## 1 Introduction  
Sensing and monitoring anywhere, anytime, and anyway in more dimensions are quite important for the coming era of digital twins and the metaverse [1]. The internet of things (IoT) is a network that provides the methods and infrastructures needed to achieve states sensing of object and relative information aggregation. The essential tendency of IoT in the future is global coverage and remote controlling with a higher data rate and access amount. Therefore, the IoT on the ground combined with the satellite network would be the most potent approach to greatly enlarging the scenarios of the IoT [2]. With the recent rapid developments in satellite networks, satellite IoT networks (S-IoT-N) have been a topic of general interest in the monitoring of mining, ocean shipping, and power transmission. The key bottleneck of the above urgent application is the enhancement of the satellite network. One of the scenarios of the sixth generation (6G) is the space-air-ground integrated network (SAGIN). The Third Generation Partnership Project (3 GPP) have proposed non-terrestrial networks (NTN) projects to promote the enhancement of satellite networks since 2019 [3]. Satellite internet networks (SIN) are regarded as the most possible evolution road, possessing characteristics of many satellites, low earth orbit (LEO), the Ku/Ka frequency, and a high data rate [4]. More advanced potential technologies are introduced in SIN evolution. There are some important tendencies:  
(a) Satellite beams (Sat-Beam) steering in a permanent position: SIN satellites have multi-Sat-Beams, which move at high speed relative to the earth. The footprint of Sat-Beams (F-Sat-Beam) are planned by constellation [5]. During the service of the observed satellite, its Sat-Beams steer in a permanent position. (b) Intra-serving beam (Ser-Beam)-hopping (Ser-BH): For the exploitation of a satellite-based phased array antenna (PAA), the F-SB can be further divided into different small footprints of serving beams (F-Ser-Beam) [6]. They can serve users at different places and in different periods through Ser-BH, which achieves occasion division multiplexing. (c) More dimensions of radio resource (RR): The introduction of combining the fifth generation (5G) new radio (NR) technologies, orthogonal frequency division multiplexing access (OFDM), and non-orthogonal multiple access (NOMA) would be exploited by SIN. Using the coverage of Sat-Beams, the users can be served by the multiplex with time, frequency, power, space, and occasion divisions [7]. Meanwhile, the requests of users are not distributed uniformly in the space domain, which causes the density of each F-Ser-Beam to be quite different. In the scope of Sat-Beams, particularly the Ser-Beams at the edge, it is hard for users to choose the long-suitable Ser-Beams for load balancing. As the granularity of RR increases, the above problem becomes more serious. The relatively shifting position between the position of Sub-Satellite Point (SSP) and the user also makes this worse. Figure 1 shows the problems of load balancing for Ser-BH in SIN. As Figure 1 shows, the first problem is how to improve the local scheme for load balancing of adjacent Ser-Beams to adapt to global scenarios. For large degradation of signal fading, the load balancing between adjacent Ser-Beams around the edge of Sat-Beams should be paid more attention to. However, there are thousands of possible scenarios for adjacent Ser-Beams with different DoU. The optimal scheme is not only to solve the problem in the case of local load balancing, but also different local scenarios around the world. The second problem is which Ser-Beams the users should be accessing to under the DoU of spatiotemporal non-uniform. The SSPs of an observed satellite are not permanent and not in the center of F-Ser-Beams, which are different from other satellites and have different times. The trajectories of SSPs are also different in the Ser-Beams serving during (SD).