## 4 Distance-Based Energy Aware Routing (DEAR) Algorithm
### Theoretical Analysis of Hotspot Problem
#### 4.1.2 Time-Based Traffic Model  
Under the time-based traffic model, each node will take turns to transmit its data to a sink node. Therefore, the key difference between time-based and event/query-based traffic model is the packet length. For example, node 1 which is the furthest from sink node only needs to transmit its data once while node \(n\) which is closest to sink node has to transmit its own data once and help forward the data (\(n-1\)) times along the line. It is worth noting that if we use the same methodology to let \(d_{I}=d_{2}=\cdots=d_{n}=d/n\), node \(n\) will become a hotspot node and die quickly since it has more traffic burden to forward. On the other hand, node 1 will have much residual energy when node \(n\) dies, which is not desirable. Thus, the objective we have here is to let \(E_{I}=E_{2}=E_{3}\cdots=E_{n}\) in order to maximize network lifetime. Since node \(i\) will transmit its own \(l\)-bits data for once and forward the traffic for (\(i-1\)) times after all nodes take turn to transmit their data to sink node. The energy consumption for node \(i\) is:  
\[E_{i}=l\cdot(E_{elec}+\varepsilon_{amp}\cdot d_{i}^{\alpha})+l\cdot(i-1)(2E_{ elec}+\varepsilon_{amp}\cdot d_{i}^{\alpha})=l\cdot(2i-1)E_{elec}+l\cdot i\cdot \varepsilon_{amp}\cdot d_{i}^{\alpha} \tag{8}\]  
Let \(E_{i}=E_{i+1}\), we can finally get:  
\[d_{i+1}=q\sqrt{\frac{-2E_{elec}+i\varepsilon_{amp}d_{i}^{\alpha}}{\varepsilon_ {amp}(i+1)}}=q\sqrt{\frac{-2 iE_{elec}+\varepsilon_{amp}d_{1}^{\alpha}}{ \varepsilon_{amp}(i+1)}} \tag{9}\]  
Since \(d_{n}>0\), namely \(\frac{-2(n-1)E_{elec}+\varepsilon_{amp}d_{1}^{\alpha}}{\varepsilon_{amp}n}>0\), it must satisfy:  
\[n<\frac{\varepsilon_{amp}d_{1}^{\alpha}}{2E_{elec}}+1\;\;\text{or}\;\;d_{1}>q \sqrt{\frac{2(n-1)E_{elec}}{\varepsilon_{amp}}} \tag{10}\]  
Based on (10), we can get the corresponding lower bound distance \(d_{1}\) when hop number \(n=[2\text{:}9]\), as is shown in Table 2. Here, \(E_{elec}=50\)\(nJ/bit\), \(\alpha=4\) and \(\varepsilon_{amp}=0.001\)\(pJ/bit/m^{4}\). Given the multi-hop number \(n\), Table 2 provides the lower bound distance value \(d_{1}\) as well as the minimal source to sink node distance \(d=\sum d_{i}\). For example, when \(d=300\), we can only use 2-hop or 3-hop route to achieve energy balancing with practical distance \(d_{1}(2)>100\) or \(d_{1}(3)>118.9\). If the hop number \(n\geq 4\), the minimal \(d=\sum d_{i}\geq 307.6\) which is contrary to \(d=300\). On the other hand, given the source to sink node distance \(d\), there might be several multi-hop routes with different hop number \(n\). There exists the highest hop number above with the minimal energy consumption for each sensor node and this is the optimal multi-hop number we need. From Table 3, we can see different set of \(d_{1}(n)\) given the source to sink node distance \(d\). For example, when \(d=800\), we can either choose 8-hop route with \(d_{1}(8)=164.8\) or choose 7-hop route with \(d_{1}(7)=170.5\). The corresponding individual distance \(d_{i}\) can be deduced from Equation (9). It is worth noting that we cannot choose 9-hop route since the minimal \(\sum d_{i}(9)=840.9>d=800\). Thus, we can choose \(n\)-hop route with \(n\in[2,8]\) when \(d=800\). Finally, we choose the highest 8-hop route with \(d_{1}(8)=164.8\) since each of the 8 nodes consumes the least energy of \(E_{i}=E_{1}=l(E_{elec}+\varepsilon_{amp}\,d_{1}^{a}(8))\). From the analysis under both event/query based and time-based traffic model, we can conclude that:  
1. Given the source to sink node distance \(d\), the optimal multi-hop number \(n\) as well as each individual distance \(d_{i},i\in[1,n]\) can be determined so that all the sensor nodes consume their energy at similar rate;
2. The event or query based model will finally become time-based traffic model when the observing time is long enough. In that case, each sensor node will be almost uniformly chosen for once among all sensor nodes from time point of view, which is similar to the time-based traffic model. 3.