# 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. Fast Mobility Networks
According to Cauchy-Schwarz inequality and inequality Equation (3), we obtain:  
\[|R[T]|\leq nW_{1}T \tag{2}\]7 \[|R[T]|\leq nW_{1}T \tag{2}\]8where Cauchy-Schwarz inequality is used in the first inequality. Hence we have:  
\[|R[T]|\leq nW_{1}T \tag{2}\]9  
Consider bit \(B\), let \(d_{B}\) denote its destination node and \(e_{B}\) denote the node storing bit \(B\). We assume that from time slot \(t_{B}\) to time slot \(t_{B}+D-1\), the minimum distance between \(d_{B}\) and \(e_{B}\) is \(L_{B}\). Then we have:  
\[\sum\limits_{B=1}^{\Lambda[T]}\frac{\Delta^{2}}{16}\left(\alpha_{B}\right)^{2 }\leq\frac{W_{1}T}{\pi} \tag{3}\]0  
For an arbitrary bit \(B\) which is in the set \(R[T]\), we have:  
\[\sum\limits_{B=1}^{\Lambda[T]}\frac{\Delta^{2}}{16}\left(\alpha_{B}\right)^{2 }\leq\frac{W_{1}T}{\pi} \tag{3}\]1  
According to [PERSON]'s formula, equality Equation (14) can be expressed as:  
\[\sum\limits_{B=1}^{\Lambda[T]}\frac{\Delta^{2}}{16}\left(\alpha_{B}\right)^{2 }\leq\frac{W_{1}T}{\pi} \tag{3}\]2  
Therefore, we obtain:  
\[\sum\limits_{B=1}^{\Lambda[T]}\frac{\Delta^{2}}{16}\left(\alpha_{B}\right)^{2 }\leq\frac{W_{1}T}{\pi} \tag{3}\]3  
Obviously, we have:  
\[\sum\limits_{B=1}^{\Lambda[T]}\frac{\Delta^{2}}{16}\left(\alpha_{B}\right)^{2 }\leq\frac{W_{1}T}{\pi} \tag{3}\]4  
Combining Equations (7) and (8), we have:  
\[\sum\limits_{B=1}^{\Lambda[T]}\frac{\Delta^{2}}{16}\left(\alpha_{B}\right)^{2 }\leq\frac{W_{1}T}{\pi} \tag{3}\]5  
Hence, we can obtain the following inequalities:  
\[\sum\limits_{B=1}^{\Lambda[T]}\frac{\Delta^{2}}{16}\left(\alpha_{B}\right)^{2 }\leq\frac{W_{1}T}{\pi} \tag{3}\]6 \[\sum\limits_{B=1}^{\Lambda[T]}\frac{\Delta^{2}}{16}\left(\alpha_{B}\right)^{2 }\leq\frac{W_{1}T}{\pi} \tag{3}\]7 \[\sum\limits_{B=1}^{\Lambda[T]}\frac{\Delta^{2}}{16}\left(\alpha_{B}\right)^{2 }\leq\frac{W_{1}T}{\pi} \tag{3}\]8 \[\sum\limits_{B=1}^{\Lambda[T]}\frac{\Delta^{2}}{16}\left(\alpha_{B}\right)^{2 }\leq\frac{W_{1}T}{\pi} \tag{3}\]9  
According to [PERSON]'s inequality, combining inequality Equation (6) and inequality Equation (9), we have:  
\[E\left[\Lambda^{d}[T]\right]\leq\frac{8\sqrt{2}}{\Delta}W_{1}T\sqrt{n}\]0 \[E\left[\Lambda^{d}[T]\right]\leq\frac{8\sqrt{2}}{\Delta}W_{1}T\sqrt{n}\]1 \[E\left[\Lambda^{d}[T]\right]\leq\frac{8\sqrt{2}}{\Delta}W_{1}T\sqrt{n}\]2  
where [PERSON]'s inequality is used in the first inequality. Let \(L=\sqrt{\frac{E\left[\Lambda^{\gamma}[T]\right]}{2\pi W_{1}nTD}}\), we can have \(E[\Lambda^{\gamma}[T]]\). \(\Box\)  
**Theorem 4**. : _In two-dimensional i.i.d. fast mobility networks, the ad hoc transmission mode capacity is:_  
\[E\left[\Lambda^{d}[T]\right]\leq\frac{8\sqrt{2}}{\Delta}W_{1}T\sqrt{n}\]3  
**Proof. ** Combining Theorems 2 and 3, we have:  
\[E\left[\Lambda^{d}[T]\right]\leq\frac{8\sqrt{2}}{\Delta}W_{1}T\sqrt{n}\]4  
In the meaning of order, the throughput capacity contributed by relaying nodes plays a dominating role compared with the throughput without relaying node. Thus, we have \(E\left[\frac{\Lambda\left[T\right]}{n}\right]\leqslant O\left(\frac{\sqrt{nD }}{n}\right)\). \(\Box\)  
Now we consider the capacity contributed by the helping nodes. In two-dimensional network model, the helping nodes are regularly placed and divide the area into \(\left(\frac{1}{\sqrt{m}}\right)\times\left(\frac{1}{\sqrt{m}}\right)\) equal-size squares. In the transmissions of helping nodes, there are three phases: uplink phase, downlink phase and HN-to-HN phase. In uplink phase, the source nodes deliver the packets to the nearest helping nodes. In this case we call the helping nodes'sources'. Then, the helping nodes transmit the packets to the helping nodes that are in the same cells with the destination nodes. In this case we call the helping nodes 'destinations'. Finally, the helping nodes deliver the packets to the destination nodes by broadcasting. Theorem 5 gives the capacity contributed by the backbone network which consists of helping nodes. **Theorem 5**.