## 4 Objective Function Under Different Transmission Modes
### Two-Hop Transmission  
The two-hop transmission scheme improves link quality by replacing long-distance direct transmission with two sequential phases. This process requires two time slots. In the first time slot, the source transmits data to the relay. The relay then forwards the data to the destination in the second time slot. The received SNRs for the two time slots are denoted, respectively, as \(\mathrm{\gamma}_{\mathrm{S,\,R}}(\mathrm{t})=\frac{\mathrm{P}_{\mathrm{t}} \times\mathrm{d}_{\mathrm{S,\,R}}^{\mathrm{d}_{\mathrm{S,\,R}}}(\mathrm{t})}{ \mathrm{N}_{0}}\) and \(\mathrm{\gamma}_{\mathrm{R,\,D}}(\mathrm{t})=\frac{\mathrm{P}_{\mathrm{t}} \times\mathrm{d}_{\mathrm{S,\,D}}^{\mathrm{d}_{\mathrm{S,\,D}}}(\mathrm{t})}{ \mathrm{N}_{0}}\). The maximum achievable instantaneous mutual information for the link is determined using the following equation:  
\[\mathrm{C}_{\mathrm{MT}}^{\varepsilon}(\mathrm{t})=\min\{\mathrm{C}_{\mathrm{ S,\,R}},\,\mathrm{C}_{\mathrm{R,\,D}}\}=\min\biggl{\{}\mathrm{Blog}_{2}\biggl{(}1+ \frac{1.5\varepsilon\,\times\,\mathrm{\gamma}_{\mathrm{S,\,R}}(\mathrm{t})}{0.2 -\varepsilon}\biggr{)},\,\,\mathrm{Blog}_{2}\,\frac{1.5\varepsilon\,\times\, \mathrm{\gamma}_{\mathrm{R,\,D}}(\mathrm{t})}{0.2-\varepsilon}\biggr{\}}. \tag{12}\]The protocol interference model mandates that all nodes within the transmitter's coverage area remain silent during ongoing transmissions. This defines the interference region of a communication link as the union of active nodes within the transmission range. Each transmission phase generates distinct interference patterns. Because the two hops operate in orthogonal time slots, their interference regions are defined as \(\mathrm{I_{S,\ R}}\ =\mathrm{I(S)}\ \bigcup\ \mathrm{I(R)}\) and \(\mathrm{I_{R,\ D}}\ =\mathrm{I(R)}\ \bigcup\ \mathrm{I(D)}\), respectively. As the two phases are temporally separated, the effective interference for the complete two-hop transmission is governed by the maximum interference region:  
\[\mathrm{I_{MT}}=\mathrm{max}(\mathrm{I_{S,\ R}},\ \mathrm{I_{R,\ D}}). \tag{13}\]  
Figure 5 illustrates the mobility relationships. In the figure, \(\mathrm{V_{S,\ R}}\) represents the relative velocity between source and relay, and \(\mathrm{V_{R,\ D}}\) denotes the relay-destination relative velocity. The link lifetimes are \(\mathrm{T_{S,\ R}}\ =\ \frac{\mathrm{Z_{S,\ R}}}{\mathrm{|V_{S,\ R}|}}\) and \(\mathrm{T_{R,\ D}}\ =\ \frac{\mathrm{Z_{R,\ D}}}{\mathrm{|V_{R,\ D}|}}\). The persistence of a link can be extrapolated to the path, as the persistence of the path is contingent upon the persistence of links from the source to the relay and from the relay to the destination. Communication can only occur when both links are simultaneously maintained. If either link is disconnected, the entire path fails. Therefore, the anticipated communication time along the path is limited by the shorter-lasting of the two links, denoted as  
\[\mathrm{T_{P}}=\mathrm{min}(\mathrm{T_{S,\ R}}\ \times\ \mathrm{A_{S,\ R}}( \mathrm{t}),\ \mathrm{T_{R,\ D}}\ \times\ \mathrm{A_{R,\ D}}(\mathrm{t})). \tag{14}\]  
Therefore, the objective function is  
\[\mathrm{g_{MT}}(\mathrm{t})=\int_{0}^{\mathrm{T_{P}}}\frac{\mathrm{C_{MT}^{ \mathrm{\epsilon}}(\mathrm{t})}}{\mathrm{I_{MT}}}\mathrm{dt}. \tag{15}\]  
The system selects the suitable transmission mode and relay node while minimizing interference to optimize link capacity. Therefore, the overall objective function for the path is as follows:  
\[\mathrm{g(i)}=\mathrm{max}(\mathrm{g_{DT^{\prime}}},\ \mathrm{g_{MT}}(\mathrm{i})). \tag{16}\]  
Given the relay candidate set \(\mathrm{r_{S,\ D}}\ =\ \{1,2,\ldots,\mathrm{m}\}\) residing within the coverage range of both source and destination, the objective function for each relay node can be computed. By selecting the relay node with the maximum value, the transmission mode and relay node that optimize network capacity can be determined. Because the objective  
Figure 5: Two-hop transmission mode.