac{k_j}{k} \right) \text{ (33.11)}$

This model has gained very good popularity because this model can be derived analytically. (This derivation is beyond the scope of these notes). However, main drawbacks of this model is that as density tends to zero, speed tends to infinity. This shows the inability of the model to predict the speeds at lower densities.

Greenberg's model is based on a logarithmic approach rather than a linear one. It suggests that as density increases, speed decreases but in a way that slows down progressively, rather than drastically. One key limitation of this model is that it implies if there is no density (meaning no cars), speed can increase without bound—something that does not occur in reality, hence it cannot accurately predict speeds when density is very low.

Think of it like a crowded coffee shop. As more people come in (increasing density), you can't move as fast because you're bumping into others. In theory, if you could remove all those people instantly, you'd zoom out of there, but real life doesn’t allow that—there are always limits to how fast you can go.

Trying to overcome the limitation of Greenberg’s model, Underwood put forward an exponential model as shown below.

$v = v_f \cdot e^{-k/k_0} \text{ (33.12)}$

The model can be graphically expressed. In this model, speed becomes zero only when density reaches infinity which is the drawback of this model. Hence this cannot be used for predicting speeds at high densities.

Underwood’s model takes a different approach, suggesting that speed decreases exponentially as density increases. A limitation here is that speed only reaches zero when density is infinitely high, which doesn't apply practically. This means that it inaccurately suggests there's always a speed, no matter how congested traffic gets, failing to reflect real-world limits on traffic speeds historically seen during extreme congestion.

Imagine a balloon filled with air. The more you squeeze it (increase density), the less air moves inside. Underwood’s model is like saying that the balloon will only collapse into nothing if you keep adding air forever—something not possible in reality because physical limits eventually stop the air from moving quickly.

Further developments were made with the introduction of a new parameter (n) to provide for a more generalized modeling approach. Pipes proposed a model shown by the following equation.

$v = v_f [1 - (k/k_j)^n] \text{ (33.13)}$

When n is set to one, Pipe’s model resembles Greenshields’ model. Thus by varying the values of n, a family of models can be developed.

Pipe’s model introduces a new variable, n, which helps create a more flexible framework to analyze speed-density relationships. By adjusting n, this model can transition from reflecting Greenshields’ linear model to potentially more complex relationships. This adaptability allows for better modeling across different traffic conditions and densities, ultimately enabling more accurate predictions of traffic behavior.

Think of a customizable recipe for a cake. Just as you can adjust ingredients to make it denser or fluffier, Pipes’ model allows traffic professionals to tweak parameters to better reflect traffic flow patterns across various conditions.

All the above models are based on the assumption that the same speed-density relation is valid for the entire range of densities seen in traffic streams. Therefore, these models are called single-regime models. However, human behaviour will be different at different densities. This is corroborated with field observations which shows different relations at different range of densities. Therefore, the speed-density relation will also be different in different zones of densities. Based on this concept, many models were proposed generally called multi-regime models. The most simple one is called a two-regime model, where separate equations are used to represent the speed-density relation at congested and uncongested traffic.

Multiregime models recognize that traffic behavior varies significantly under different density conditions. Unlike previous models that apply a single formula for all conditions, multi-regime models route their analysis through multiple categories or 'regimes' of density, allowing for distinct speed-density relationships to be defined for congested versus uncongested conditions. By doing so, these models can provide more accurate insights into traffic dynamics.

Think of a classroom: students behave very differently when the class is full and quiet (congested) compared to when they’re waiting in line at a coffee shop (uncongested). Just as a teacher would apply different strategies depending on class behavior, traffic models need distinct approaches for various traffic conditions.

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