# Capacity of Heterogeneous Mobile Wireless Networks with \(D\)-Delay Transmission Strategy
## 6 Capacity of Heterogeneous Mobile Wireless Networks
### Capacity of Three-Dimensional i.i.d. Fast Mobility Networks  
Inequality Equation (1) and inequality Equation (2) which hold in two-dimensional networks are also suitable for three-dimensional networks. However, in three-dimensional i.i.d fast mobility model, we should modify inequality Equation (3) to analyze the network capacity of the networks. **Theorem 12**: _In three-dimensional i.i.d fast mobility model, we have:_  
\[\sum_{B=1}^{\Lambda[T]}\frac{\Delta^{3}}{24}(\alpha_{B})^{3}\leq\frac{W_{1}T }{\pi} \tag{22}\]  
**Proof. ** The proof progress is similar with Theorem 1. \(\Box\)  
**Theorem 13**: _In three-dimensional i.i.d fast mobility model, we can obtain that the bounds on the number of bits directly delivered to destination nodes by ad hoc transmission mode:_  
\[E\left[\Lambda^{d}[T]\right]\leq\frac{6\sqrt[3]{6}}{\Delta}\sqrt[3]{W_{1}^{2 }T^{2}n}\]  
**Proof. ** According to the Cauchy-Schwarz inequality, we can obtain the following inequality:  
\[\left(\sum_{B=1}^{\Lambda^{d}[T]}\alpha_{B}\right)^{3} \leq \left(\sum_{B=1}^{\Lambda^{d}[T]}1\right)\left(\sum_{B=1}^{ \Lambda^{d}[T]}\alpha_{B}^{3}\right) \tag{23}\] \[\leq\Lambda^{d}[T]\frac{24W_{1}T}{\pi\Delta^{3}}\]  
which implies:  
\[E\left[\sum_{B=1}^{\Lambda^{d}[T]}\alpha_{B}\right]\leq\sqrt[3]{\Lambda^{d}[T] }\left(\sqrt[3]{\frac{24W_{1}T}{\pi\Delta^{3}}}\right)\]  
Since the node distribution is i.i.d in a three-dimensional space, according to the knowledge of Probability Theory, we have that:  
\[\Pr\left(dist(X_{i},X_{j})(t)\leq L\right)=\frac{4\pi}{3}L^{3}\]  
According to the inequality above, we conclude that:  
\[E\left[\sum_{t=1}^{T}\left(\sum_{i=1}^{n-1}1_{dist(X_{1},X_{i+1})(t)\leq L} \right)\right]=\frac{4\pi}{3}TnL^{3}\]  
In single time slot, each node can carry at most \(W_{1}\) bits. Hence we have:  
\[\sum_{B=1}^{\Lambda^{d}[T]}1_{\alpha_{B}\leq L}\leq W_{1}\sum_{t=1}^{T}\sum_{ i=1}^{n-1}1_{dist(X_{i},X_{i+1})(t)\leq L}\]  
Taking expectation on the inequality above, we have:  
\[E\left[\Lambda^{d}[T]\right]-E\left[\sum_{B=1}^{\Lambda^{d}[T]}1_{\alpha_{B}> L}\right]\leq\frac{4\pi}{3}W_{1}nTL^{3} \tag{24}\]  
Using the Taylor formula, we can obtain the following inequality:  
\[\Pr\left(L_{B}^{\prime}\leq L\right)=\frac{4\pi}{3}DL^{3}+o(\frac{4\pi}{3}DL^{3}) \leq\frac{4\pi}{3}DL^{3}\]  
Then we can get the bound on the expectation of the number of bits which satisfy \(L_{B}^{\prime}\leq L\):  
\[E\left[\Lambda^{d}[T]\right]\leq\frac{6\sqrt[3]{6}}{\Delta}\sqrt[3]{W_{1}^{2 }T^{2}n}\]0  
Since \(\sum\limits_{B\in R\left[T\right]}1_{L_{B}^{\prime}\leq L}\geq\sum\limits_{B= 1}^{\Lambda^{\prime}\left[T\right]}1_{\alpha_{B}\leq L}\), we have:  
\[E\left[\Lambda^{d}[T]\right]\leq\frac{6\sqrt[3]{6}}{\Delta}\sqrt[3]{W_{1}^{2 }T^{2}n}\]1  
Furthermore, we have:  
\[E\left[\Lambda^{d}[T]\right]\leq\frac{6\sqrt[3]{6}}{\Delta}\sqrt[3]{W_{1}^{2 }T^{2}n}\]2 \[E\left[\Lambda^{d}[T]\right]\leq\frac{6\sqrt[3]{6}}{\Delta}\sqrt[3]{W_{1}^{2 }T^{2}n}\]3 \[E\left[\Lambda^{d}[T]\right]\leq\frac{6\sqrt[3]{6}}{\Delta}\sqrt[3]{W_{1}^{2 }T^{2}n}\]4 \[E\left[\Lambda^{d}[T]\right]\leq\frac{6\sqrt[3]{6}}{\Delta}\sqrt[3]{W_{1}^{2 }T^{2}n}\]5  
Combining inequality Equation (25) and inequality Equation (26), we can obtain the following inequalities:  
\[E\left[\Lambda^{d}[T]\right]\leq\frac{6\sqrt[3]{6}}{\Delta}\sqrt[3]{W_{1}^{2 }T^{2}n}\]6 \[E\left[\Lambda^{d}[T]\right]\leq\frac{6\sqrt[3]{6}}{\Delta}\sqrt[3]{W_{1}^{2 }T^{2}n}\]7 \[E\left[\Lambda^{d}[T]\right]\leq\frac{6\sqrt[3]{6}}{\Delta}\sqrt[3]{W_{1}^{2 }T^{2}n}\]8  
where [PERSON]'s inequality is used in the first inequality. Let \(L=\sqrt[3]{\frac{E\left[\Lambda^{\prime}\left[T\right]\right]}{2\pi W_{1}nT}}\), we can prove Theorem 14. Combining Theorems 13 and 14, we can obtain the capacity contributed by _ad hoc_ transmission mode:  
**Theorem 15**. : _In three-dimensional i.i.d. fast mobility networks, the ad hoc transmission mode capacity is:_  
**Proof.