# AEF: Adaptive En-Route Filtering to Extend Network Lifetime in Wireless Sensor Networks
## 3 Proposed Work
### Energy-Hole Model and Discussions  
In static sink-based WSNs, events or messages are replayed to the \(BS\) using a multihop many-to-one communication model. This results in uneven energy consumption loads. The energy-hole problem mathematical description is discussed in this section. Based on the energy-hole problem presented in Reference [46], we present a modified mathematical model for semi-rings where the \(BS\) is located at the middle bottom of the sensor field.  
A square sensor field of uniformly and randomly deployed sensors in an area of \(\mathrm{F_{w}\times F_{h}}=\left(\mathrm{F\times F}\right)\mathrm{m^{2}}\) is shown in Figure 4, where \(F=M\times r\) meters. In a uniformly and randomly deployed sensor network, the node density is uniform in the entire network:  
\[p=\frac{N}{A_{sn}} \tag{5}\]  
where \(N\) is the total number of sensor nodes and \(A_{sn}\) is the coverage area of the sensor network. \(BS\) is located at the bottom center of the sensor field; the whole area can be divided into \(\frac{M}{2}\) bands with step sizes of \(r\) meters. The ring 0 is a small semi-ring of radius \(r\) meters, and ring 1 has a radius of \(2r\) meters and so on. We assume that messages can move from one ring to the next using one hop for simplicity, although in practice, in order to cross a single ring, multiple hops may be traversed. It can be observed that most communication load frequencies are on nodes around the \(BS\) and on  
Figure 3: First-order radio model for energy consumption analysis.  
critical paths as compared to nodes farther from the \(BS\). Inner rings (in particular, ring 0, as it needs to relay all outer rings data to the \(BS\) in addition to replying ring data) also need to transmit their own messages. Therefore, intuitively, the rate of energy consumption load on rings closer to the \(BS\) is faster as compared to outer rings. Thus, as the ring 0 node's energy level reaches zero, the \(BS\) cuts off from the entire network, known as the energy-hole or hot-spot problem. Thus per node communication load (i) in semi-ring zero (\(s_{ring0}\)) that is half of the load in full ring is illustrated by Equation (6).  
\[\mathbf{l}_{s_{ring0}}=\frac{1}{2}(\frac{total\;network\; traffic}{number\;of\;nodes\;in\;s_{ring0}})=\frac{\rho(Mr)^{2}b}{2\rho\pi r^{2}}= \frac{M^{2}}{2\pi}b \tag{6}\]  
Similarly, for \(s_{ring1}\), per node communication load is given by Equation (7)  
\[\mathbf{l}_{s_{ring1}}=\frac{1}{2}(\frac{total\;load\;from\; outside\;s_{ring0}}{number\;of\;nodes\;in\;s_{ring0}1})=\frac{\left(\rho(Mr)^{2}-\pi r^{2} \right)b}{2\rho\left(\pi r^{2}-\pi r^{2}\right)}=\frac{\left(\frac{M^{2}}{\pi }-1\right)}{6}b \tag{7}\]  
and more generally, the \(i_{th}\) semi-ring load can be represented by Equation (8)  
\[\mathbf{l}_{s_{ring_{i}i_{th}}}=\frac{\left(\rho(Mr)^{2}-\pi\left(ir^{2} \right)\right)b}{2\rho\left(\pi(i+1)r)^{2}-\pi\left(ir\right)^{2}\right)}= \frac{\left(\frac{M^{2}}{\pi}-i^{2}\right)}{2(2i+1)}b \tag{8}\]  
where \(i=0,1,\ldots,(\frac{M}{2}-1)\).  
There is considerable difference in energy consumption in different semi-rings as can be observed from the above mathematical model. With proceeding inner rings, the rate of energy consumption increases around the \(BS\).  
Figure 4: Energy-hole problem due to a faster rate of energy consumption around the base station (BS) and critical paths.