## 3 System Model and Problem Statement
### Problem Statement  
The MRP algorithm uses the ACO algorithm to search for multiple paths after cluster formation. The process of ants moving will result in multiple paths forming between CH and sink. After multiplepaths are formed, data will be transferred along the multiple paths. The model of data transmission in MRP is shown in Figure 1.  
From Figure 1, we conclude that MRP can maximize the network lifetime in two ways. One is to reduce the transmitted data by clustering, which can use data aggregation to reduce energy consumption. The other is to use multiple paths to achieve load balancing (_i.e._, it can avoid frequently using the path in which the minimum energy node is located).  
Based on the above introduction, we can describe the objective of MRP as follows: maximizing the network lifetime (\(T_{net}\)), while minimizing the energy consumption between the CH and sink, which can be formulated as follows:  
\[\max T_{net},\ \min\sum_{\{i,j\in V,(i,j)\in E\}}e_{ij} \tag{3}\]  
_Theorem 1:_ Network lifetime is associated with residual energy of node, the energy consumption and the number of hops in path \(p\).  
In order to prove the theorem, we use a simplified model of energy consumption, in which energy consumption is the same in each node.  
_Prove:_\(N\) is the set of discovered paths between the CH and sink. The energy consumption of path \(p\) ( \(p\) \(\in N\) ) is the sum of the energy expended at each sensor node along the path. If (\(n_{1}\),\(n_{2}\),...,\(n_{m}\)) denotes the sequence of nodes along path \(p\), the total energy consumption _E(p)_ is given by  
\[E(p)=\sum_{k=1}^{m-1}(E_{r}+E_{cpu}+E_{r})=(E_{r}+E_{cpu}+E_{r})\times(m-1) \tag{4}\]  
where \(E_{r}\) and \(E_{r}\) represent the energy consumption of receiving or transmitting \(L\)-bit data, respectively. \(E_{cpu}\) is the energy consumption used in these jobs, such as computation, sensing events, _etc_.  
From [3], we have  
\[E_{cpu}<<E_{r}+E_{r} \tag{5}\]  
Therefore, we have  
\[E(p)=(E_{r}+E_{r})\times(m-1) \tag{6}\]  
In (6), because the value of (m-1) is equal to the number of hops between the CH and sink, we have  
\[E(p)=(E_{r}+E_{r})\times h_{CH}(p) \tag{7}\]  
Figure 1: Data Transmission Model in MRP.  
where \(h_{\text{CH}}\)(p) is the number of hops in path \(p\). Thus, \(E_{\text{r}}+E_{\text{t}}\) can be given by  
\[E_{r}+E_{t}=E(p)/h_{\text{CH}}(p) \tag{8}\]  
Since the working time _T(p)_ of path \(p\) is partly determined by the minimum energy node in path \(p\), _T(p)_ is given by  
\[T(p)=E_{\text{min}}(p)/(E_{r}+E_{t}) \tag{9}\]  
where \(E_{\text{min}}(p)\) is the current energy of the minimum energy node in path \(p\).  
Using (8) and (9), we have  
\[T(p)=E_{\text{min}}(p)\times h_{\text{CH}}(p)/E(p) \tag{10}\]  
We define the network lifetime \(T_{\text{net}}\) as the time when the first node in the network runs out of energy. Then, \(T_{\text{net}}\) is given by  
\[T_{\text{net}}=\min_{p\in N}T(p)=\min_{p\in N}E_{\text{min}}(p)\times h_{\text {CH}}(p)/E(p) \tag{11}\]  
Therefore, \(T_{\text{net}}\) is associated with the residual energy of the minimum energy node, energy consumption in a path and hop distance to sink.  
According to (3) and (11) it is obvious that maximizing the network lifetime \(T_{\text{net}}\) is equivalent to maximizing the minimum _T(p)_. We have  
\[\max\underset{p\in N}{\min}T(p) \tag{12}\]  
From (12), we infer that MRP needs to prolong working time of the minimum energy node and reduce the energy consumption in a path in order to maximize the network lifetime.