# A Type of Low-Latency Data Gathering Method with Multi-Sink for Sensor Networks
## 3 Network Model  
As mentioned above, in sensor networks, random mobility and the fixed mobility schemes for data gathering are unable to balance energy consumption of nodes and the path for data gathering is not the optimal. While, the controlled mobility scheme in data gathering performs well in prolongingnetwork lifetime as well as reducing time delay and could efficiently mitigate the energy hole problem. Thus, it is adopted as the mobile strategy in LDGM.  
It is assumed that the network is a rectangular area whose length and width is \(M\) and \(L\) respectively. Nodes are deployed randomly in the area and meet the following conditions:  
1. Nodes are in static and high-density deployment. Total number and density of them is defined as \(n\) and \(\rho\) respectively.
2. All nodes except the Sink have the same initial energy \(E_{0}\). The transmission power as well as the sensing radius of each node is also adjustable.
3. There are several mobile Sinks in the network whose energy and storage space are unlimited.  
To establish the trajectories of the mobile Sinks and to reduce the amount of redundant data sensed by adjacent nodes, the network is divided into several virtual disks with radius \(R\). Each of them is called \"Data Gathering Unit\" (DGU). Centers of disks (white dots) are distributed on the transverse or longitudinal line, as shown in Figure 1.  
According to [1; 2; 3], data (especially the multimedia data) gathered by nodes closed to each other often show some similarity in the same period. Thus, to decrease the amount of the redundant data, the definition of the \"Data Gathering Area\" (DGA) is proposed as follows.  
As shown in Figure 1, the vertical distance from \(n_{3}\) to the line segment \(n_{1}n_{2}\) is just \(R\). Thus, the region consisting of DGU \(n_{1}\), \(n_{2}\) and \(n_{3}\) is defined as a DGA, just as the gray area in Figure 1.  
On the other hand, to reduce the probability of collecting redundant data, it is necessary to minimize area of the overlapping region between these DGAs. Thus, in Figure 2, DGU \(n_{3}\), \(n_{4}\) and \(n_{5}\) need to meet the requirement of the minimum coverage model and \(O\) is the point where these DGUs intersect. The distance between the centers of these adjacent DGUs is\(\sqrt{3}R\).  
In LDGM, the virtual DGUs are uniformly distributed in the network. For the centers of some DGUs, if their vertical coordinates are the same, these DGUs are regarded as in the same layer. Centers with the minimum vertical coordinate value are just located at the first layer. The number of DGUs in the first layer is defined as \(Num(L)\) and \(Num(C)\) is the number of the layers. Then, total number of DGAs (defined as \(Num(R)\)) is  
_Case 1_: \(Num(C)\) is even and \(2\times Num(L)\%3=0\) (Figure 2).  
\[Num\left(R\right)=\frac{2 Num\left(L\right)}{3}\times\frac{Num\left(C\right)} {2} \tag{1}\]  
Figure 1: Data gathering unit in the network.  
Case 2: \(Num(C)\) is even and \(2\times Num(L)\%3=1\) (Figure 3).  
\[Num\left(R\right)=\left(\frac{2 Num\left(L\right)-1}{3}+1\right)\times\frac{Num \left(C\right)}{2} \tag{2}\]  
Case 3: \(Num(C)\) is even and \(2\times Num(L)\%3=2\) (Figure 4).  
\[Num\left(R\right)=\left(\frac{2 Num\left(L\right)-2}{3}+1\right)\times\frac{Num \left(C\right)}{2} \tag{3}\]  
Case 4: Distribution of DGAs (_Case 3_).  
Figure 4: Distribution of DGAs (_Case 3_).  
Figure 3: Distribution of DGAs (_Case 2_).  
_Case 4_: \(Num(C)\) is odd (Figure 5).  
\[Num\left(R\right)=\left\{\begin{array}{ll}\frac{2 Num\left(L\right) \cdot 1}{3}\times P&\sigma=0\\ \left(\frac{2 Num\left(L\right)\cdot 1}{3}+1\right)\times P&\sigma=1\\ \left(\frac{2 Num\left(L\right)\cdot 2}{3}+1\right)\times P&\sigma=2\\ \end{array}\right. \tag{4}\]  
\[P=\frac{Num\left(C\right)+1}{2}\text{ and }\sigma=2\times Num\left(L\right) \%3\]