shown in black, that the ARZ model possesses. By construction, each velocity curve in the ARZ model is merely a vertical translation of the LWR velocity curve. Hence, the homogeneous ARZ model possesses the same number of parameter functions (namely: a single one) as the LWR model.

In line with Remark \[rem:dependent\_h\_U\], i.e., by choosing $U_\text{eq}(\rho) = U(\rho) = U(0)-h(\rho)$, we can extend model to the inhomogeneous case, yielding $$\label{eq:aw_rascle_zhang_model_w_inhomogeneous}
\begin{split}
\rho_t+(\rho u)_x &= 0\;, \\
w_t+uw_x &= \tfrac{1}{\tau}{\left(U(0)-w\right)}\;, \\
\text{where~}u &= w-h(\rho)\;,
\end{split}$$ which adds a temporal relaxation of each vehicle’s empty road velocity $w$ towards a uniform value $U(0)$. In other words: the dynamics, that can be on any velocity curve $u_w(\rho)$, are driven towards the LWR velocity function $U(\rho)$, i.e., the red curves in Fig. \[fig:fd\_lwr\_arz\]—unless the system is driven away from equilibrium by another effect, such as boundary conditions (see §\[subsubsec:relaxtion\_GARZ\_LWR\]).

Generalized ARZ Model {#subsec:garz}
---------------------

The interpretation of the ARZ model as possessing a family of velocity curves (see ) reveals a fundamental shortcoming of the model: due to the additive relationship between velocity, empty road velocity, and hesitation, there is not a unique maximum density $\rho_\text{max}$, at which the flow stagnates. On the contrary, 