## 5 Properties Analysis of the Functions \(\varphi_{m}^{s}\)
### Single-Resource-Block Maritime User Allocation
([PERSON]); software, [PERSON] ([PERSON]); validation, [PERSON] ([PERSON]); formal analysis, [PERSON] ([PERSON]) and [PERSON] ([PERSON]); data curation, [PERSON] ([PERSON]); writing--original draft preparation, [PERSON] ([PERSON]); writing--review and editing, [PERSON] ([PERSON]) and [PERSON]; supervision, [PERSON] ([PERSON]); project administration, [PERSON] ([PERSON]), [PERSON] ([PERSON]) and Y.Z. ; funding acquisition, [PERSON] ([PERSON]) and Y.Z. All authors have read and agreed to the published version of the manuscript. This research was supported in part by the NSF of China under Grant 61661018, Hainan Provincial NSF under Grants 2019 RC036, 619 MS029, and 620 MS020, and Hainan High Education Research Program Hnky2019-8. (Corresponding author: [PERSON]). Not applicable. We thank Hainan University, Binjiang College of UNIST, the NSF of China under Grant 61661018, Hainan Provincial NSF under Grants 2019 RC036, 619 MS029, and 620 MS020, and Hainan High Education Research Program Hnky2019-8 for their funding. Last but foremost, we thank [PERSON] for his guidance on this research. The authors declare no conflict of interest. ## Appendix A

The derivation process from Equation (7) to Equation (12) is as follows:

Figure 3:0

Figure 3:1

Figure 3:2

Figure 3:3

Equation (14) is derived from the definition of Shannon formula in Equation (5). Then, we move \(\beta_{s}(n)\) into the logarithm in Equation (15). Equation (16) combines the denominator of the \((n-1)\)th term with the numerator of the \(n\)th term to complete the derivation. Finally, we obtain the equivalent optimization objective \(W^{\star}\) by redefining the variables. In Equation (12), the constant term is \(C=\sum\limits_{s\in S}\alpha_{\beta_{s}(T)}log_{2}(\frac{1}{\widetilde{\sigma }_{\beta_{s}(T)}^{s}})\). Because \(\widetilde{p}_{1}^{s}=\sum\limits_{m=1}^{T}P_{\beta_{s}(m)}^{s}=\sum\limits_{ t\in T}P_{t}^{s}\), Conditions \(w1^{\star}\) and \(w2^{\star}\) are derived from \(w1\) and \(w2\), respectively. In the same manner, we can get Conditions \(w4^{\star}\) and \(w5^{\star}\) according to \(w4\) and \(P_{\beta_{c}(n)^{*}}^{s}(n\in\{1,\ldots,T\},s\in S)\). The set of valid maritime users in \(w3^{*}\) is redefined as \(A_{s}*\triangleq\{\widetilde{p}_{n}^{s}\geq\widetilde{p}_{n+1}^{s},n\in\{1, \ldots,T\}\}\). Appendix B We define \(P_{s}^{*}\) as the optimal power budget on resource block \(s\) in problem \(W_{MRB}\). Let \(a_{s}*\triangleq\left\lfloor\frac{P*}{p}\right\rfloor\delta\) denote the discrete power budget. We give the following derivation:

Figure 3:4

Figure 3:5

Figure 3:6

Considering that \(W_{MCKP}{}^{*}\) is the optimal solution of the problem \(W_{MCKP}\) with discrete step size \(\delta\), we can get \(W_{MCKP}{}^{*}\geq\sum_{s\in S}W_{s}(a_{s}*)\) in Equation (10). In [32], it was proved that \(W_{s}(p^{s})=max_{A_{s}*}\{W_{s}(A_{s}*,p^{s})\}\),\(W_{s}(A_{s}*,p^{s})\) is concave and twice differentiable. Therefore, we can further deduce that \(W_{s}\) is below the maximum slope tangent of \(W_{s}(A_{s}*,p^{s})\). According to Equation (56) and the fact that \(p_{s}*-a_{s}*\leq\delta\) from the definition of \(\delta\), we derive the Equation (12). ## References

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