Module 4 : Macroscopic And Mesoscopic Traffic Flow Modeling
Lecture 17 : Traffic Flow Modeling Analogies
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Formulation

Assuming that the vehicles are flowing from left to right, the continuity equation can be written as

$\displaystyle \frac{\partial k(x,t)}{\partial t} + \frac{\partial q(x,t)}{\partial x} = 0$ (1)

where $ x$ denotes the spatial coordinate in the direction of traffic flow, $ t$ is the time, $ k$ is the density and $ q$ denotes the flow. However, one cannot get two unknowns, namely $ k(x,t)$ by and $ q(x,t)$ by solving one equation. One possible solution is to write two equations from two regimes of the flow, say before and after a bottleneck. In this system the flow rate before and after will be same, or

$\displaystyle k_1v_1=k_2v_2$ (2)

From this the shock wave velocity can be derived as

$\displaystyle v(t_o)_p = \frac{q_2-q_1}{k_2-k_1}$ (3)

This is normally referred to as Stock's shock wave formula. An alternate possibility which Lighthill and Whitham adopted in their landmark study is to assume that the flow rate $ q$ is determined primarily by the local density $ k$, so that flow $ q$ can be treated as a function of only density $ k$. Therefore the number of unknown variables will be reduced to one. Essentially this assumption states that k(x,t) and q (x,t) are not independent of each other. Therefore the continuity equation takes the form

$\displaystyle \frac{\partial k(x,t)}{\partial t} + \frac{\partial q (k(x,t))}{\partial x} = 0$ (4)

However, the functional relationship between flow $ q$ and density $ k$ cannot be calculated from fluid-dynamical theory. This has to be either taken as a phenomenological relation derived from the empirical observation or from microscopic theories. Therefore, the flow rate $ q$ is a function of the vehicular density k; $ q = q(k)$. Thus, the balance equation takes the form

$\displaystyle \frac{\partial k(x,t)}{\partial t} + \frac{\partial q (k(x,t))}{\partial x} = 0$ (5)

Now there is only one independent variable in the balance equation, the vehicle density $ k$. If initial and boundary conditions are known, this can be solved. Solution to LWR models are kinematic waves moving with velocity

$\displaystyle \frac{dq(k)}{dk}$ (6)

This velocity $ v_k$ is positive when the flow rate increases with density, and it is negative when the flow rate decreases with density. In some cases, this function may shift from one regime to the other, and then a shock is said to be formed. This shock wave propagate at the velocity

$\displaystyle v_s = \frac{q(k_2) - q(k_1)}{k_2-k_1}$ (7)

where $ q(k_2)$ and $ q(k_1)$ are the flow rates corresponding to the upstream density $ k_2$ and downstream density $ k_1$ of the shock wave. Unlike Stock's shock wave formula there is only one variable here.