1) $\vec{\ell_{IM}}$, $\vec{\ell_{OM}}$

2) $\vec{\ell_{I}}$, $\vec{\ell_{O}}$

\begin{align} f_t &= v \mapsto \left(\frac{v}{d_{max}}\right)^\epsilon v \\ \vec{t_{I}} &= map \left(f_t, \vec{\ell_{I}}\right) \\ \mathbb{T}_c &= sum \left(\vec{t_{I}}\right) \end{align}

3)

\begin{align} \vec{i_{I}}, \vec{\delta_{I}} &= \left(\vec{t_{I}}^{\prime} - \vec{t_{I}}\right).sumby\left(\vec{v_{IC}}\right) \\ \vec{i_{O}}, \vec{\delta_{O}} &= \left(\vec{t_{O}}^{\prime} - \vec{t_{O}}\right).sumby\left(\vec{v_{OC}}\right) \\ \vec{\delta_T} &= 0, \vec{\delta_T} \in \mathbb{R}^m \\ \vec{\delta_T}\left[\vec{i_{I}}\right] &\gets \vec{\delta_T}\left[\vec{i_{I}}\right] + \vec{\delta_{I}} \\ \vec{\delta_T}\left[\vec{o_{I}}\right] &\gets \vec{\delta_T}\left[\vec{o_{I}}\right] + \vec{\delta_{O}} \\ \end{align}

3)

\begin{align} \vec{u_{IC}}, \vec{v_{IC}} &= u_C, v_C \\ \vec{u_{OC}}, \vec{v_{OC}} &= u_C, v_C \\ \vec{\ell_{IM}} &= \texttt{max_incoming...}\left(\vec{b_I}, \vec{b_r}, \vec{u_{CI}}, \vec{v_{CI}}, \vec{L_{CI}}, \vec{p}\right) \\ \vec{\ell_{OM}} &= \texttt{max_incoming...}\left(\vec{b_O}, \vec{b_r}, \vec{u_{CO}}, \vec{v_{CO}}, \vec{L_{CO}}, \vec{p}\right) \\ \end{align}

\begin{align} f_{Id} &= (u, v) \mapsto \begin{cases}delay(\vec{p}[u], \vec{p}[v]), & if \vec{b_r}[u] = 1\\ \vec{\ell_{IM}}[u] + delay(\vec{p}[u], \vec{p}[v]), & otherwise\end{cases} \\ f^{\prime}_{Id} &= (u, v) \mapsto \begin{cases}delay(\vec{p}[u], \vec{p^{\prime}}[v]), & if \vec{b_r}[u] = 1\\ \vec{\ell_{IM}}[u] + delay(\vec{p}[u], \vec{p^{\prime}}[v]), & otherwise\end{cases} \\ f_{Od} &= (u, v) \mapsto \begin{cases}delay(\vec{p}[u], \vec{p}[v]), & if \vec{b_r}[u] = 1\\ \vec{\ell_{OM}}[v] + delay(\vec{p}[u], \vec{p}[v]), & otherwise\end{cases} \\ f^{\prime}_{Od} &= (u, v) \mapsto \begin{cases}delay(\vec{p^{\prime}}[u], \vec{p}[v]), & if \vec{b_r}[u] = 1\\ \vec{\ell_{OM}}[v] + delay(\vec{p^{\prime}}[u], \vec{p}[v]), & otherwise\end{cases} \\ \vec{\ell_{I}} &= map \left(f_{Id}, \left(\vec{u_{IC}}, \vec{v_{IC}}\right) \right) \\ \vec{\ell^{\prime}_{I}} &= map \left(f^{\prime}_{Id}, \left(\vec{u_{IC}}, \vec{v_{IC}}\right) \right) \\ \vec{\ell_{O}} &= map \left(f_{Od}, \left(\vec{u_{OC}}, \vec{v_{OC}}\right) \right) \\ \vec{\ell^{\prime}_{O}} &= map \left(f^{\prime}_{Od}, \left(\vec{u_{OC}}, \vec{v_{OC}}\right) \right) \\ \end{align}

\begin{align} \vec{\ell_{I}} &= map \left(f_{Id}, \left(\vec{u_{IC}}, \vec{v_{IC}}\right) \right) \\ \vec{\ell^{\prime}_{I}} &= map \left(f^{\prime}_{Id}, \left(\vec{u_{IC}}, \vec{v_{IC}}\right) \right) \\ \vec{\ell_{O}} &= map \left(f_{Od}, \left(\vec{u_{OC}}, \vec{v_{OC}}\right) \right) \\ \vec{\ell^{\prime}_{O}} &= map \left(f^{\prime}_{Od}, \left(\vec{u_{OC}}, \vec{v_{OC}}\right) \right) \\ \end{align}