# Throughput Capacity of Selfish Wireless _Ad Hoc_ Networks with General Node Density
## 3 Achievable Rate
### The Rate for Transporting a Packet  
To achieve the rate of transmit a packet, we use the Time Division Multiplex (TDM) strategy. The idea of the TDM strategy is that when a node transmits along a path, other nodes which are far away can simultaneously transmit without causing excessive interference, as shown in Figure 4.  
The following Theorem introduces the fact that the rate can be obtained _w.h.p._ on the path simultaneously. The theorem is stated in slightly more general terms considering nodes at distance \(d\) in the edge percolation grid, where \(d\) is not the Euclidean distance but the number of \(d\) squares away. **Theorem 3**: _In each square, for any integer \(d>0\), the rate that source-destination pair can be obtained is_  
\[R(d)= \begin{cases}\Omega\left(\left(\frac{d}{\sqrt{\zeta}p(n)}\right)^{-\alpha -2}\right)&\text{if }dl(n)=\Omega(1)\\ \Omega\left(d^{-2}\right)&\text{if }dl(n)=O(1)\end{cases}  \tag{6}\]  
As depicted in Figure 4, we partition the network into squares and divide the time frame into \(k^{2}\) successive slots, where \(k=2(d+1)\). Then, the disjoint set of squares of \(s_{i}\) can transmit simultaneously. Figure 3: There are at least \(\delta\log m(n)\) disjoint highways in each slab. And the three phase routing scheme is illustrated. For a specific square \(s_{i}\), a node in square \(s_{i}\) transmits toward a destination node located in a square at distance \(d\) away. At the same time slot, there are four closest squares located at Euclidean distance at least \(l(n)(d+1)\) from the receiver, the next eight closest squares are at least \(l(n)(3d+3)\) Euclidean distance, and so on. By extending the sum of the interference of the network area, we can calculate the upper bound of the interference at the receiver as  
\[I(d,n)\leq\sum_{j=1}^{\infty}4j\cdot P(l(n)(2j-1)(d+1))^{-\alpha}\leq P(l(n)(d +1))^{-\alpha}\times\sum_{j=1}^{\infty}4j(2j-1)^{-\alpha} \tag{7}\]  
where \(P\) is the transmission power, and we notice that this sum is converged if the path loss exponent \(\alpha>2\). Thus, we can get that \(I(d,n)=O\left(\left(dl(n)\right)^{-\alpha}\right)\). Next we will give a lower bound of the signal received from the transmitter. According to the interference mode, we notice that the distance between transmitter and receiver is at most \(\sqrt{2}l(n)(d+1)\). Hence, the lower bound of the signal \(S(d)\) at the receiver is  
\[S(d,n)\geq P\left(\sqrt{2}(d+1)l(n)\right)^{-\alpha} \tag{8}\]  
Next, By General Physical Model [4], combining the interference and the receive signal, we can achieve transmission rate at receiver located \(d\) squares away is  
\[R(d)=\lim_{n\rightarrow\infty}\log\left(1+\frac{S(d,n)}{N_{0}+I(d,n)}\right)= \begin{cases}\Omega\left(\left(dl(n)\right)^{-\alpha}\right)&\text{if }dl(n)= \Omega(1)\\ \Omega\left(1\right)&\text{if }dl(n)=O(1)\end{cases}  \tag{9}\]  
This means that there is a threshold on the rate, which is related to \(d\) and square length \(l(n)\). In addition, based on the TDM strategy we adopted, there are \(k^{2}=4(d+1)^{2}\) time slots in our TDM strategy. Thus, the actual rate available needs to be divided by \(k^{2}\). Correspondingly,  
\[R(d)= \begin{cases}\Omega\left(l^{-\alpha}(n)d^{-\alpha-2}\right)&\text{if }dl(n)= \Omega(1)\\ \Omega\left(d^{-2}\right)&\text{if }dl(n)=O(1)\end{cases}  \tag{10}\]  
Figure 4: The time division multiplex (TDM) strategy and the case \(d=3\). Gray squares can transmit simultaneously. Notice that around each grey square there is a “silent” region of squares that are not allowed to transmit in the given time slot. Equation (10) acts as the groundwork to compute the rate in this work. Since node in this paper is distributed with general node density and nodes are possibly selfish, it makes it more complicated than previous works. In particular, we partition the square dynamically to ensure there exists an AN in each square _w.l.p._ The rate between transmitter and receiver is associated with the node density \(\zeta\) and probability of altruist \(p(n)\).