## 3 System Model
### Computing Model
#### 3.5.2 LEO Satellite Computing Latency  
Analogous to the UAV computing latency, when IoTD \(k\) offloads its MLLM request to a LEO satellite, the process involves IoTD \(k\) initially transmitting multimodal data to the LEO satellite \(n\) through a ground-to-satellite link. If an MLLM task is transmitted from accessible satellite \(n\) to satellite \(n^{\prime}\) via a multi-hop path through inter-satellite links (ISL), the transmission latency of ISL needs to be considered. Let the path be a sequence of satellites \(s_{1},s_{2},\ldots,s_{h}\), where \(s_{1}=n\) and \(s_{h}=n^{\prime}\). Then, the upload transmission latency from IoTD \(k\) to the satellite \(n^{\prime}\) via accessible satellite \(n\) can be expressed as \[T_{k,n,n^{\prime}}^{np}(t)=\frac{u_{k}(t)}{R_{kn}^{np}(t)}+\frac{d_{k,n}}{c}+ \mathbb{I}_{k,n,n^{\prime}}(t)\sum_{i=1}^{h-1}\left(\frac{d_{s_{i},s_{i+1}}}{c}+ \frac{u_{k}(t)}{R_{ISL}}\right), \tag{20}\]  
where \(c\) is the speed of light for propagation latency, \(d_{s_{i},s_{i+1}}\) is the distance between satellites \(s_{i}\) and \(s_{i+1}\), \(H_{n,n^{\prime}}\) is the number of routing hops, and \(R_{ISL}\) is the transmission rate of the ISL. Here, \(\mathbb{I}_{k,n,n^{\prime}}(t)\) is the indicator function of whether or not the accessible satellite \(n\) and offloaded satellite \(n^{\prime}\) are the same, which is defined as  
\[\mathbb{I}_{k,n,n^{\prime}}(t)= \begin{cases}1&\text{if }\ n\
eq n^{\prime},\\ 0&\text{otherwise}.\end{cases}  \tag{21}\]  
Consequently, the energy consumption for the upload transmission from IoTD \(k\) to LEO satellite \(n^{\prime}\) via accessible satellite \(n\) is expressed as  
\[ \begin{split} E_{k,n}^{np}(t)&=P_{k}(t)\Bigg{(}\frac{u_{k} (t)}{R_{\text{Tor}}^{np}(t)}+\frac{d_{k,n}}{c}\Bigg{)}\\ &\quad+\mathbb{I}_{k,n,n^{\prime}}(t)\sum_{i=1}^{h-1}P_{ISL}\bigg{(} \frac{d_{s_{i},s_{i+1}}}{c}+\frac{u_{k}(t)}{R_{ISL}}\bigg{)},\end{split}  \tag{22}\]  
where \(P_{ISL}\) is the transmission power of the ISL. When the MLLM task is executed on the LEO satellite, the inference latency for IoTD \(k\) at LEO satellite \(n\) is calculated as  
\[T_{k,n}^{exc}(t)=\frac{v_{n}\cdot o_{k}(t)}{f_{n}^{gpu}}, \tag{23}\]  
where \(v_{n}\) represents the number of FLOPs used by one inference step of MLLM on LEO satellite \(n\) per output word, and \(f_{n}^{gpu}\) is the FLOPS of LEO satellite \(n\). The inference energy consumption for IoTD \(k\)'s task executed on LEO satellite \(n\) is  
\[E_{k,n}^{exc}(t)=P_{n}^{exc}T_{k,n}^{exc}(t), \tag{24}\]  
where \(P_{n}^{exc}\) represents the device power of LEO satellite \(n\). The latency associated with transmitting the token results from LEO satellite \(n^{\prime}\) to IoTD \(k\) through accessible satellite \(n\) is expressed as  
\[ \begin{split} T_{k,n^{\prime},n}^{dl}(t)&=\mathbb{I }_{k,n,n^{\prime}}(t)\sum_{i=1}^{h-1}\Big{(}\frac{d_{s_{i},s_{i+1}}}{c}+\frac{ d^{token}o_{k}(t)}{R_{ISL}}\Big{)}\\ &\quad+\frac{d^{token}o_{k}(t)}{R_{km}^{dl}(t)}+\frac{d_{k,n}}{c} \,,\end{split}  \tag{25}\]  
where \(R_{km}^{dl}(t)\) represents the downlink transmission rate of the ground-to-satellite link. The energy consumption for the resulting transmission from LEO satellite \(n^{\prime}\) to IoTD \(k\) through accessible satellite \(n\) is given by  
\[ \begin{split} E_{k,n}^{dl}(t)&=\mathbb{I}_{k,n^{ \prime},n}(t)\sum_{i=1}^{h-1}P_{ISL}\Bigg{(}\frac{d_{s_{i},s_{i+1}}}{c}+\frac{ d^{token}o_{k}(t)}{R_{ISL}}\Bigg{)}\\ &\quad+P_{n}\Bigg{(}\frac{d^{token}o_{k}(t)}{R_{km}^{dl}(t)}+ \frac{d_{k,n}}{c}\Bigg{)},\end{split}