## 3 Analysis  
The method for formulating aggregate throughput optimal flow rate allocation as an optimization problem, for random topologies, when interference is treated as noise, is presented in detail in our prior work [24]. In this study we consider the case where receivers may employ SIC, thus, we will introduce some modifications in the aforementioned framework. For that reason, we present all necessary notations in Table 1 along with the final form of the corresponding optimization problem. The set \(\mathcal{V}\) denotes the nodes and its cardinality is denoted by \(|\mathcal{V}|=N\). We assume \(m\) flows \(f_{1},f_{2},...,f_{m}\), that need to forward traffic to destination node \(D\). The analysis that follows can also be applied for the case where multiple flows have different destination nodes. The set \(\mathcal{K}=\{r_{1},r_{2},...,r_{m}\}\) represents the \(m\) disjoint paths employed by these flows. \(|r_{i}|\) is used to denote the number of links in path \(r_{i}\). \(\mathcal{I}_{i,j}\) is the set of nodes that cause interference to packets sent from \(i\) to \(j\) and its size is denoted by \(\mathcal{L}_{i,j}=|\mathcal{I}_{i,j}|\). In addition, \(\mathit{Src}(r_{k})\) is used to denote the source node of the \(k\)_th_ flow employing path \(r_{k}\), while \(r(i)\) returns the index of the path where node \(i\) belongs. Moreover, \(q_{\mathit{Src}(r_{k})}\) denotes the transmission probability of the originator of the \(k\)_th_ flow. \(\mathcal{T}_{i,j}\) and \(\mathcal{T}_{r_{k}}\) denote the average throughput, measured in packets per slot, achieved by link \((i,j)\) and flow \(f_{k}\) forwarded over path \(r_{k}\), respectively. Let also \(\mathcal{I}_{i,j}[n]\) denote the id of the \(n\)_th_ interfering node for link \((i,j)\). For each node \(i\), \(q_{i}\) denotes its transmission probability, given that there is a packet available for transmission in its queue. As already discussed, for flow originators it indicates the rate at which flow is injected on a path, while for relay nodes it is assumed fixed to a specific value. The average throughput for a link \((i,j)\), \(\mathcal{T}_{i,j}\), can be expressed through (10). Average throughput for that link, \(\mathcal{T}_{i,j}\), is expressed through the probability of a successful packet reception over that link and is denoted by (10). Estimating thus a link's \((i,j)\) average throughput, requires enumerating all possible subsets of active transmitters. Assuming that all nodes contribute with interference to transmissions over link \((i,j)\) and a network with \(N\) nodes, all such subsets of interfering nodes for \((i,j)\) are \(2^{\mathcal{L}_{i,j}}\). In (10), \(l\) enumerates \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Definition** \\ \hline \(\mathcal{V}\) & Set of node, \(|\mathcal{V}|=N\) \\ \hline \(q_{i}\) & Transmission probability for node \(i\) given there is a packet available for transmission \\ \hline \(f_{1},f_{2},...,f_{m}\) & \(m\) flows \\ \hline \(r_{i}\) & Path \(i\) is employed by flow \(f_{i}\) \\ \hline \(\mathcal{K}=\{r_{1},r_{2},...,r_{m}\}\) & Set of node-disjoint paths \\ \hline \(|r_{i}|\) & Number of links in path \(r_{i}\) \\ \hline \(\mathcal{I}_{i,j}\) & Interfering nodes for link \((i,j)\) \\ \hline \(\mathcal{I}_{i,j}[n]\) & Id of \(n\)_th_ interfering node for link \((i,j)\) \\ \hline \(\mathcal{L}_{i,j}=|\mathcal{I}_{i,j}|\) & Number of nodes that interfere with transmissions on \((i,j)\) \\ \hline \(\mathit{Src}(r_{k})\) & Source node of the \(k\)_th_ flow \\ \hline \(q_{\mathit{Src}(r_{k})}\) & Transmission probability of the originator of the \(k\)_th_ flow \\ \hline \(\mathcal{T}_{i,j}\) & Average throughput for \((i,j)\) (Packets/slot) \\ \hline \(\mathcal{T}_{r_{k}}\) & Average throughput for \(k\)_th_ flow (Packets/slot) \\ \hline \(\mathcal{T}_{\mathit{agg}}\) & Average throughput aggregate flow \\  & throughput (Packets/slot) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations.