# 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  
We will first present some important inequalities firstly, which are shown in the following theorem:  
**Theorem 1**: _In two-dimensional mobile networks, we have the following inequalities:_  
\[\Lambda[T]\leq nW_{1}T \tag{1}\]  
\[|R[T]|\leq nW_{1}T \tag{2}\]  
\[\sum\limits_{B=1}^{\Lambda[T]}\frac{\Delta^{2}}{16}\left(\alpha_{B}\right)^{2 }\leq\frac{W_{1}T}{\pi} \tag{3}\]  
**Proof. ** For one node, the largest number of bits it can deliver in one time slot is \(W_{1}\) bits. Therefore, in \(T\) time slots, all the nodes in the network can take at most \(nW_{1}T\) bits, which are necessarily larger than the packets successfully transmitted to the destination nodes from source nodes, as shown in inequality Equation (1). It is also larger than the number of bits which are stored at relay nodes. Hence we have equality Equation (2). \(\Box\)  
Now we investigate the throughput capacity contributed by _ad hoc_ transmission mode. First we will analyze the number of bits directly transmitted to the source nodes without relaying in \(T\) time slots, which is denoted by \(\Lambda^{d}[T]\). We have the following theorem:  
**Theorem 2**: _In two-dimensional i.i.d fast mobility model, without considering relaying progress, we can bound the number of bits directly delivered to destination nodes as:_  
\[E\left[\Lambda^{d}[T]\right]\leq\frac{8\sqrt{2}}{\Delta}W_{1}T\sqrt{n}\]  
**Proof. ** Consider inequality Equation (3), according to the Cauchy-Schwarz inequality, we have:  
\[\left(\sum\limits_{B=1}^{\Lambda^{d}[T]}\left(\alpha_{B}\right) \right)^{2} \leq\left(\sum\limits_{B=1}^{\Lambda^{d}[T]}1\right)\left(\sum \limits_{B=1}^{\Lambda^{d}[T]}\left(\alpha_{B}\right)^{2}\right)\] \[\leq\Lambda^{d}[T]\frac{16 WT}{\pi\Delta^{2}}\]  
which can be expressed by:  
\[E\left[\sum\limits_{B=1}^{\Lambda^{d}[T]}\alpha_{B}\right]\leq E\left[\sqrt{ \Lambda^{d}[T]}\right]\left(\sqrt{\frac{16 WT}{\Delta^{2}\pi}}\right) \tag{4}\]  
The inequality above gives a bound on the expected distance which the bits take. For two arbitrary nodes \(X_{i}\) and \(X_{j}\), let \(dist(X_{i},X_{j})(t)\) denote their distance in time slot \(t\). Based on the assumption that the mobility model are i.i.d, we have:  
\[\Pr\left(dist(X_{i},X_{j})(t)\leq L\right)=\pi L^{2}\]  
In \(T\) time slots, we have:  
\[E\left[\sum\limits_{t=1}^{T}\left(\sum\limits_{i=1}^{n-1}1_{dist(X_{1},X_{i+1} )(t)\leq L}\right)\right]=Tn\pi L^{2}\]The network can carry at most \(W_{1}\) bits in single time slot, hence we can obtain:  
\[\sum\limits_{B=1}^{\Lambda^{d}[T]}1_{\alpha_{B}\leq L}\leq W_{1}\sum\limits_{t=1 }^{T}\sum\limits_{i=1}^{n-1}1_{dist(X_{i},X_{i+1})(t)\leq L}\]  
According to the inequalities above, we have:  
\[|R[T]|\leq nW_{1}T \tag{2}\]0  
By [PERSON]'s inequality, combining inequality Equation (4) and inequality Equation (5), we can obtain the following inequality:  
\[|R[T]|\leq nW_{1}T \tag{2}\]1 \[|R[T]|\leq nW_{1}T \tag{2}\]2 \[|R[T]|\leq nW_{1}T \tag{2}\]3 \[|R[T]|\leq nW_{1}T \tag{2}\]4  
where [PERSON]'s inequality is used in the first inequality. Let \(L=\sqrt{\frac{E\left[\Lambda^{d}[T]\right]}{2\pi W_{1}nT}}\), we obtain:  
\[|R[T]|\leq nW_{1}T \tag{2}\]5  
Simplify the inequality above, we can obtain \(E[\Lambda^{d}[T]]\). \(\Box\)  
Theorem 2 has given number of bits directly delivered to destination nodes. Now we consider the scenario where packets can be relayed in the networks. Theorem 3: _In two-dimensional i.i.d fast mobility model, if packet relaying exists, the bound on the number of bits transmitted to the destination number through relay nodes is:_  
\[|R[T]|\leq nW_{1}T \tag{2}\]6  
where \(D\) is delay constant. Proof: In \(T\) time slots, let \(\Lambda^{r}[T]\) denote the number of bits which are transmitted from relay nodes to destination nodes in \(T\) time slots, we have \(\Lambda[T]=\Lambda^{r}[T]+\Lambda^{d}[T]\).