# Capacity of Heterogeneous Mobile Wireless Networks with \(D\)-Delay Transmission Strategy
## 6 Capacity of Heterogeneous Mobile Wireless Networks
### Capacity of Two-Dimensional i.i.d. Slow Mobility Networks  
Similar to the fast mobility model, we can calculate the throughput capacity of slow mobility model. First we present some important inequalities, as shown in Theorem 7:  
**Theorem 7**. : _In two-dimensional i.i.d. slow mobility networks, we have the following inequalities:_  
\[\sum_{B=1}^{\Lambda\left[T\right]}\sum_{h=1}^{H\left(B\right)}1\leqslant nW_{ 1}T \tag{16}\]\[\sum\limits_{B=1}^{\Lambda[T]}\sum\limits_{h=1}^{H(B)}\frac{\Delta^{2}}{16}\left( \alpha_{B}^{h}\right)^{2}\leqslant\frac{W_{1}T}{\pi} \tag{17}\]  
**Proof. ** Similarly with the proof of Theorem 1, we can prove the theorem above. \(\Box\)  
First we consider the case that all the packets are directly transmitted to the destination nodes without relaying. Theorem 8: _In two-dimensional i.i.d. slow mobility networks, if relaying progress is not allowed, the number of bits directly delivered to destination nodes is bounded as follows:_  
\[E\left[\Lambda^{d}[T]\right]\leqslant\frac{4\sqrt{2}}{3\sqrt{\Delta^{2}}}W_{1 }T\sqrt[3]{n^{2}}\]  
**Proof. ** According to Theorem 1 and the Cauchy-Schwarz inequality, we have:  
\[\left(\sum\limits_{B=1}^{\Lambda[T]}\sum\limits_{h=1}^{H(B)}\alpha _{B}^{h}\right)^{2} \leqslant\left(\sum\limits_{B=1}^{\Lambda[T]}\sum\limits_{h=1}^{H( B)}1\right)\left(\sum\limits_{B=1}^{\Lambda[T]}\sum\limits_{h=1}^{H(B)}\left( \alpha_{B}^{h}\right)^{2}\right)\] \[\leqslant nW_{1}T\sum\limits_{B=1}^{\Lambda[T]}\sum\limits_{h=1}^ {H(B)}\left(\alpha_{B}^{h}\right)^{2}\] \[\leqslant\frac{16 nW_{1}^{2}T^{2}}{\Delta^{2}\pi}\]  
where the Cauchy-Schwarz inequality is used in the first inequality. Since \(\sum\limits_{h=1}^{H(B)}\alpha_{B}^{h}\geqslant L_{B}\), we have that:  
\[\sum\limits_{B=1}^{\Lambda[T]}L_{B}\leqslant\frac{4\sqrt{n}W_{1}T}{\Delta \sqrt{\pi}} \tag{18}\]  
Similar to the analysis in Theorem 2, the following inequality can be obtained:  
\[E\left[\Lambda^{d}[T]\right]-E\left[\sum\limits_{B=1}^{\Lambda^{d}[T]}1_{L_{B }>L}\right]\leqslant W_{1}\pi nTL^{2} \tag{19}\]  
Combining inequality Equation (18) and inequality Equation (19), we have:  
\[\frac{4\sqrt{n}W_{1}T}{\Delta\sqrt{\pi}} \geqslant E\left[\sum\limits_{B=1}^{\Lambda^{d}[T]}L_{B}\right]\] \[\geqslant LE\left[\sum\limits_{B=1}^{\Lambda^{d}[T]}1_{L_{B}>L}\right]\] \[\sum\limits_{B=1}^{\Lambda[T]}\sum\limits_{h=1}^{H(B)}\frac{\Delta^{2}}{16}\left( \alpha_{B}^{h}\right)^{2}\leqslant\frac{W_{1}T}{\pi} \tag{17}\]0  
Let \(L=\sqrt{\frac{E\left[\Lambda^{d}[T]\right]}{2\pi W_{1}nT}}\), we can conclude that:  
\[\sum\limits_{B=1}^{\Lambda[T]}\sum\limits_{h=1}^{H(B)}\frac{\Delta^{2}}{16}\left( \alpha_{B}^{h}\right)^{2}\leqslant\frac{W_{1}T}{\pi} \tag{17}\]1Simplify the inequality above, we can obtain Theorem 8. \(\circ\)  
**Theorem 9**. : _In two-dimensional i.i.d. slow mobility network model, if packet relaying is allowed, the bound on the number of bits which are transmitted successfully from relay nodes to destination nodes is as follows:_  
\[\sum\limits_{B=1}^{\Lambda[T]}\sum\limits_{h=1}^{H(B)}\frac{\Delta^{2}}{16}\left( \alpha_{B}^{h}\right)^{2}\leqslant\frac{W_{1}T}{\pi} \tag{17}\]2  
Proof. : Similar to the steps in capacity analysis of network capacity of two-dimensional i.i.d. fast mobility networks, we have:  
\[\sum\limits_{B=1}^{\Lambda[T]}\sum\limits_{h=1}^{H(B)}\frac{\Delta^{2}}{16}\left( \alpha_{B}^{h}\right)^{2}\leqslant\frac{W_{1}T}{\pi} \tag{17}\]3  
Therefore, we can obtain the following inequalities:  
\[ \begin{array}{ll}E\left[\sum\limits_{B=1}^{\Lambda^{\prime}[T]}L_{B}\right] &\geqslant E\left[\sum\limits_{B=1}^{\Lambda^{\prime}[T]}L_{B}1_{L_{B}>L} \right]\\ &\geqslant E\left[\sum\limits_{B=1}^{\Lambda^{\prime}[T]}1_{L_{B}>L}\right]L \\ &\geqslant\left(E\left[\Lambda^{\prime}[T]\right]-E\left[\sum\limits_{B=1}^{ \Lambda^{\prime}[T]}1_{L_{B}<L}\right]\right)L\\ &\geqslant E\left[\Lambda^{\prime}[T]\right]L-nW_{1}T\pi DL^{3}\end{array}