## 3 System Model and Problem Formulation  
### Scenario Description  
In this paper, we consider a mega-constellation of Ka-band LEO satellites deployed at 1050 km from earth. There is a GEO high throughput satellite above one of the orbits of the LEO system and within the coverage area of LEO satellite, there are some GEO ground stations which share the same spectrum with LEO systems. Each LEO satellite is equipped with a steerable phased array antenna, a fixed signaling antenna and a regenerative telecommunication transceiver. The fixed signaling beam can cover the entire satellite footprint and is responsible for the signaling transmission including the uploading of ground user request. The steerable multi-beam phased array antenna can generate \(K\) spot beams for transmitting traffic data, which can be directed to any area of \(\pm 50^{\circ}\), and the satellite can flexibly and dynamically allocate on-board power and bandwidth to these spot beams. There are \(N(N>K)\) cells in each satellite footprint, each spot beam can cover one cell and all sub-satellite coverage areas can be served by \(K\) beams in a time-division multiplex manner. In this paper, we divide time into time-segments and assume that the relative position of the LEO satellite and the ground are fixed within the time-segment. Each time-segment is further divided into T time slots, which is the minimum time unit for beam dwelling. The satellite receives the terminal request through the signaling beam, then the service spot beam can hop from one cell to another according to the traffic demand distribution and allocate the beam resources. Assuming that the satellite can provide \(N\) queues to store the arriving traffic of each cell and each request can only survive for \(T_{ttl}\) time due to the limited queue length. \(T_{ttl}\) is defined as Time To Live, which is an integer multiple of the time slot length, and packets that cannot be satisfied within this time will be discarded. The whole scenario is shown in Figure 1, and in this paper, we mainly focus on the downlink resource allocation. In the following context, to avoid confusion, a lit cell and a serving beam are essentially the same concept. In the beam hopping scenario of this paper, the resources that need to be flexibly allocated are communication time slots, bandwidth and power. According to the real-time traffic demand distribution on the ground, we first determine which \(K\) cells are illuminated by the \(K\) beams in each time slot to serve them, which is called an illumination pattern in beam hopping. Each beam can flexibly allocate the bandwidth and power. The total bandwidth on the satellite is \(B_{tot}\), which is divided into \(m\) sub-bands of equal bandwidth, and each beam can use one or more consecutive sub-bands. The total power available on the satellite is denoted as \(B_{tot}\), which means the sum of the power used by all spot beams lit in the same time slot cannot exceed this limitation. For the protection of higher-priority GEO systems, the transmitting power of LEO satellite beams will be limited to ensure that the signal power received by the GEO ground stations is controlled within the interference threshold. There are two main challenges for resource scheduling problem in BH scenario. On the one hand, the distribution of satellite terminals in different ground areas is different and the bandwidth requirements are also different, thus facing an extremely uneven space-time distribution of service access requests. Due to the rapid movement of satellites, the terminal needs to switch frequently between different beams and the satellite nodes. On the other hand, when allocating three-dimensional resources of time slot, frequency and power at the same time, the search space is very large, and the allocation of resources in each dimension has strong coupling, so it cannot be decoupled and solved separately.