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chapter1[1] - Chapter 1 Continuum Hypothesis of Traffic...

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Chapter 1 Continuum Hypothesis of Traffic Flow 1.1 Three basic variables in traffic problem We discuss the one straight lane traffic problem: a7 a6 a4 a5 a7 a6 a4 a5 a7 a6 a4 a5 a7 a6 a4 a5 a45 x direction of traffic Figure 1.1.1 One straight lane traffic Traffic velocity : u ( x, t ) ( unit: length/time ) The velocity or speed, u ( x, t ) is well-defined in general physics. For traffic problem it is surely that 0 u ( x, t ) U max , where the maximum velocity, U max is determined by the condition of the car, the quality of the road and the driver etc., but is finally determined by the local Transportation Department. Remark (Euler’s configuration and Lagrangian configuration) As illustrated in Figure 1.1.2 , our observation is always to follow the position x , at which the car may be different at different time. This approach is called the Euler’s configuration. At time t 1 Car B Car A a7 a6 a4 a5 a7 a6 a4 a5 a7 a6 a4 a5 a7 a6 a4 a5 u ( x, t 1 ) a45 x At time t 2 Car B Car A a7 a6 a4 a5 a7 a6 a4 a5 a7 a6 a4 a5 a7 a6 a4 a5 u ( x, t 2 ) a45 x Figure 1.1.2 Euler’s configuration Another approach is to follow the car , which changes its position from time to time as shown in Figure 1.1.3 . It is called the Lagrangian configuration.

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2 . At time t 1 Car A a7 a6 a4 a5 a7 a6 a4 a5 a7 a6 a4 a5 a7 a6 a4 a5 u ( x 1 , t 1 ) a45 x At time t 2 Car A a7 a6 a4 a5 a7 a6 a4 a5 a7 a6 a4 a5 a7 a6 a4 a5 u ( x 2 , t 2 ) a45 x Figure 1.1.3 Lagrangian configuration In this subject we always use the Euler’s configuration. Traffic density : ρ ( x, t ) ( unit: number of cars/length ) The traffic density at time t is defined by density No. of cars distance or precisely ρ ( x, t ) = lim Δ x 0 No. of cars at time t in ( x, x + Δ x ) Δ x Even though the argument, Δ x 0 is not practical, the above definition is commonly accepted as a continuum model for the traffic flow. It is also clear that one must have 0 ρ ( x, t ) ρ max , where the constant ρ max may be fund by the following argument. The maximum density is reached, if and only if there is no space between cars. The Figure 1.1.4 illustrates this situation. a7 a6 a4 a5 a7 a6 a4 a5 a7 a6 a4 a5 a7 a6 a4 a5 a54 a54 ( 1 2 + 1 2 ) length of one car a45 x Figure 1.1.4 Maximum traffic density The above argument leads to that ρ max = 1 length of one car
3 Traffic flux : q ( x, t ) ( unit: number of cars/time ) The traffic flux at position x is defined by flux No. of cars

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chapter1[1] - Chapter 1 Continuum Hypothesis of Traffic...

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