# Space Habitat Data Centers--For Future Computing
## 5 Performance Formulation
### Computing Platform Access Latency  
This section develops the performance model describing the access latency associated with using the computing resources aboard computing platforms, i.e., data centres. The formulation is different from that associated with using the space edge computing platform found in [PERSON] et al. [19]. In the formulation here, data centres are considered to be suitable as seen in [44] and not edge-computing nodes (existing in [PERSON] et al. [19]) are considered. The formulation extends the consideration of space edge computing nodes to that of accessing computing resources aboard data centres located in the stratosphere and in space (SHDC). The consideration of the SHDC here extends [44] where the suitability of a space-based data centre is recognized but without any mathematical formulation. The novelty of the formulation in comparison to existing work in [PERSON] et al. [19] is that it considers that satellites process their data on data centres instead of space edge computing platforms. However, it is recognized by [PERSON] et al. [44] that the use of space-based computing platforms is beneficial. This is because data centres host a significant number of servers and have more computing resources in comparison to the space edge-computing node. The small satellite \(St_{d}\) can access computing resources from platforms sited in different locations i.e., terrestrial, stratosphere or space (i.e, SHDC). Let \(\mathbf{\hat{f}}\) be the set of stratosphere-based computing platforms:  
\[\mathbf{\hat{f}}=\left\{\mathbf{\hat{f}}_{1},\mathbf{\hat{f}}_{2},\ldots,\mathbf{\hat{f}}_{V}\right\} \tag{17}\]  
The altitude of the small satellite \(St_{d}\), SHDC \(\theta_{j}\), space vehicle, \(\delta_{e}\) stratosphere-based computing platform \(\mathbf{\hat{f}}_{z},\mathbf{\hat{f}}_{z}\in\mathbf{\hat{f}}\) are \(h(St_{d})\), \(h\!\left(\theta_{j}\right)\), \(h(\delta_{e})\) and \(h\!\left(\mathbf{\hat{f}}_{z}\right)\) respectively. In addition, let \(Th\!\left(St_{d},q,t_{y}\right)\), \(q\in\left\{\theta_{j},\mathbf{\hat{f}}_{z},\ \delta_{e},\alpha_{a}^{i}\right\}\) denote the link speed between the small satellite and computing platform entity \(q\) at the epoch \(t_{y}\). The amount of data from small satellite \(St_{d}\) requiring access to computing resource aboard the computing platform entity \(q\) is denoted \(D\!\left(St_{d},t_{y}\right)\). The computing platform access latency for stratosphere computing platforms, \(\Gamma_{1}\) is given as  
\[\Gamma_{1}=\sum_{y=1}^{Y}\sum_{z=1}^{V}\sum_{d=1}^{D}\!\left(\frac{D\!\left(St _{d},t_{y}\right)}{Th\!\left(St_{d},\mathbf{\hat{f}}_{z},t_{y}\right)}+\frac{\left( h(St_{d})-h\!\left(\mathbf{\hat{f}}_{z}\right)\right)}{3\!\times\!10^{8}}\right) \tag{18}\]  
It is also feasible that the small satellite accesses a stratosphere computing platform via other stratosphere computing platforms through inter-platform links. This is necessary when small satellite has transmit power constraints. The compute resource (or platform) access latency \(\Gamma_{1}^{\prime}\) is given as  
\[\Gamma_{1}^{\prime}=\sum_{y=1}^{Y}\sum_{u=1}^{V}\sum_{d=1}^{D}\!\left(\frac{D \!\left(St_{d},t_{y}\right)}{Th\!\left(St_{d},\mathbf{\hat{f}}_{u},t_{y}\right)}+ \frac{D\!\left(\mathbf{\hat{f}}_{u},\mathbf{\hat{f}}_{u+x},t_{y}\right)}{Th\!\left(\bm {\hat{f}}_{u},\mathbf{\hat{f}}_{u+x},t_{y}\right)}+\frac{\left(h(St_{d})-h\!\left( \mathbf{\hat{f}}_{z}\right)\right)}{3\!\times\!10^{8}}+\frac{\left|h\!\left(\mathbf{ \hat{f}}_{u}\right)-h\!\left(\mathbf{\hat{f}}_{u+x}\right)\right|}{3\!\times\!10^ {8}}\right) \tag{19}\]  
\(\mathbf{\hat{f}}_{u}\in\mathbf{\hat{f}}\), \(\mathbf{\hat{f}}_{u+x}\in\mathbf{\hat{f}}\), \(\mathbf{\hat{f}}_{u}\
eq\), \(\mathbf{\hat{f}}_{u+x}\). \(D\!\left(\mathbf{\hat{f}}_{u},\mathbf{\hat{f}}_{u+x},t_{y}\right)\) and \(Th\!\left(\mathbf{\hat{f}}_{u},\mathbf{\hat{f}}_{u+x},t_{y}\right)\) are the size of data transmitted and inter-platform link speed between the \(u\)th and \((u+x)\)th stratosphere computing platforms at the epoch \(t_{y}\) respectively.