# An Energy-Efficient Two-Stage Cooperative Routing Scheme in Wireless Multi-Hop Networks
## 2 System Model and Existing Schemes
### Existing Schemes  
This subsection introduces serval existing routing schemes including MTE [9], PC\(-l\)[10] which focus on minimizing the end-to-end transmission energy consumption, and FA [18], FACR [19] which are designed to maximize the network lifetime. The main difference between these schemes lies in the definition of link cost \(\mathcal{C}_{i,j}\) and the transmission model in each hop. **MTE transmission scheme**[9] defines \(\mathcal{C}_{i,j}^{\text{MTE}}\) as the minimum transmission power cost to achieve a packet transmission between node \(i\) and \(j\),  
\[\mathcal{C}_{i,j}^{\text{MTE}}=\min\Big{\{}\left|w_{i}\right|^{2}\Big{|}\gamma _{j}\geq\gamma_{\min}\Big{\}}=\frac{\gamma_{\min}\sigma^{2}}{\left|h_{i,j} \right|^{2}}. \tag{7}\]  
**PC\(-l\) transmission scheme**[10] introduces the multipoint-to-point cooperative model as Equation (6). In this case, the link cost is designed as the minimum transmission power cost to form the cooperative link \((\mathcal{T},j)\)  
\[\mathcal{C}_{\mathcal{T},j}^{\text{PC}}=\min\Bigg{\{}\sum\limits_{t\in \mathcal{T}}\left|w_{t}\right|^{2}\Bigg{|}\gamma_{j}\geq\gamma_{\min}\Bigg{\}} =\frac{\gamma_{\min}\sigma^{2}}{\sum\limits_{t\in\mathcal{T}}\left|h_{t,j} \right|^{2}}, \tag{8}\]  
where the last equation comes from the Cauchy-Schwartz inequality. The PC\(-l\) scheme starts from establishing a non-cooperative route \(r^{(1)}=\left\{V_{s},r_{2}^{(1)},\cdots,V_{d}\right\}\) with Equations (1) and (7). Then the source \(V_{s}\) sends data to next node \(r_{2}^{(1)}\) along the path and combines the first two nodes into a super node \(\mathcal{T}=\left\{V_{s},r_{2}^{(1)}\right\}\). Next, the link costs between the super node and other nodes are calculated with Equation (8), and the route is updated to \(r^{(2)}=\left\{\mathcal{T},r_{2}^{(2)},\cdots,V_{d}\right\}\). This process continues and at the \(k^{\text{th}}\) hop, the super node will contain all last \(\min\{l,k\}\) nodes, i.e., \(\mathcal{T}=\left\{r_{2}^{(k-l)},r_{2}^{(k-l+1)},\cdots,r_{2}^{(k-1)}\right\}\) if \(k>l\), or \(\mathcal{T}=\left\{V_{s},\cdots,r_{2}^{(k-1)}\right\}\). All nodes in \(\mathcal{T}\) transmit the same data to \(r_{2}^{(k)}\) with the cooperative beamforming model. It is obvious that MTE is a special case of PC-\(l\) with \(l=1\). **FA transmission scheme**[18] aims at improving the network lifetime and considers the normalized residual energy of transmitting node in calculating the link cost, i.e.,  
\[\mathcal{C}_{i,j}^{\text{FA}}=\min\Bigg{\{}\left|w_{i}\right|^{2x_{1}}\cdot \frac{E_{\text{ini}}^{2}}{E_{i}^{3x}}\Bigg{|}\gamma_{j}\geq\gamma_{\min} \Bigg{\}}=\Bigg{(}\frac{\gamma_{\min}\sigma^{2}}{\left|h_{i,j}\right|^{2}} \Bigg{)}^{x_{1}}\cdot\frac{E_{\text{ini}}^{x_{2}}}{E_{i}^{3x}}. \tag{9}\]  
where \(E_{i}\) is the residual energy of node \(i\) at current time, \(E_{\text{ini}}\) is the initial energy of node \(i\), and \((x_{1},x_{2},x_{3})\) are parameters that govern the effect of normalized residual energy. It can be seen from Equations (7) and (9) that MTE is a special case of FA with \((x_{1},x_{2},x_{3})=(1,0,0)\). **FACR transmission scheme**[19] introduces cooperative transmission into FA to further improve the network lifetime. The procedure of FACR is exactly the same as PC-\(l\) except that the link cost is replaced by  
\[\mathcal{C}_{\mathcal{T},j}^{\text{CR}}=\min\Bigg{\{}\sum\limits_{t\in \mathcal{T}}\frac{E_{\text{ini}}}{E_{t}}\left|w_{t}\right|^{2}\Bigg{|}\gamma_{ j}\geq\gamma_{\min}\Bigg{\}}=\frac{\gamma_{\min}\sigma^{2}}{\sum\limits_{t\in \mathcal{T}}\frac{E_{t}}{E_{\text{ini}}}\left|h_{t,j}\right|^{2}}. \tag{10}\]  
It should be noted that in the cooperative routing schemes like PC-\(l\) and FACR, nodes in transmitter set \(\mathcal{T}\) should have the CSI related to the receiver node to perform the cooperative beamforming. This implies that \(\mathcal{T}\) is within the neighbor set of receiver. PC-\(l\) and FACR can considerably improve the network performance compared with their non-cooperative version MTE and FA.