## 4 Network Capacity
The four components of \(A_{i,T}^{\chi}\) can be used to populate the following matrices to aid in implementation: \(\varphi_{i}\left(d,p,t\right)\in\Phi_{i}\left(t\right)^{M\times Q}\), \(r_{i}\left(d,p,t\right)\in R_{i}\left(t\right)^{M\times Q}\), \(s_{i}\left(d,p,t\right)\in S_{i}\left(t\right)^{M\times Q}\) and \(\eta_{i}\left(d,p,t\right)\in\mathrm{H}_{i}\left(t\right)^{M\times Q}\), where \(M=\left|D_{i}\right|\), \(Q=\underset{d\in D_{i}}{\mathrm{max}}\left(\left|P_{i}^{d}\right|\right)\), and \(\left|\cdot\right|\) denotes the number of elements in the set. (1) Availability: The first component of the source node capacity model is based on the existence of path \(p\). Additionally, this is dependent on the line-of-sight of each link in the path \(p\) as a function of the orbital dynamics of the satellites, the position of sensor nodes, the minimum elevation visibility constraints and time. The availability matrix is denoted by \(\Phi_{i}\left(t\right)^{M\times Q}\), consisting of elements \(\varphi_{i}\left(d,p,t\right)\in\left\{0,1\right\}\), \(\forall d\in D_{i}\), \(p\in P_{i}^{d}\), \(t\in\left[t_{0},t_{0}+T\right]\). Where an available path \(p\) between the source node \(v_{i}\) and the destination node \(d\) is expressed as \(\varphi_{\mathit{idp}}\left(t\right)=1\) and when the path is not available or does not exist, the corresponding element in the matrix is assigned to zero. (2) Transfer rate: The data transfer rate matrix is denoted by \(R_{i}\left(t\right)^{M\times Q}\), where the data transfer rate between source node \(v_{i}\) and destination node \(d\) in the path \(p\) is \(r_{\mathit{idp}}\left(t\right)\), \(d\in D_{i}\), \(p\in P_{i}^{d}\), \(t\in\left[t_{0},t_{0}+T\right]\). Generally, the data transfer rates are selected based on expected channel performance (_i.e._, constrained by the minimum signal-to-noise (SNR) requirements) and can be updated during the operation of the network [40,41]. The optimal link rate distributions may be selected by network scheduling algorithm \(\chi\) to maximize the throughput. (3) Establishment of the path: Governed by the network scheduling algorithm \(\chi\in\mathrm{X}\), a path \(p\) may or may not be wanted even if it is available. The establishment matrix is denoted by \(S_{i}\left(t\right)^{M\times Q}\), and the wanted path \(p\) between source node \(v_{i}\) and destination node \(d\) is denoted by \(s_{\mathit{idp}}\left(t\right)=1\), where \(d\in D_{i}\), \(p\in P_{i}^{d},t\in\left[t_{0},t_{0}+T\right]\). If network scheduling algorithm \(\chi\) does not allow \(p\) to transfer data, \(s_{\mathit{idp}}\left(t\right)=0\). (4) Path efficiency: Successful data transfer from source node \(v_{i}\) to destination node \(d\) is influenced by all of the nodes in the path \(p\). Nodes in the path may not maintain perfect links due to some reasons, e.g., antenna slewing and acquisition maneuvers, unknown noise that degrades the SNR and system failures. A node \(v_{r}\in p\) that always operates perfectly has an efficiency factor \(\eta_{r}\left(t\right)=1\), \(\forall t\in\left[t_{0},t_{0}+T\right]\). While a node \(v_{r}\in p\) is available on average 95% of the time, \(\eta_{r}\left(t\right)=0.95\), \(\forall t\in\left[t_{0},t_{0}+T\right]\). The path efficiency is defined as \(\eta_{i}\left(d,p,t\right)\overset{\Delta}{=}\underset{v_{r}\in p}{\prod} \eta_{r}\left(t\right)\). The path efficiency matrix is denoted by \(\mathrm{H}_{i}\left(t\right)^{M\times Q}\), and the efficiency of path \(p\) between \(v_{i}\) and \(d\) is denoted by \(\eta_{\mathit{idp}}\left(t\right)\).