# Minimization of _n_th Order Rate Matching in Satellite Networks with One to Many Pairings
## 2 Dynamic Capacity Allocation
### Capacity Allocation Problem and Proposed Mechanism
Specifically, the values of rate-matching objectives based on the pairs of Exhaustive (\(\mathrm{obj}_{\mathrm{Exh}}\)) and Proposed (\(\mathrm{obj}_{\mathrm{Prop}}\)) schemes are used for the relative error expressed as \(\left|\mathrm{obj}_{\mathrm{Exh}}-\mathrm{obj}_{\mathrm{Prop}}\right|\Big{/} \mathrm{obj}_{\mathrm{Exh}}\). This error is computed in each simulation scenario and then is averaged. For the average relative error, a useful performance indicator is the maximum among the minimum relative errors through the different GWs-UEs configurations. Considering the 1 st and 2 nd order RM in Figure 1, this is about 14% for the 3 \(\times\) 9 and 28.4% for the 4 \(\times\) 6 GWsxUEs configurations, respectively, while the iterative proposed process has fast convergence, as shown from the average iterations of Algorithm 1 that are low. In case of HTS systems, with a larger number of GWs/UEs where a pairing allocation decision has to be completed in short time, it is practical to use an additional termination criterion with the maximum number of iterations in Algorithm 1. Moreover, in Figure 3, without any loss of generality for 3 \(\times\) 6 and 3 \(\times\) 9 systems, we depict that for both odd (first line subplots) and even (second line subplots) values of \(n\), an increase of the latter, from 3 to 5 and 4 to 6, respectively, results in an increase of the best (minimum) relative error for each separate system, which has also been observed for all the rest of the systems. Indicatively, for the 3 \(\times\) 9 system, the best relative error is about 28% and 82% for \(n=3\) and \(n=5\), respectively, and 48% and 130% for \(n=4\) and \(n=6\), respectively. A possible reason is that more concave terms appear in BE as \(n\) increases, which are approximated by linearizations in the convexified step of the CCP method; hence, more approximations are made, leading to worse performance compared with the optimal Exhaustive scheme. Finally, in Figure 2, the average value of [PERSON]'s fairness index [30] is shown, which is defined in each simulation scenario as \(\left(\sum\limits_{\mathrm{j=1}}^{\mathrm{N}}w_{\mathrm{j}}\right)^{2}\Bigg{/} \left(\mathrm{N}\sum\limits_{\mathrm{j=1}}^{\mathrm{N}}w_{\mathrm{j}}^{2}\right)\), where \(w_{j}\) is computed for each UE based on the absolute difference presented on the LHS of the first line in (2). In each separate GWs-UEs system (i.e., in each of the four subplots), observing the performance of the optimal Exhaustive scheme, represented by the blue lines, the minimization of higher than \(n=1\) order RM leads to higher fairness among UEs. The UEs' fairness is denoted as the absolute difference between their requested capacities and the offered capacities of gateways that are matched with them. For practical systems, even the increase from \(n=1\) to \(n=2\) results in a solution with much more fairness, and our fast convergent approach achieves a relative error less than 30%. This effect can be explained by the fact that a higher \(n\) focuses on the minimization of the higher absolute differences in the minimization process of the initial problem, resulting in more similar absolute differences among the UEs. Moreover, this trend is also observed in most cases of the Proposed schemes (black and red lines), which confirms the robustness of this important remark. In conclusion, in an optimization problem that does not contain the 'fairness', because of its minimization structure, the increase of \(n\) can indirectly assist the increase of UEs' fairness. This is a valuable observation for the network's operators, because this factor can be exploited as a fairness controller in terms of rate matching not only in investigated satellite but also in other wireless systems.