\rho_t+(\rho u)_x &= 0\;, \\
(u+h(\rho))_t+u(u+h(\rho))_x &= 0\;,
\end{split}$$ where we call $h(\rho)$ the *hesitation function*.[^1] We assume that $h'(\rho)>0$ and use the gauge $h(0) = 0$. The addition of a relaxation term (analogous to the Payne-Whitham model) yields the inhomogeneous ARZ model [@Greenberg2001; @Rascle2002] $$\label{eq:aw_rascle_zhang_model_inhomogeneous}
\begin{split}
\rho_t+(\rho u)_x &= 0\;, \\
(u+h(\rho))_t+u(u+h(\rho))_x &= \tfrac{1}{\tau}{\left(U_\text{eq}(\rho)-u\right)}\;.
\end{split}$$ We call $U_\text{eq}(\rho)$ the *desired velocity function* or the *equilibrium velocity function*, and $\tau$ the *relaxation time scale*.

As one can easily verify, the homogeneous ARZ model possesses no mechanism to make drivers move when starting with all vehicles at rest, i.e., $u(x,0) = 0$. In turn, the inhomogeneous ARZ model does. We therefore expect the homogeneous ARZ model to yields reasonable results only when the traffic flow is close to its equilibrium state, i.e., $u\approx U_\text{eq}(\rho)$. Yet, in general the inhomogeneous ARZ model has the potential to yield more realistic predictions.

The conservative form of is given by $$\label{eq:aw_rascle_zhang_model_conservative}
\begin{split}
\rho_t+{\left(q-\rho h(\rho)\right)}_x &= 0\;, \\
q_t+{\left(\tfrac{q^2}{\rho}-h(\rho)q\right)}_x &= \tfrac{1}{\tau}{\left(Q_\text{eq}(\rho)+\rho h(\rho)-q\right)}\;,
\end{split}$$ where the two conserved variables are $\rho$ and $q = \rho (u+h(\rho))$, and $Q_\text{eq}(\rho) = \rho U_\text{eq}(\rho)$ is called the *equilibrium curve* that the momentum density $\rho u$ relaxes to.

\[rem:dependent\_h\_U\] Various authors 