# Capacity of Heterogeneous Mobile Wireless Networks with \(D\)-Delay Transmission Strategy
## 4 Main Results
_Reliable Broadcasting Channel:_ Let \(L_{1}\) denote the transmission radius of source node. Hence, the source can cover an area of \(\pi(L_{1})^{2}\). For simplicity, we omit the guard zone \(\Delta\). Hence, there are at most \(\dfrac{1}{\pi\left(L_{1}\right)^{2}}\) transmitting pairs in the network in a time slot. On average, every source node has \(P_{1}\) fraction of time to broadcast, where:  
\[P_{1}=\dfrac{1}{\pi\left(L_{1}\right)^{2}n}\]  
Therefore, for reliable broadcast channel, its throughput is:  
\[\dfrac{W_{1}}{\pi\left(L_{1}\right)^{2}n}\]  
There exist \(\pi(L_{1})^{2}\)\(n\) mobile nodes in a disk with radius \(L\) on average. Hence, each broadcast can transmit packets \(\pi(L_{1})^{2}\)\(n\) nodes as duplicate copies. _Unreliable Relay Channel:_ We assume that the transmission radius for relay nodes is \(L_{2}\). Since the nodes are distributed in a square of unit area, we assume that \(L_{2}\) is far less than \(1\). For a duplicated packet, the probability that it fail to reach the destination node within \(D\) time slots is:  
\[P_{2}=\left(1-\pi\left(L_{2}\right)^{2}\right)^{D}\]  
As what analyzed above, a sent packet is delivered to \(\pi L_{2}^{2}n\) relay node on average. Hence there are \(\pi L_{2}^{2}n\) duplicate copies. Therefore, within \(D\) time slots, the probability that there are no packets fallen into the area covered by its destination node is:  
\[P_{3}=\left(1-\pi\left(L_{2}\right)^{2}\right)^{\pi\left(L_{1}\right)^{2}nD}\]  
_Reliable Receiving Channel:_ Now we consider the packets delivered from relay nodes to destination nodes. Since \(L_{2}\) is a common transmission radius which all the relay nodes use, the number of transmitting relay-destination pairs in a time slot is no more than \(\dfrac{1}{\pi\left(L_{2}\right)^{2}}\). As the highest transmission rate is \(W_{1}\) for relay nodes, the per-node reliable receiving channel can be expressed by  
\[\dfrac{W_{1}}{\pi\left(L_{2}\right)^{2}n}\]  
As shown in Figure 5, for a source-destination pair in the virtual channel model, the maximum throughput is:  
\[\lambda=\underset{L_{1},L_{2}}{\text{max}}\ \text{min}\left\{\dfrac{W_{1}}{\pi \left(L_{1}\right)^{2}n}\left(1-\left(1-\pi\left(L_{2}\right)^{2}\right)^{ \pi\left(L_{1}\right)^{2}nD}\right),\dfrac{W_{1}}{\pi\left(L_{2}\right)^{2}n}\right\}\]  
Figure 5: The virtual channel representation for the two-dimensional i.i.d. fast mobility network model. Next we consider the two-dimensional i.i.d. networks with slow mobility. In virtual channel systems, the relay channel and the receiving channel are the same with the analysis above. Hence we only need to consider the broadcasting channel. For each broadcasting channel, the throughput capacity is:  
\[\frac{W_{1}}{\pi L_{1}\sqrt{n}}\]  
Hence, the virtual channel systems are as shown in Figure 4:0  
We can conclude that for a source-destination pair, the maximum throughput is:  
Figure 4:1  
Now we consider the three-dimensional i.i.d. fast mobility networks. Different from the analysis above, all the nodes are distributed in a three-dimensional space in this network model. Hence, the broadcasting channel is:  
Figure 4:2  
The probability that there are no packets fallen into the area covered by its destination node within \(D\) time slots is:  
Figure 4:3  
The receiving channel is:  
Figure 4:4  
Figure 7 shows the virtual channel model for three-dimensional i.i.d. fast mobility networks, and the throughput capacity is:  
Figure 4:5  
Figure 4:6 The virtual channel representation for the two-dimensional i.i.d. slow mobility network model. Next we turn to the three-dimensional i.i.d. slow mobility network model. The relay channel and the receiving channel have been obtained in the analysis of three-dimensional fast mobility networks. As described in [38], the broadcasting channel in three-dimensional slow mobility can be expressed by:  
Figure 4:7  
As shown in Figure 8, we can obtain the throughput capacity:  
Figure 4:8