chapter3[1] - Chapter 3 Applications and Advanced Topics In...

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Chapter 3 Applications and Advanced Topics In this chapter we will study several real triffic problems. For some of them, an introductory model is proposed to stimulate students to create their own models to investigate these problems . 3.1 After an accident Suppose that one traffic accident happened at the position x = 0. How to model the traffic after the accident? How the traffic accident effects special traffic problems afterwards? Introductory modelling (I) : Sometimes the driver may meet a suddenly stop or slow down without a reasonable traffic condition. In fact, this is due to a suddenly stop or slow down at a previous time and at the position far away from the driver. Thus the driver should understand this is because a shock happened sometimes before and is propagated to this position now. To investigate this phenomena, the density just after the accidently stop is modelled by ρ 0 ( x ) = ρ max · exp ( - x 2 ) . a45 x a54 ρ 0 ρ max Figure 3.1.1 Initial density after an accident Introductory modelling (II) : The above modelling seems reasonable for x 0, and may not be practical for the density for x > 0, since the density is usually zero for x > 0. The density just after the accident may be modelled by ρ 0 ( x ) = ρ opt x ( -∞ , - L ) ρ max x ( - L, 0 ) 0 x ( 0 , - ∞ )
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2 a45 x a54 ρ 0 ρ max ρ 0 - L The length L depends on the time period within that the traffic was blocked due to the accident and ρ 0 depends on the traffic condition during the accident time.
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3 3.2 Traffic lights The traffic light is one of the most attractive topics in the study and applications of traffic problems. How long should the green/red light be kept?
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