# Minimization of _n_th Order Rate Matching in Satellite Networks with One to Many Pairings
## 1 Introduction
### Motivation and Related Literature  
The challenging wireless networks' ecosystem, with strict demands for higher data rates, not only requires their performance evaluation through new channel model frameworks [11] but also makes the investigation in the resource management field an important ally toward the users' quality of service (QoS) satisfaction. Specifically, in terms of satellite communication (SatComs) systems [12, 13], there are many scientific works presentingmethods based on optimization, game theory and recently artificial intelligence (AI) techniques, targeting the appropriate allocation of resources including power, time, bandwidth and user pairing. Especially, in [14], a terrestrial-satellite network (TSN) with hot air balloons at different heights and elevation angles that relay the signals between satellites and ground stations is presented. The maximization of network energy efficiency is assumed subject to the optimization of the transmission power, the serving period of satellites and the required number of lasers per satellites at the relays. The resource allocation problem is modeled and solved by geometric programming with Taylor series approximation. Moreover, in [15], a broadband GEO multi-beam SatCom system is presented, and the goal of resource management is to satisfy the beam traffic demand by making use of the minimum transmit power and utilized bandwidth. The scope is the maximization of spectrum utilization based on the spectrum reuse considering tolerable interference. Due to the non-convexity of joint power and carrier assignment optimization problem, the authors propose the solution of two sub-problems, i.e., carrier and power assignment, respectively. These are solved using successive convex approximation (SCA). The same mathematical technique is applied in [16] to a satellite and aerial-integrated network to support IoT devices. Especially, the maximization of the sum rate is depicted, and the use of first-order Taylor expansion is made. Finally, another use of SCA is shown in [17], where a secure beamforming scheme for rate-splitting multiple access cognitive TSN is investigated in the presence of multiple eavesdroppers. The intention is the maximization of the secrecy-energy efficiency of the earth station (ES) with respect to the constraints on the ES secrecy rate, the cellular users' rate requirements and transmit power budgets of the satellite and base station. The objective and constraints are non-convex and are solved iteratively using convex conversions. Regarding the application of game theory, in [18], the cooperative transmission and resource allocation in a cloud-based integrated TSN is studied as a game-based resource allocation problem. The operator offers two levels of services of different QoS and price and the maximization of the utility of the operator, which is composed by the Stackelberg game between the operator and users, the evolutionary game between all users, and the energy minimization problem is implemented. Furthermore, in terms of AI, techniques such as convolutional networks [19], long-short-term memory (LSTM) networks [20] and deep reinforcement learning (DRL) [21; 22; 23] have already been used. In [19], a convolutional neural networks' approach is presented for flexible payload management in very high-throughput satellite systems. The authors use a realistic traffic model and suggest a cost function aiming to minimize both the error between the offered capacity and the required capacity and the amount of resources used in the satellite. Additionally, in [20], a LSTM prediction-based resource-matching scheme for low earth orbit (LEO) SatComs is presented. The traffic is predicted, and a resource allocation method is proposed to efficiently match power and spectrum resources. The DRL framework is considered in [21] for dynamic resource allocation in a multi-beam satellite system, where image-like tensors are exploited to extract traffic spatial and temporal features.