## 3 Problem Formulation
### Mathematical Formulation  
The aforementioned problem can be modeled through mathematical programming. The followings are parameters and decision variables; corresponding notations are defined in Tables 1 and 2. \begin{table}
\begin{tabular}{l l} \hline \hline
**Given Parameters** & \\ \hline
**Notation** & **Definition** \\ \hline \(V\) & The set of all nodes in the network \\ \(S\) & The set of multicast source nodes \\ \(D_{s}\) & The set of destination nodes for each source \(s\) \\ \(L\) & The set of all links (\(i,j\)) in the network, \(\forall i,j\in V,i\
eq j\) \\ \(L_{v}\) & The set of wireless links associated with node \(v,\ \forall v\in V\). \\ \(P_{sd}\) & The set of paths from source \(s\) to its destination \(d\) \\ \(\delta_{nl}\) & 1 if link \(l\) is on the path \(p\), and 0 otherwise \\ \hline \end{tabular}
\end{table} 
Table 1: _Cont._ \begin{table}
\begin{tabular}{l l} \hline \hline
**Decision Variable** & \\ \hline
**Notation** & **Description** \\ \hline \(x_{p}\) & 1 if path \(p\) is selected for the multicast group \(s\) to destination \(d\), and 0 otherwise. \\ \(y_{ul}\) & 1 if link \(l\) is in the multicast group \(s\), and 0 otherwise. \\ \(t_{sd}\) & 1 if link \(l\) is used by destination d of multicast group \(s\), and 0 otherwise. \\ \(c_{ul}\) & The capacity of link \(l\) in the multicast group \(s\). \\ \(c_{l}\) & The allocated capacity of each link \(l\) \\ \(g_{nl}\) & The traffic rate of link \(l\) in the multicast group \(s\). \\ \(t_{s}\) & The end-to-end delay per path in the multicast group \(s\). \\ \(T\) & The end-to-end delay for each multicast group. \\ \hline \end{tabular}
\end{table} 
Table 2: Notation descriptions of decision variables. **Objective Function:**  
\[Z_{{}_{I\!P1}}=\min_{s\in S}\,T,\] (IP 1)  
**Subject to:**  
**Delay Constraints**  
\[\beta_{s}t_{s}=T \forall s\in S \tag{1}\] \[\sum_{l\in L}h_{sdl}D_{s}(c_{s},g_{sl})=t_{s} \forall s\in S,d\in D_{s} \tag{2}\]  
**Routing Constraints**  
\[\sum_{p\in P_{s}^{\prime}}x_{p}\partial_{pl}=h_{sdl} \forall s\in S,\forall d\in D_{s},l\in L \tag{3}\] \[\sum_{p\in P_{s}^{\prime}}x_{p}=1 \forall s\in S,d\in D_{s}\] (4) \[\sum_{d\in D_{s}}h_{sdl}\leq\left|D_{s}\right|y_{sl} \forall s\in S,l\in L \tag{5}\]  
**Capacity and Traffic Constraints**  
\[g_{sl}=h_{sdl}y_{sd}<c_{sl} \forall s\in S,\forall d\in D_{s},l\in L \tag{6}\] \[g_{sl}\in\left[0,\max_{d\in D_{s}}r_{sd}\right] \forall s\in S,l\in L\] (7) \[\sum_{s\in S}c_{sl}\leq c_{sl} \forall l\in L\] (8) \[Z_{{}_{I\!P1}}=\min_{s\in S}\,T,\]0  
**Tree Constraints**  
\[Z_{{}_{I\!P1}}=\min_{s\in S}\,T,\]1 \[Z_{{}_{I\!P1}}=\min_{s\in S}\,T,\]2 (11) \[Z_{{}_{I\!P1}}=\min_{s\in S}\,T,\]3  
**Integer Constraints**  
\[Z_{{}_{I\!P1}}=\min_{s\in S}\,T,\]4 \[Z_{{}_{I\!P1}}=\min_{s\in S}\,T,\]5 (14) \[Z_{{}_{I\!P1}}=\min_{s\in S}\,T,\]6  
Explanation of Objective Function:  
The objective function (IP 1) is to minimize the end-to-end inter-delay in group \(T\). The group \(T\) is obtained by multiplying any end-to-end intra-delay \(t_{s}\) corresponding to the group weight \(\beta_{s}\). Therefore, \(T\) is also restricted both by the intra-group and inter-group end-to-end delay fairness. Explanation of Constraints:  
Constraint (1.1) confines the end-to-end inter-delay fairness among groups. In (1.1), taking the given weights of all groups into account, we multiply the end-to-end intra-delay for each group by the given weights \(\beta_{s}\). The result should be equal to the inter-delay decision variable, T. Thus, we can confirm 100% inter-delay fairness in the whole multiple-multicast wireless sensor network. Constraint (1.2) confines the intra-group end-to-end delay fairness. Considering the to-be-determined link capacity, \(c_{sl}\), and the traffic flow, \(g_{sl}\), for a link \(l\) belonging to multicast group \(s\), we can obtain the minimal end-to-end delay among each path, which should be equal to the intra-delay variable \(t_{s}\) on the right side of the equation, thus achieving 100% intra-delay fairness.