# Space Habitat Data Centers--For Future Computing
## 3 Problem Description  
The paper considers data centre operators seeking to process space data and site data centres near water sources. Let \(\alpha\) be the set of terrestrial data centre operators. \[\alpha=\{\alpha_{1},\alpha_{2},\ldots,\alpha_{A}\} \tag{1}\]  
The set of terrestrial data centres deployed by the _ath_ operator, \(\alpha_{a},\alpha_{a}\)\(\epsilon\)\(\alpha\) is given by  
\[\alpha_{a}=\left\{\alpha_{a}^{1},\alpha_{a}^{2},\ldots,\ \alpha_{a}^{1}\right\} \tag{2}\]  
The cooling indicator of the _ith_ Earth-based terrestrial data centre from the _ath_ operator \(\alpha_{a}^{i},\ \alpha_{a}^{i}\)\(\epsilon\)\(\alpha_{a}\) is denoted \(I\left(\alpha_{a}^{i}\right)\)\(\epsilon\)\(\{0,1\}\). The indicator values \(I\left(\alpha_{a}^{i}\right)=0\) and \(I\left(\alpha_{a}^{i}\right)=1\) signifies that data centre \(\alpha_{a}^{i}\) is not and is water cooled respectively. In addition, let \(\beta\left(\alpha_{a}^{i},\ t_{y}\right)\), \(\ t_{y}\epsilon\)\(t\); \(t=\{t_{1},\ t_{2},\ldots,\ t_{Y}\}\) be the water footprint of the Earth-based terrestrial data centre \(\alpha_{a}^{i}\) at the epoch \(t_{y}\). Furthermore, let \(\gamma\) be the set of ground locations of terrestrial data centres. \[\gamma=[\gamma_{1},\gamma_{2},\ldots,\ \gamma_{B}] \tag{3}\]  
The location indicator of data centre \(\alpha_{a}^{i}\) at location \(\gamma_{b},\ \gamma_{b}\,\epsilon\,\gamma\) at epoch \(t_{y}\) is denoted \(I\big{(}\alpha_{a}^{i},\gamma_{b},t_{y}\big{)}\epsilon\,\{0,1\}\). The data centre \(\alpha_{a}^{i}\) is not located and is located at \(\gamma_{b}\) at epoch \(t_{y}\) when \(I\big{(}\alpha_{a}^{i},\gamma_{b},t_{y}\big{)}=0\) and \(I\big{(}\alpha_{a}^{i},\gamma_{b},t_{y}\big{)}=1\) respectively. In the discussion here, terrestrial data centres utilise Earth's water resources. The Earth's water resource being used is at a temperature that is suitable for data centre cooling. The use of terrestrial data centres poses a challenge to water security when  
\[\sum_{a=1}^{A}\sum_{i=1}^{I}\sum_{b=1}^{B}\sum_{y=1}^{Y}I\big{(}\alpha_{a}^{i} \big{)}I\big{(}\alpha_{a}^{i},\gamma_{b},t_{y}\big{)}\beta\big{(}\alpha_{a}^{i },\ t_{y}\big{)}\ \geq\ \sum_{b=1}^{B}w(\gamma_{b}) \tag{4}\]  
where \(w(\gamma_{b})\) is the amount of water resources available at the \(bth\) Earth-based terrestrial location \(\gamma_{b}\). Let \(\phi\) be the set of applications requiring access to water resources such that  
\[\phi=\big{\{}\phi_{1},\phi_{2},\ldots,\phi_{C}\big{\}} \tag{5}\]  
In addition, let \(w(\phi_{c},\gamma_{b})\); \(\phi_{c}\,\epsilon\,\phi\) be the water footprint of the \(cth\) application \(\phi_{c}\) at location \(\gamma_{b}\). The demand for water by other application gives rise to water access challenges given the condition  
\[\left(\sum_{a=1}^{A}\sum_{i=1}^{I}\sum_{b=1}^{B}\sum_{y=1}^{Y}I\big{(}\alpha_{ a}^{i}\big{)}I\big{(}\alpha_{a}^{i},\gamma_{b},t_{y}\big{)}\beta\big{(}\alpha_{a}^{i },\ t_{y}\big{)}\right)+\left(\sum_{c=1}^{C}\sum_{b=1}^{B}w(\phi_{c},\gamma_{b })\right)\geq\sum_{b=1}^{B}w(\gamma_{b}). \tag{6}\]  
The relation in (6) holds true under different conditions such as  
\[\left(\sum_{a=1}^{A}\sum_{i=1}^{I}\sum_{b=1}^{B}\sum_{y=1}^{Y}I\big{(}\alpha_ {a}^{i}\big{)}I\big{(}\alpha_{a}^{i},\gamma_{b},t_{y}\big{)}\beta\big{(}\alpha _{a}^{i},\ t_{y}\big{)}\right)>\left(\sum_{c=1}^{C}\sum_{b=1}^{B}w(\phi_{c}, \gamma_{b})\right) \tag{7}\]  
\[\left(\sum_{a=1}^{A}\sum_{i=1}^{I}\sum_{b=1}^{B}\sum_{y=1}^{Y}I\big{(}\alpha_ {a}^{i}\big{)}I\big{(}\alpha_{a}^{i},\gamma_{b},t_{y}\big{)}\beta\big{(}\alpha _{a}^{i},\ t_{y}\big{)}\right)\equiv\left(\sum_{c=1}^{C}\sum_{b=1}^{B}w(\phi_{ c},\gamma_{b})\right) \tag{8}\]  
If (6) and (7) holds true, the data centre water footprint is less than that of other applications. The data centre water demand overwhelms water supply in the considered Earth locations and is described as case C1. The case where (6) and (8) holds true is one in which data centre water footprint is roughly equal to that of other applications and is described as case C2.