# Capacity of Heterogeneous Mobile Wireless Networks with \(D\)-Delay Transmission Strategy
## 7 Capacity Analysis  
In the Section above we have obtained the throughput capacity of the four types of mobile networks. In this section we analyze the impact of some important parameters such as \(n\), \(D\) and \(m\) to the network capacity. Firstly we list the capacity contributed by _ad hoc_ transmission mode as follows: in two-dimensional i.i.d. fast mobility model, we have \(\lambda_{A}(n)=O\left(\sqrt{\frac{D}{n}}\right)\); in two-dimensional i.i.d. slow mobility model, we have \(\lambda_{A}(n)=O\left(\sqrt[3]{\frac{D}{n}}\right)\); in three-dimensional i.i.d. fast mobility model, we have \(\lambda_{A}(n)=O\left(\sqrt[3]{\frac{D}{n^{2}}}\right)\); and in three-dimensional i.i.d. slow mobility model, we have \(\lambda_{A}(n)=O\left(\sqrt[3]{\frac{D}{n^{2}}}\right)\). Figure 9 shows \(\lambda_{A}(n)\) of the four types of networks. The relationship of \(\lambda_{A}(n)\) and \(n\) is shown in Figure 10. We find that fast mobility networks have much higher capacity than the slow mobility networks, especially in the case that the number of mobile nodes is small. The reason is that in fast mobility networks, the source nodes have more opportunities to communicate with the destination nodes directly. If the number of mobile nodes is small, the fast mobility is one of the most important factors which influence the network capacity. As \(n\) goes to infinity, the number of mobile nodes offers many opportunities for source nodes to meet destination nodes, even in slow mobility networks. Hence, if the number of mobile nodes is large enough, the capacities of the four types of networks are similar. Figure 9: **(a)**\(\lambda_{A}\) of two-dimensional i.i.d. fast mobility model; **(b)**\(\lambda_{A}\) of three-dimensional i.i.d. fast mobility model; **(c)**\(\lambda_{A}\) of two-dimensional i.i.d. slow mobility model; **(d)**\(\lambda_{A}\) of three-dimensional i.i.d. slow mobility model. Let \(D=\Omega(n^{2})\), we have \(\lambda_{A}=\Theta(1)\). In this case, the four types of mobile networks have the same capacity, which is \(\Theta(1)\). It means that in delay tolerant networks, the network capacity can achieve constant order, and the price is that the delay will go to infinity. Of course, the delay constraint \(D\) is approximated in the analysis above. Concretely, in two-dimensional mobile network model, \(D=\Omega(n)\); in three-dimensional mobile network model, \(D=\Omega(n^{2})\). However, since \(n\) is assumed to be large enough, \(D\) is not limited in both the two cases. Let \(D=O(n)\), we have \(\lambda_{A}=0\). In this case, the capacity will decrease to zero as \(n\) goes to infinity. Concretely, in two-dimensional mobile network model, \(D=O(n)\); in three-dimensional network model, \(D=O(n^{2})\). It implies that if the delay is assumed to be very small, the number of bits transmitted by _ad hoc_ transmission mode is very little. Our analysis is shown in Figure 11.  
When \(D\) falls in between \(O(n)\) and \(\Omega(n)\), there is a positive correlation between \(\lambda_{A}\) and the delay constraint. Theorems 5 and 16 offer the throughput capacity of backbone networks. Figure 12 shows the relationship of throughput capacity and the number of helping nodes. Figure 11: **(a)** The relationship of \(\lambda_{A}\) and \(D\) in two-dimensional network model; **(b)** The relationship \(\lambda_{A}\) of and \(D\) in three-dimensional network model. Figure 10: Relationship of \(\lambda_{A}(n)\) and the number of normal nodes. Combining the analysis above, we reach the following conclusions:  
1. In two-dimensional networks: \[\lambda\left(n,m\right)=\left\{ \begin{array}{cc}\Theta\left(1\right)+\Theta \left(\frac{\sqrt{m}W_{2}}{n}\right),&\text{if }D=\Omega\left(n\right)\\ \Theta\left(\frac{\sqrt{m}W_{2}}{n}\right),&\text{if }D=O\left(n\right) \end{array} \right.\]  
If the delay is allowed to be larger than some fixed constant \(D=\Omega(n)\), the capacity is \(\Theta\left(1\right)+\Theta\left(\frac{\sqrt{m}W_{2}}{n}\right)\). In this condition the network capacity can achieve constant order.