# Satellite-Aided Consensus Protocol for Scalable Blockchains
## 5 Throughput Analysis  
### Network Capacity Limits  
In the context of blockchain, throughput means the amount of information, e.g., transactions, gets confirmed per second. We would like to characterize the maximum throughput of the satellite-aided consensus protocol. Let us say the maximum block size limit is \(B\) bits, and each round corresponds to \(\Delta\) seconds. Hence, since at most one valid block can be generated in each round, the maximum throughput is \(B/\Delta\) bps. However, the presence of the adversary and the outage of satellite links degrade the blockchain throughput. Based on the growth rate of the benign chain in Equation (1), the throughput of the satellite-aided consensus is  
\[\mathsf{T}_{\mathrm{CPTY}}=\frac{hB}{\Delta}=p\left(1-\beta\right)\frac{B}{ \Delta}. \tag{5}\]Let C(bps) be the capacity of the terrestrial mining network. Obviously, it is impossible to deliver a block with size \(B\) within the interval \(\Delta\), if \(B/\Delta>C\). Hence, the values of \(B\) and \(\Delta\) are constrained by \(B/\Delta\leq C\). The maximum throughput of a satellite-aided consensus under capacity limits is given by  
\[\mathsf{T}_{\text{CPTY}}\leq p\left(1-\beta\right)C\triangleq\mathsf{T}_{ \text{CPTY}}^{m}. \tag{6}\]  
As a benchmark, we present the throughput of PoW-based consensus borrowed from a recent work [11]. In PoW, the block generation obeys a Poisson process with mean \(\lambda\), wherein \(\beta\) fraction of miners are adversarial and the rest are benign. During the period \(\tau\), the probability of no benign block is \(\exp\left(-\lambda(1-\beta)\tau\right)\). Hence, the growth rate of the benign chain (at least one benign block during \(\tau\)) is \(1-\exp\left(-\lambda(1-\beta)\tau\right)\), and the throughput is  
\[\mathsf{T}_{\text{PoW}}=\left(1-\exp\left(-\lambda\left(1-\beta\right)\tau \right)\right)B/\tau. \tag{7}\]  
As the block generation rate \(\lambda\) cannot exceed the limit \(C/B\), the limiting throughput \(\mathsf{T}_{\text{PoW}}\) is thus bounded by  
\[\mathsf{T}_{\text{PoW}} \leq\left(1-\exp\left(-\frac{\tau C}{B}\left(1-\beta\right) \right)\right)\frac{B}{\tau}\] \[\leq\left(1-\exp\left(\beta-1\right)\right)C\triangleq\mathsf{T}_ {\text{POW}}^{m}. \tag{8}\]  
The equality is achieved if and only if \(\lambda=C/B=1/\tau\). By comparing \(\mathsf{T}_{\text{CPTY}}^{m}\) in Equation (6) with \(\mathsf{T}_{\text{POW}}^{m}\) in Equation (8), we have \(\mathsf{T}_{\text{CPTY}}^{m}>\mathsf{T}_{\text{POW}}^{m}\) if \(p\left(1-\beta\right)\geq 1-\exp\left(\beta-1\right)\). We thus define \(y\left(\beta\right)\triangleq\frac{1-\exp\left(\beta-1\right)}{1-\beta}\) as the threshold of \(p\). If the quality of multicast channel is better than a given threshold related to \(\beta\), i.e., \(p>y\left(\beta\right)\), then the satellite-aided consensus protocol has a larger maximum throughput than PoW. Note that \(y\left(\beta\right)\) is monotonically increasing in \(\beta\) within \(0\leq\beta<1\). To show this, for any \(\beta<1\), we have  
\[y^{\prime}\left(\beta\right)=\frac{\left(\beta-2\right)\exp\left(\beta-1 \right)+1}{\left(\beta-1\right)^{2}}>\frac{-1+1}{\left(\beta-1\right)^{2}}=0,\]  
where the inequality is because \(\left(\beta-2\right)\exp\left(\beta-1\right)\) is lower bounded by \(\left(\beta-2\right)\exp\left(\beta-1\right)|_{\beta=1}=-1\). As a result, for any \(\beta\leq 0.5\), if \(p>y\left(\beta=0.5\right)\approx 0.787\), then the proposed consensus protocol has a higher throughput than PoW as \(\mathsf{T}_{\text{CPTY}}^{m}>\mathsf{T}_{\text{POW}}^{m}\) always holds. As a benchmark, the outage probability of China's Beidou navigation satellite system is about 2% [42], i.e., \(p\approx 98\%\), which is much higher than the threshold \(y\left(\beta=0.5\right)\). If \(p<y\left(\beta=0\right)=1-\exp\left(-1\right)\), then \(\mathsf{T}_{\text{CPTY}}^{m}<\mathsf{T}_{\text{PoW}}^{m}\). **Remark 2**: _The above conclusion implies that the scalability of satellite-aided blockchains is highly related to the quality of the satellite multicast channels.