# Capacity of Heterogeneous Mobile Wireless Networks with \(D\)-Delay Transmission Strategy
## 4 Main Results  
In this section, we present the main results of this paper. In the study on capacity of mobile networks, many mobility models are proposed to model the mobility of networks. In this paper, we consider four mobility models: two-dimensional i.i.d. fast mobility network model, two-dimensional i.i.d. slow mobility network model, three-dimensional fast mobility network model and three-dimensional slow mobility network model. Our main results are summarized as follows:  
1. Two-dimensional i.i.d. fast mobility network model: it is shown that with a delay constraint \(D\), the capacity is \(\lambda\left(n,m\right)=O\left(\sqrt{\frac{D}{n}}\right)+\Theta\left(\frac{ \sqrt{m}W_{2}}{n}\right)\), where \(O\left(\sqrt{\frac{D}{n}}\right)\) is the throughput capacity contributed by _ad hoc_ transmission mode and \(\Theta\left(\frac{\sqrt{m}W_{2}}{n}\right)\) is the capacity contributed by helping transmission mode. 2. Two-dimensional i.i.d. slow mobility network model: It is shown that with a delay constraint \(D\), the capacity is \(\lambda\left(n,m\right)=O\left(\sqrt[3]{\frac{D}{n}}\right)+\Theta\left(\frac{ \sqrt{m}W_{2}}{n}\right)\), where \(O\left(\sqrt[3]{\frac{D}{n}}\right)\) is the throughput capacity contributed by _ad hoc_ transmission mode and \(\Theta\left(\frac{\sqrt{m}W_{2}}{n}\right)\) is the capacity contributed by helping transmission mode. 3. Three-dimensional i.i.d. fast mobility network model: It is shown that with a delay constraint \(D\), the capacity is \(\lambda\left(n,m\right)=O\left(\sqrt[3]{\frac{D}{n^{2}}}\right)+\Theta\left( \frac{\sqrt[3]{m^{2}}W_{2}}{n}\right)\), where \(O\left(\sqrt[3]{\frac{D}{n^{2}}}\right)\) is the capacitycontributed by _ad hoc_ transmission mode and \(\Theta\left(\frac{\sqrt[3]{m^{2}}W_{2}}{n}\right)\) is the capacity contributed by helping transmission mode. 4. Three-dimensional i.i.d. slow mobility network model: It is shown that with a delay constraint \(D\), the capacity is \(\lambda\left(n,m\right)=O\left(\sqrt[4]{\frac{D}{n^{2}}}\right)+\Theta\left( \frac{\sqrt[3]{m^{2}}W_{2}}{n}\right)\), where \(O\left(\sqrt[4]{\frac{D}{n^{2}}}\right)\) is the capacity contributed by _ad hoc_ transmission mode and \(\Theta\left(\frac{\sqrt[3]{m^{2}}W_{2}}{n}\right)\) is the capacity contributed by helping transmission mode. 5. **Intuitive Analysis** In [38], the authors proposed the virtual channel model. We adopt this idea to have some heuristic arguments on _ad hoc_ transmission mode. A virtual channel is a logic representation of a packet transmission from source node to destination node. According to the virtual channel model, there are two main constraints on the network capacity:  
1. Interference: In the process of packet transmissions, the near nodes interfere with each other, which limit the throughput capacity. 2. Mobility: Since source nodes, destination nodes and relay nodes are mobile all the time, the packets are not ensured to reach their destination nodes before the delay deadline. In a virtual channel mode, a packet is assumed to be transmitted from its source node to its destination node by a two-hop transmission. First, it is delivered from the source node to the relay node around the source. Then, it is delivered from the relay node to the destination node. Hence, in _ad hoc_ transmission mode, for a successful packet delivery, we consider three virtual channels, as shown in Figure 4:  
_Reliable Broadcasting Channel_: The source nodes transmit the packets to the relay nodes around it by broadcasting channel;  
_Untreliable Relay Channel_: The relay nodes move to the neighborhood of the destination;  
_Reliable Receiving Channel_: The relay nodes deliver the packets to the destination nodes. All the packets delivered to the destinations by _ad hoc_ transmission mode are via the three virtual channels. Using the virtual channel model, we present the intuitions on the capacity of our network model. First the two-dimensional i.i.d. fast mobility model is considered:  
Figure 4: The virtual channel model in _ad hoc_ transmission mode.