p\right)^2}\;, \quad
b = \sqrt{1+\left(\lambda(1-p)\right)^2}\;,\text{~and}\quad
y = \lambda \left(\rho/\rho_\text{max}-p\right).\end{aligned}$$ Each flow rate function $Q_{\alpha,\lambda,p}(\rho)$ in this family vanishes for $\rho = 0$ and $\rho = \rho_\text{max}$. The three free parameters allow for controlling three important features of $Q_{\alpha,\lambda,p}(\rho)$: the value of maximum flow rate $Q_\text{max}$ (mainly determined by $\alpha$), the critical density $\rho_\text{c}$ (mainly controlled by $p$), and the “roundness" of the curve, i.e., how rapidly the slope transitions from positive to negative near $\rho_\text{c}$ (dominated by $\lambda$).

Third, from this three-parameter family of flow rate curves, the one is selected that is the closest to the data points $(\rho_j,Q_j),\;j=1,\dots n$ in a least-squares sense, i.e. we solve $$\label{eq:LSQ}
\min_{\alpha,\lambda,p}\; \sum_{j=1}^n
(Q_{\alpha,\lambda,p}(\rho_{j})-Q_{j})^2\;.$$ In the right panel of Fig. \[fig:fd\_lwr\_arz\], the resulting least-squares fit to the given gray data points, called $Q_\text{eq}(\rho)$, is depicted by the red curve. The red curve in the left panel represents the resulting velocity function $U_\text{eq}(\rho) = Q_\text{eq}(\rho)/\rho$.

As described in §\[subsec:ARZ\_generalization\_of\_LWR\], the ARZ model generalizes the LWR model to a one-parameter family of velocity curves $u_w(\rho) = U_\text{eq}(\rho)+(w-U_\text{eq}(0))$ (black curves in the left panel of Fig. \[fig:fd\_lwr\_arz\]) and flow rate curves $Q_w(\rho) = Q_\text{eq}(\rho)+(w-U_\text{eq}(0))\rho$ (right panel of Fig. \[fig:fd\_lwr\_arz\]). Due to this property, the ARZ 