# A Green Computing Business Aggregation Strategy for Low Earth Orbit Satellite Networks
## 3 Problem Modeling  
Based on the operational modes of satellites, the working state of \(S_{i}\) can be divided into three states: active, idle, and sleep. In its active state, the satellite performs business transmission and computation functions, consuming the highest power, with average power consumption for transmission, reception, and computation modules denoted as \(P_{T}^{a}\), \(P_{R}^{a}\), and \(P_{C}^{a}\), respectively. In the idle state, the satellite maintains a low-power mode for communication and computation modules, with power levels \(P_{T}^{\mathit{pile}}\), \(P_{R}^{\mathit{pile}}\), and \(P_{C}^{\mathit{pile}}\). In the sleep state, the computation module is turned off to save power, with \(P_{C}^{\mathit{peler}}=0\), though communication functions remain active for rapid response [9; 10]. In specific scenarios, such as low business volume at night or in sparsely populated regions, certain satellites are underutilized. To enhance energy efficiency, a business aggregation strategy can be employed to consolidate businesses on a subset of satellites, allowing others to enter a sleep state. As illustrated in Figure 2, the LSNs concentrate business processing on a subset of service satellites, allowing other satellites to enter a sleep state. We define the set of sleeping satellites as \(\mathbb{S}^{S}=\{S_{1},\dots,S_{u},\dots,S_{U}\}\) and working satellites as \(\mathbb{S}^{W}=\{S_{1},\dots,S_{v},\dots,S_{V}\}\), where \(S_{u}\) and \(S_{v}\) are any sleep satellites and working satellites, respectively, and \(U\) and \(V\) represent the number of sleeping and working satellites, respectively, such that \(U+V=M\). The energy consumption for transferring a business \(\mathbb{B}_{uj}\) from a sleeping satellite \(S_{u}\) to a working satellite \(S_{v}\) is:  
\[E_{T}^{ujv}=(P_{T}^{a}+P_{R}^{a})\cdot\tau_{ujv} \tag{10}\]  
where \(\mathbb{B}_{uj}\) denotes the \(j\)-th business of the \(u\)-th sleeping satellite, and \(\tau_{ujv}\) is the transmission duration of \(\mathbb{B}_{uj}\) from the \(u\)-th sleeping satellite to the \(v\)-th working satellite. The total communication energy for aggregating businesses from \(\mathbb{S}^{S}\) to \(\mathbb{S}^{W}\) is:  
\[E_{T}=\sum_{u=1}^{U}\sum_{j=1}^{N_{u}}E_{T}^{ujv} \tag{11}\]  
Figure 1: Layered architecture of LSNs showing control, business, and user layers. where \(N_{u}\) is the number of businesses of the \(u\)-th sleep satellite. The energy savings for a sleeping satellite \(S_{u}\) are:  
\[E_{\mathrm{s}}^{u}=P_{\mathrm{C}}^{idle}\cdot\tau^{\prime} \tag{12}\]  
where \(\tau^{\prime}\) is the duration of the sleep period. The total energy savings for all sleeping satellites \(\mathbb{S}^{S}\) are:  
\[E_{\mathrm{s}}=\sum_{u=1}^{U}E_{\mathrm{s}}^{u} \tag{13}\]  
The net energy saved by business aggregation is:  
\[\Delta E=E_{s}-E_{T} \tag{14}\]  
After receiving businesses from sleeping satellites, the working satellite \(S_{v}\) updates its business queue. Suppose business \(\mathbb{B}_{uj}\) is assigned a sequence number \(N_{a}\) upon insertion; then the updated queue for \(S_{v}\) is \(B_{v}^{\prime}=\{\mathbb{B}_{v1},\ldots,\mathbb{B}_{vN_{v}},\ldots,\mathbb{B} _{vN_{u}},\ldots,\mathbb{B}_{vN_{v}^{\prime}}\}\), where \(N_{v}^{\prime}\) represents the total number of businesses in \(S_{v}\) after aggregation.