## 5 Properties Analysis of the Functions \(\varphi_{m}^{s}\)
### Single-Resource-Block Maritime User Allocation
In Figure 6, we compare the WAR optimization performance of three multi-resource-block joint power and maritime user allocation schemes (GRAD heuristic algorithm, MCKP-DP scheme, and DP-FPTA approximation problem). The offshore communication system serves 80 offshore users, including 10 resource blocks. Furthermore, the total power budget provided by BS increases from 5 W to 50 W. It can be seen from the simulation result that the GRAD with an accuracy of 0.0001 can be defined as the reference standard for optimal WAR. At the same time, the MCKP-DP scheme (the number of discrete items \(J=1000\)) is comparable to the optimization performance of the GRAD program. When

Figure 4: Weighted achievable rate of RBPUA and RBUA schemes vs. power budget in NOMA or OMA communications. Figure 5: Computation complexity of RBPUA and RBUA schemes vs. number of users in NOMA or OMA communications. the coastal BS provides a power budget of 50 W, the WAR of the above two schemes in the NOMA system is increased by 4.53% compared with the OMA performance. For a NOMA maritime system with a 50 W power budget, the performance of our proposed DP-FPTA with a 0.01 approximate accuracy is 1.11% higher than that of the DP-FPTA with a 0.1 accuracy. In particular, the DP-FPTA scheme with an error of 0.01 almost achieves the optimized performance of GRAD and MCKP-DP. Furthermore, Figure 7 exhibits the corresponding histogram for convenience; therefore, we can clearly distinguish the WAR simulation results for each optimization scheme. Figure 8 provides the histograms for the numerical changes of WAR as the number of maritime users increases from 10 to 80. The simulation result shows that our communication system obtains the maximum WAR with 50 users. When maritime users continue to increase beyond this saturation state, the heavy load of channel reuse worsens user interference, resulting in a decline in system data rates. Moreover, in the saturation state, both GRAD and MCKP-DP solutions (number of discrete items: \(J=1000\)) achieve an almost equivalent optimization performance of 9.421 \(\times\) 10\({}^{7}\) bit/s. Their WAR optimization results

Figure 6: Weighted achievable rate of the three multi-subchannel joint optimization schemes vs. total power budget (line chart). Figure 7: Weighted achievable rate of the three multi-subchannel joint optimization schemes vs. total power budget (histogram): (**a**) WAR of DP-PFTA with different \(\varepsilon\) and (**b**) WAR of MCKP-DP and GRAD algorithms. are higher than those of OMA by 7.47%. In this saturated NOMA system, the DP-FPTA with an 0.01 approximate accuracy achieves a WAR performance of 9.420 \(\times\) 10\({}^{7}\) bit/s comparable to that of GRAD and MCKP-DP. Meanwhile, it promotes the WAR by 1% over the approximate scheme with an accuracy of 0.1. Figure 9 displays the optimization result of WAR for the DP-FPTA algorithm versus the increase of \(4S/\varepsilon\). Taking the optimization performance of the MCKP-DP scheme as a reference and \(J=1000\) as the total number of discrete items on each resource block, we normalize the \(x\)-axis by \(\left.\right|J_{s}\left|\right.=4S/\varepsilon\). Evaluating the subset \(Js\) of projects on each resource block makes it more convenient to simulate the relationship between WAR and approximation accuracy \(\varepsilon\). Our system, equipped with 10 resource blocks and a power budget of 10 W, serves 80 maritime users. In Figure 9, we also propose that the minimum performance index of the DP-FPTA is at least \((1-\varepsilon)\cdot W_{MCKP}*\). The result indicates that the WAR is higher than the performance index in both OMA and NOMA marine systems. As the number of items increases, the DP-FPTA-optimized performance gradually approaches the MCKP-DP performance curve. Moreover, when \(4S/\varepsilon=500\) (\(\varepsilon=0.08\)), the WAR of our approximate scheme will reach 99.55% of the MCKP-DP scheme.