# A Beam Hopping Scheme Based on Adaptive Beam Radius for LEO Satellites
## 3 Model
### Traditional Beam-Hoppping Model  
Considering a high throughput multi-beam satellite system with \(N_{B}\) beams. In order to adapt to the changing traffic distribution and the increasing demand for Internet traffic, this paper adapts the on-demand resource allocation and full frequency reuse strategies. To provide on-demand services more flexibly for users, we allow beams to hop within the field of view under the constraint of interference angle. In addition, we focus on the gateway-satellite-user information stream of a broadband multi-beam satellite system. The beam hopping model based on the above considerations is shown in Figure 1.  
In the Figure 1, we can see that: according to the beam hopping time plan (BHTP) formulated by the ground network control center (NCC), the on-board computer guides the beam to point to the cell planned to be served in the schedule; meanwhile, a portion of the data packets arriving at the gateway station are transmitted to the destination terminals via the forward link, while the remaining portion is temporarily buffered in the message cache queue at the gateway station. After the end of the current time slot, the on-board computer will control the beam to point to the next cell to be served according to the BHTP, and so on. In this article, we define \(N_{B}\) as the number of user beams, which is the channel of line of sight communication, \(N_{cell}\) is the number of cells in the satellites' field of view. The hopping windows is \(T_{H}\) which is equally divided into \(N_{T}\) slots, a time slot is \(T_{slot}=\frac{N_{B}}{N_{T}}\) long in time. In each of time slots, NCC should allocate power to each cell. The power that \(cell_{i}\) acquires in time slot t denoted as \(P_{cell_{i},t}\). Randomly distributed users randomly generate service demands, and the length of the data packet whose destination user is \(user_{i}\) is denoted as \"\(pck^{user_{i}}len\)\" The arriving time of the data packet is \"\(pck^{user_{i}}.start_{time}\)\". Similarly, the leaving time of the data packet is \(pck^{user_{j}}.end_{time}\). The delay of the above packet is denoted as \(D\) equals to \(pck^{user_{j}}.end_{time}-pck^{user_{j}}.start_{time}\). Moreover, the total delay of \(cell_{i}\) is \(D_{cell_{i}}\) which equals to \(D_{cell_{i}}=\sum_{pck\in cell_{i}}D\). The system delay is \(Delay=\sum_{i=0}^{N_{cell}}D_{cell_{i}}\). The total service demand of the \(user_{j}\) in the kth time slot is defined as \(D_{user_{j}k}\), and the throughput provided by the communication system for the user is \(R_{user_{j}k}\). The service demand of \(cell_{i}\) is \(D_{t}^{cell_{i}}\) equaling to \(\sum_{j}D_{user_{j}t}\), the offered throughput to \(cell_{i}\) is \(R_{t}^{cell_{i}}\). To avoid wasting communication resources, the actual rate provided should be equal to the cell's bandwidth demand. Therefore, actual throughput is used to measure the degree of service satisfaction for service cells. The actual throughput, denoted as \(Throughput\), is equal to \(\sum_{t=0}^{99}\sum_{i=0}^{N_{cell}}\ min(D_{t}^{cell_{i}},R_{t}^{cell_{i}})\). It is worth noting that the effective throughput throughput is obtained by summing the effective throughput of all service cells across all time slots. The delay sensitivity factor is \(\rho\), and the virtual demand traffic of a packet is:  
\[pck^{user_{i}}.len\times\rho^{pck^{user_{i}}.end_{time}-pck^{user_{i}}.start_{ time}}\]  
and the virtual demand traffic of \(cell_{i}\) is:  
\[VirD_{t}^{cell_{i}}=\sum_{pck\in cell_{i}}pck^{user_{i}}.len\times\rho^{l-pck^{ user_{i}}.start_{time}}.\]  
We introduce virtual traffic demand that not only considers the significance of broadband packets but also takes into account the importance of delay packets that are measured by \(\rho\) and the time to live. Both of them guarantee the users' QoS by participating in cell selection. The above parameters are somewhat complex. Table 1 below provides a clear presentation of the symbols and definitions of these parameters for better understanding.