2.4 Traffic flow

Lighthill and Whitham (1955) and Richards (1956) modelled road traffic by the density \(\rho (x, t)\) of cars, in cars per kilometre, with flux \(q = \rho v(\rho )\). The speed falls from \(v_{\max }\) on an empty road to \(0\) at the jam density \(\rho _{\max }\); the simplest choice is \(v = v_{\max }(1 - \rho /\rho _{\max })\), giving the conservation law \[ \rho _t + q(\rho )_x = 0, \qquad q(\rho ) = v_{\max }\rho \Bigl (1 - \frac {\rho }{\rho _{\max }}\Bigr ) . \] The flux is greatest, \(q_{\max } = \frac 14v_{\max }\rho _{\max }\), at \(\rho _c = \frac 12\rho _{\max }\): the capacity of the road. The characteristic speed \(q'(\rho ) = v_{\max }(1 - 2\rho /\rho _{\max })\) is the speed at which disturbances travel, which is less than the speed of the cars \(v(\rho )\), and negative in dense traffic: a disturbance in heavy traffic travels backwards, against the flow, as every driver who has braked for no visible reason knows.

At a red light, cars arriving at density \(\rho _l\) meet stopped cars at \(\rho _r = \rho _{\max }\). Since \(q'(\rho _l) > q'(\rho _{\max })\), this is a shock, the back of the queue, and by Rankine–Hugoniot it moves at speed \[ s = \frac {q(\rho _{\max }) - q(\rho _l)}{\rho _{\max } - \rho _l} = -v_{\max }\frac {\rho _l}{\rho _{\max }} , \] backwards. When the light turns green, the jam at \(\rho _{\max }\) on the left faces an empty road on the right, and the solution is a rarefaction fan in which the density thins out continuously: the front cars accelerate away while the cars further back are still waiting for the fan to reach them.

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