## 4 Formulating the Expression of the Total Number of Transmissions and Receptions
### Formulating \(N_{r}^{p}\)  
**Proposition 1**. : _The number of receptions in Stage 2 \(N_{r}^{P}\) is:_  
\[N_{r}^{P}=N_{s}N\pi r_{t}^{2}-C_{N_{s}}^{2}\pi^{2}Nr_{t}^{4}. \tag{23}\]  
Proof. : According to the procedures of Stage 2, \(N_{r}^{P}\) equals the number of neighboring nodes around all the source nodes \(N_{s,nei}\) minus the number of nodes \(N_{r2}\) located in the overlapping communication region of the two sources nodes. This relation occurs because each node receives just one packet and the number of receptions for those nodes is counted twice, thus \(N_{r}^{P}\) can be represented as follows:  
\[N_{r}^{P}=N_{s,nei}-N_{r2}. \tag{24}\]  
The average number of neighboring nodes for all source nodes \(N_{s,nei}\) is expressed as follows:  
\[N_{s,nei}=N_{s}N\frac{\pi r_{t}^{2}}{S}=N_{s}N\pi r_{t}^{2}. \tag{25}\]  
In Figure 5, the red circle denotes the communication region and \(S_{2}\) represents the shaded area jointly covered by the two source nodes. A and B are two intersections. When the distance between two source nodes \(d(O,O^{\prime})\) satisfies \(0<d(O,O^{\prime})\leq 2r_{t}\), \(N_{r2}\) exists. Thus, the probability \(p_{L}\) of an existing communication between the two nodes is expressed as follows:  
\[p_{L}=p\{d(O,O^{\prime})\leq 2r_{t}\}=\frac{\pi(2r_{t})^{2}}{S}=4\pi r_{t}^{2}. \tag{26}\]  
In the \(N_{s}\) source nodes, an average of \(N_{L}\) nodes pairs satisfy the condition in Equation (26) (i.e., \(N_{L}\) source nodes pairs can communicate with each other). The expressions for \(N_{L}\) and \(N_{r2}\) are, respectively, as follows:  
\[N_{L}=C_{N_{s}}^{2}.p_{L}=C_{N_{s}}^{2}.4\pi r_{t}^{2}. \tag{27}\]  
\[N_{r2}=N_{L}\times N\times\frac{\overline{S}_{2}}{S}. \tag{28}\]  
Because the nodes are uniformly distributed and \(0<d(O,O^{\prime})\leq 2r_{t}\), the probability \(p\{d\leq x\}\) is equal to  
\[F_{1}(x)=p\{d\leq x\}=\frac{\pi x^{2}}{\pi(2r_{t})^{2}}=\frac{x^{2}}{4r_{t}^{2}}. \tag{29}\]  
Thus, the probability density function (PDF) \(f_{1}(x)\) is  
1. Case 1: As presented in Figure 7, there are two broadcasting nodes \(\mathbf{n_{t1}^{q-1}}\) and \(\mathbf{n_{t2}^{q-1}}\) in the \((q-1)\)th forwarding, while their communication ranges are represented by the two red circles. This case can be divided into two situations via the distance \(\mathbf{d}\Big{(}\mathbf{n_{t1}^{q-1}},\mathbf{n_{t2}^{q-1}}\Big{)}\): (a) \(\mathbf{0}<\mathbf{d}\Big{(}\mathbf{n_{t1}^{q-1}},\mathbf{n_{t2}^{q-1}}\Big{)}<\mathbf{r_{t}}\); and (b) \(\mathbf{r_{t}}<\mathbf{d}\Big{(}\mathbf{n_{t1}^{q-1}},\mathbf{n_{t2}^{q-1}}\Big{)}<\mathbf{2r_{t}}\). Taking the first situation as an example, if node \(\mathbf{n_{t}^{q}}\) is located in the black area \(\mathbf{S_{4}}\), the nodes in the shadow area \(\mathbf{S_{5}}\) can receive packets from nodes \(\mathbf{n_{t}^{q}}\) or \(\mathbf{n_{t2}^{q-1}}\), thus the number of receptions of those nodes is counted twice, and that value should be subtracted from \(\mathbf{N_{t1}^{q}}\). Suppose that the number of receiving nodes in areas such as \(\mathbf{S_{5}}\) is \(\mathbf{N_{t2}^{q}}\) and that the number in areas such as \(\mathbf{S_{7}}\) is \(\mathbf{N_{r3}^{q}}\). 2. Case 2: Similarly, there are two transmitting nodes \(\mathbf{n_{t1}^{q}}\) and \(\mathbf{n_{t2}^{q}}\) in the \(q\)th forwarding, whose communication ranges are represented by two black circles in Figure 8. This case can be divided into two situations via the distance \(\mathbf{d}\Big{(}\mathbf{n_{t1}^{q}},\mathbf{n_{t}^{q-1}}\Big{)}\): (a) \(\mathbf{0}<\mathbf{d}\Big{(}\mathbf{n_{t1}^{q}},\mathbf{n_{t}^{q-1}}\Big{)}<\mathbf{r_{t}}\); and (b) \(\mathbf{r_{t}}<\mathbf{d}\Big{(}\mathbf{n_{t1}^{q}},\mathbf{n_{t}^{q-1}}\Big{)}<\mathbf{2r_{t}}\). Taking the first situation as an example, when node \(\mathbf{n_{t2}^{q}}\) is distributed in the black area \(\mathbf{S_{8}}\), the nodes located in shadow area \(\mathbf{S_{9}}\) receive one of the packets broadcasted by node \(\mathbf{n_{t1}^{q}}\) or \(\mathbf{n_{t2}^{q}}\).