## 4 Network Capacity  
In this section, we develop a mathematical model that assesses the network capacity of an arbitrary sensor and satellite network as mentioned above. Denote the above network by \(G\left(V_{N},E\right)\). The network capacity is the total amount of data that can be exchanged across a network over a finite time. Let \(v_{i}\in V_{N}\) be a source node, and let \(b_{i,j}\) be the \(j\)-th bit transmitted from \(v_{i}\) to its destination \(d\left(v_{i},j\right)\). In unicast scenarios, \(d\left(v_{i},j\right)\) represents a single destination; and for multicast, \(d\left(v_{i},j\right)\) represents the set of all destinations. Let \(A_{i,T}^{\chi}\) be the amount of bits transmitted by \(v_{i}\) and which successfully reached their respective destinations during the time interval \(\left[t_{0},t_{0}+T\right]\), where \(t_{0}\) is the start time. We take \(A_{i,T}^{\chi}\) as the capacity of source node \(v_{i}\). \(\chi\in\mathrm{X}\) in \(A_{i,T}^{\chi}\) denotes the spatial and temporal network scheduling algorithm, and \(\mathrm{X}\) denotes the set of all scheduling algorithms. If the same bit is transmitted from a source \(v_{i}\) to multiple destinations \(D\left(v_{i},j\right)\), _i.e._, multicast. All of the bits are the same, so only one bit is counted in the calculation of \(A_{i,T}^{\chi}\). Then, we assume that the network is stable \(\forall\chi\in\mathrm{X}\). A network is stable if and only if for any fixed \(N\), each node in the network has an infinite queue to transmit, and the queue length in any relay node storing packets in transit does not grow to infinity as \(T\rightarrow\infty\). That is, with an infinite queue to transmit, the long-term incoming data rate into the network equals the outgoing data rate. We further assume that there is no traffic loss for queue overflow. The transport capacity of an arbitrary sensor and satellite network \(G\left(V_{N},E\right)\) when using the spatial and temporal network scheduling algorithm \(\chi\), denoted by \(C_{G}^{\chi}\), is defined as:  
\[C_{G}^{\chi}\overset{\Delta}{=}\frac{\sum_{i=1}^{N}A_{i,T}^{\chi}}{T} \tag{1}\]  
Additionally, the maximum transport capacity of the network is defined as:  
\[C_{G}\overset{\Delta}{=}\max_{\chi\in\mathrm{X}}C_{G}^{\chi} \tag{2}\]  
Obviously, for any \(\chi\), we have \(C_{G}\geq C_{G}^{\chi}\). The amount of bits successfully transmitted by the source (the capacity of) is defined as:  
\[A_{i,T}^{\chi}=\sum\limits_{d\in D_{i}}\int_{t=t_{0}}^{T+t_{0}}\left(\sum \limits_{p\in P_{i}^{t}}\phi_{i}\left(d,p,t\right)r_{i}\left(d,p,t\right)s_{i} \left(d,p,t\right)\eta_{i}\left(d,p,t\right)\right)\mathrm{d}t \tag{3}\]  
In Equation (3), \(D_{i}\) is the set of all destination nodes of \(v_{i}\) in \(\left[t_{0},t_{0}+T\right]\), and \(P_{i}^{d}\) is the set of all paths from \(v_{i}\) to the corresponding destination node \(d\). \(\varphi_{i}\left(d,p,t\right)\) repeats the availability of the path \(p\) (existence of a line-of-sight between nodes in the path) from the source node \(v_{i}\) to the destination node \(d\) at time \(t\). The dynamic data transfer rate is denoted by \(r_{i}\left(d,p,t\right)\) and is characteristic of the IWSSN systems. The establishment of the path \(p\) from \(v_{i}\) to \(d\) is driven by the spatial and temporal network scheduling algorithm \(\chi\) and is denoted by \(s_{i}\left(d,p,t\right)\). \(\eta_{i}\left(d,p,t\right)\) is the efficiency function of path \(p\) scheduled by \(\chi\). The total amount of bits successfully transmitted by source \(v_{i}\) is computed by summing the integrated data transfer rates to each destination node over the full time period \(t\in\left[t_{0},t_{0}+T\right]\).