Abstract
The continuity equation, which holds for all macroscopic models, describes the rate of change of the density in terms of gradients (or differences) of the flow. The model is closed by specifying flow or local speed. In this chapter we discuss the simpler approach in which the flow is given as a static function of the density, i.e., by a fundamental diagram. The models of this class of first-order models which are also called Lighthill–Whitham–Richards models differ only in the functional form of the fundamental diagram and in their mathematical representation.
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Notes
- 1.
Speed-density and flow-density plots are one of the most important visualizations of aggregated traffic data and have already been discussed in Sects. 4.2 and 4.4. Strictly speaking, the fundamental diagram describes the one-dimensional manifold of steady states parameterized as a function of the density. However, in several publications, the (scattered) flow-density data itself is often incorrectly referred to as the fundamental diagram as well.
- 2.
In the mathematical literature, this widening of the downstream transition zone is called a dispersion fan.
- 3.
Personal experience from the authors tells us that there is considerable dispersion in the downstream front of the “mega-jam” that is formed by the participants of marathon, inline-skating or cross-country skiing events after the starter’s gun. However, in vehicular traffic, this dispersion is very limited.
- 4.
In order to obtain the same result for \(I>1\) we would substitute \(n\) with \(n/I\) in the following equations. Furthermore, we would replace all densities and speeds by their respective effective values as defined in Sect. 7.1
- 5.
This is in contrast to the vehicle index where the first (must downstream) vehicle has the lowest index.
- 6.
In explicit integration schemes, the new state, i.e., all densities \(\rho _k(t+\varDelta t)\), are given in terms of the old state \(\{\rho _k(t)\}\). Implicit methods are characterized by relations between the old and new states that cannot be easily solved for the new state. In traffic-flow models, only explicit methods play a role.
- 7.
The numerical complexity \({\fancyscript{C}}\) indicates the number of multiplications or other operations on a computer which are necessary to obtain a certain approximate solution. Typically, the absolute value is irrelevant and the numerical complexity is given in terms of a scaling relation, here \({\fancyscript{C}}\propto \varDelta x^{-2}\).
- 8.
With the exception of numerical diffusion caused by the discretization of the CTM, cf. Sect. 9.5.
- 9.
A prerequisite for using LWR models for pedestrian flow are unidirectional pedestrian streams. This is satisfied in the arguably most prominent application example, namely the model-assisted planning and organization of the flow of pilgrims at the Hajj in Mecca, Saudi Arabia.
- 10.
- 11.
Strictly speaking, this determines the dynamic capacity which differs from the static capacity by the capacity drop. However, LWR models do not describe the phenomenon of a capacity drop.
- 12.
Since the evolution of a traffic breakdown takes 10 min and more, higher flows are possible over shorter time periods. However, LWR models cannot describe traffic instabilities, so these flow peaks are irrelevant for this model class.
- 13.
This agrees with most observations: Jams are observed at the begin or upstream of a bottleneck but rarely within the zone of reduced capacity.
- 14.
In a more coarse-grained picture, one would model a signalized intersection by a permanent bottleneck whose capacity is given by the maximum number of passing vehicles per signal cycle.
- 15.
In Germany, there is anecdotic evidence that behaviorally-induced bottlenecks have caused traffic breakdowns on otherwise completely homogeneous road sections.
- 16.
Remarkably, in models with dynamic speed (second-order macroscopic models and most microscopic models), even a local increase of capacity can lead to a breakdown (“more is less”) which is mediated by the speed perturbations associated with any local change together with traffic flow instabilities.
- 17.
Part of off-ramp induced congestions, however, have a more trivial reason: If the off-ramp itself is congested (caused, e.g., by an insufficient capacity on the secondary road network at junctions or by congestion on the target highway at interchanges), the off-ramp queue can spill over and obstruct a lane of the original road.
- 18.
In order to arrive at this rule, we must require the first CFL condition (8.27) to be satisfied. Since the road sections generally are several kilometer long, this rarely is an issue.
- 19.
Here and in the whole subsection, the indices of flows and densities refer to the traffic states, and not to the road sections. Specifically, in Fig. 8.22, we have three road sections and four traffic states.
- 20.
Yellow/amber phases are associated to the red or green phases as appropriate for the respective country and average driving behavior.
- 21.
This really happens as is observed by one of the authors in his home city.
- 22.
Here, the limits of a collective macroscopic description become obvious: A single vehicle cannot “smear out” by diffusion effects.
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Further Reading
Further Reading
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Lighthill, M.J., Whitham, G.B.: On kinematic waves: II. A theory of traffic on long crowded roads. Proceedings of the Royal Society A 229 (1955) 317–345
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Richards, P.: Shock waves on the highway. Operations Research 4, (1956) 42–51
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Daganzo, C.F.: The cell transmission model: A dynamic representation of highway traffic consistent with the hydrodynamic theory. Transportation Research Part B: Methodological 28, (1994) 269–287
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Daganzo, C.F.: The cell transmission model part II: Network traffic. Transportation Research Part B: Methodological 29, (1995) 79–93
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Daganzo, C.F.: Urban gridlock: Macroscopic modeling and mitigation approaches. Transportation Research Part B: Methodological 41, (2007) 49–62
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Helbing, D.: A section-based queueing-theoretical traffic model for congestion and travel time analysis in networks. Journal of Physics A: Mathematical and General 36, (2003) L593–L598
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Treiber, M., Kesting, A. (2013). The Lighthill–Whitham–Richards Model. In: Traffic Flow Dynamics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32460-4_8
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