# Space Habitat Data Centers--For Future Computing
## 5 Performance Formulation
### Computing Platform Access Latency
\(h\!\left(\mathbf{\hat{f}}_{u}\right)\) and \(h\!\left(\mathbf{\hat{f}}_{u+x}\right)\) are the altitude of the \(u\)th and \((u+x)\)th stratosphere computing platforms respectively. The compute resource access latency for the terrestrial cloud computing platforms \(\Gamma_{2}\) is given as  
\[\Gamma_{2}=\sum_{y=1}^{Y}\sum_{i=1}^{I}\sum_{d=1}^{D}\left(\frac{D\big{(}St_{d},t _{y}\big{)}}{Th\big{(}St_{d},\alpha_{a}^{i},t_{y}\big{)}}+\frac{(h(St_{d}))}{3 \times 10^{8}}\right). \tag{20}\]  
It is also feasible that the small satellite accesses the terrestrial cloud computing platform via high-altitude platform (HAP). This becomes necessary when there is transmit power limitation aboard small satellites. The compute resource access delay, in this case, is denoted \(\Gamma_{2}^{\prime}\) and given as  
\[\Gamma_{2}^{\prime}=\sum_{y=1}^{Y}\sum_{a=1}^{A}\sum_{d=1}^{D}\sum_{z=1}^{Z} \left(\left(\frac{D\big{(}St_{d},\
ot{\epsilon}_{z},t_{y}\big{)}}{Th\big{(}St _{d},\
ot{\epsilon}_{z},t_{y}\big{)}}+\frac{D\big{(}\
ot{\epsilon}_{z},\alpha_ {a}^{i},t_{y}\big{)}}{Th\big{(}\
ot{\epsilon}_{z},\alpha_{a}^{i},t_{y}\big{)} }\right)\right)+\left(\frac{(h(St_{d}))}{3\times 10^{8}}\right). \tag{21}\]  
\(Th\big{(}\
ot{\epsilon}_{z},\alpha_{a}^{i},t_{y}\big{)}\) is the link throughput between the stratosphere cloud platform \(\
ot{\epsilon}_{z}\) and the \(i\)th terrestrial cloud platform of the _ath_ operator. \(D\big{(}St_{d},\
ot{\epsilon}_{z},t_{y}\big{)}\) and \(D\big{(}\
ot{\epsilon}_{z},\alpha_{a}^{i},t_{y}\big{)}\) are the size of data transmitted from the small satellite to the HAP and from the HAP to the terrestrial data centre respectively. The parameters \(\Gamma_{1},~{}\Gamma_{1}^{\prime},~{}\Gamma_{2}\) and \(\Gamma_{2}^{\prime}\) describe the latency associated with computing resource access latency in the context of existing work. Terrestrial cloud computing platform access is supported in references [19; 44] when small satellites have compute resource constraints. In the proposed scheme, small satellites access idle computing resources on space vehicles and the SHDC. Let \(\Gamma_{3}\) and \(\Gamma_{4}\) denote the compute resource access delay for the space vehicle and the SHDC. \[\Gamma_{3}=\sum_{y=1}^{Y}\sum_{e=1}^{E}\sum_{d=1}^{D}\left(\frac{D\big{(}St_{d },t_{y}\big{)}}{Th\big{(}St_{d}~{},\emptyset_{e},t_{y}\big{)}}+\frac{\big{|}h (St_{d})-h(\emptyset_{e})\big{|}}{3\times 10^{8}}\right), \tag{22}\]  
\[\Gamma_{4}=\sum_{y=1}^{Y}\sum_{j=1}^{J}\sum_{d=1}^{D}\left(\frac{D\big{(}St_{ d},t_{y}\big{)}}{Th\big{(}St_{d},\emptyset_{j},t_{y}\big{)}}+\frac{\big{|}h (St_{d})-h\big{(}\emptyset_{j}\big{)}\big{|}}{3\times 10^{8}}\right). \tag{23}\]  
The scenarios in \(\Gamma_{3}\) and \(\Gamma_{4}\) do not consider the context where data is forwarded from the space vehicle to the SHDC. This is feasible when a new SHDC is deployed and it is necessary to transmit data from the space vehicle to the SHDC. In this case, the compute resource access latency is denoted \(\Gamma_{4}^{\prime}\) and given as  
\[\Gamma_{4}^{\prime}=\sum_{d=1}^{D}\sum_{e=1}^{E}\sum_{j=1}^{J}\sum_{y=1}^{Y} \left(\frac{D\big{(}St_{d},t_{y}\big{)}}{Th\big{(}St_{d},\emptyset_{e},t_{y} \big{)}}\right)+\left(\frac{D\big{(}\emptyset_{e},\theta_{j},t_{y}\big{)}}{Th \big{(}\emptyset_{e},\theta_{j},t_{y}\big{)}}\right)+\left(\frac{\big{|}h(St_{ d})-h(\emptyset_{e})\big{|}}{3\times 10^{8}}\right)+\left(\frac{\big{|}h \big{(}\theta_{j}\big{)}-h(\emptyset_{e})\big{|}}{3\times 10^{8}}\right). \tag{24}\]