# Satellite-Aided Consensus Protocol for Scalable Blockchains
## 5 Throughput Analysis
### Propagation Delay  
In the above analysis, we considered the capacity limits of \(B/\Delta\leq C\). As shown in [29], the spreading delay across, e.g., 90% nodes, via Internet could be quite long. Hence, we define \(i(t)\) as the ratio of informed nodes at time \(t\) for a specific network. Take \(t=\Delta\) as an example, where the round slot is \(\Delta\) seconds. Hence, \(i_{\Delta}\triangleq i(t=\Delta)\) is the ratio of nodes receiving the block in the last round. After the block of last round is published, \(1-i_{\Delta}\) fraction of nodes will not receive that block in the next round, potentially leading to orphaned blocks as the case in Figure 1(c). However, such orphaned blocks lower the chain growth rate and also affect the blockchain throughput, even without an adversary. (As a rule of thumb, if two or more forks occur, the next winner will always follow the longest known chain. If these forks are at the same height, the next winner will follow the one which achieves this height earlier). Now things get complicated. In Figure 3a, starting from a consistent state, the number of possible blockchain states grows exponentially after just three rounds. Note that the ratio of informed nodes after one round, i.e., \(i_{\Delta}\), is the most critical term affecting orphaning. To characterize this complicated problem, we focus on the factor \(i_{\Delta}\) and assume \(i\left(k\Delta\right)=1\) for \(k=2,3,...\) without affecting the spirit of our results. These countless states can be classified into four categories. We can abstract their states via the minimum height gap between the longest chain and other active forks. In Figure 3a, we start from a consistent state, denoted as \(S_{\infty}\), since there exists a unique active chain. If the next winner receives the last block, it is still in state \(S_{\infty}\). Otherwise, it jumps to the state \(S_{0}\) with a newly-generated fork. The next block generated on any fork will lead to a longer main chain by one block, i.e., state \(S_{1}\). Then, in the next round, if the winner follows the longest branch, the blockchain returns to a consistent state \(S_{\infty}\), otherwise it jumps back to the tied state \(S_{0}\). For any state, if the next winner misses the oracle or is an adversary, no block is generated or released in this round, and the chain goes to the state \(S_{\times}\). One can see that these four states form a Markov chain, as shown in Figure 3b. We can mathematically write the probability transition matrix as follows:  
\[\mathbf{Q}=\left( \begin{array}{ccccc}S_{0}:&0&h&0&1-h\\ S_{1}:&h\left(1-i_{\Delta}\right)&0&hi_{\Delta}&1-h\\ S_{\infty}:&h\left(1-i_{\Delta}\right)&0&hi_{\Delta}&1-h\\ S_{\times}:&0&0&h&1-h\end{array} \right),\]  
where \(h\) is the growth rate of the benign chain defined in Equation (1). Let \(\mathbf{w}\) be the steady-state distribution of the considered Markov chain. The expression of \(\mathbf{w}\) can be obtained from \(\mathbf{w}\mathbf{Q}=\mathbf{w}\):  
\[\mathbf{w}=\left( \begin{array}{c}\frac{h^{2}\left(1-i_{\Delta}\right)}{ \left(1-i_{\Delta}\right)h+1}\end{array} \left| \begin{array}{c}\frac{h^{3} \left(1-i_{\Delta}\right)}{\left(1-i_{\Delta}\right)h+1}\end{array} \right| \begin{array}{c}\frac{h-h^{3}\left(1-i_{\Delta}\right)}{\left(1-i_{\Delta} \right)h+1}\end{array} \right|1-h\end{array}\right).\]  
Note that, these four states have different probabilities to extend a new block in the longest chain. By comprehensively considering these states, we can obtain the growth rate:  
\[h_{\text{PROP}}=w_{0}h+w_{1}hi_{\Delta}+w_{\infty}hi_{\Delta}+w_{\times}h= \frac{h}{1+\left(1-i_{\Delta}\right)h}.\]  
As a result, the blockchain throughput with information propagation delay can be expressed as  
\[\mathsf{T}_{\text{PROP}}=\frac{h_{\text{PROP}}B}{\Delta}\leq\frac{h}{1+\left( 1-i_{\Delta}\right)h}C\triangleq\mathsf{T}_{\text{PROP}}^{m}. \tag{9}\]  
Figure 3: Blockchain evolution affected by information propagation delay. (**a**) The number of chain states skyrockets after three rounds.