# Wireless May Benefit Blockchain +
## 5 Numerical Results
### Results and Discussion  
#### 5.2.1 Scalability  
We start with presenting the results of R, which has been defined as Equation (2) in Section 4.1. We recall from the section that this paper adopts S for a Zookeeper network as the scalability benchmark, which is presented as a black solid line in all of Figures 2-5 and is compared to the R's of PoW, PoS, and PoC.  
Figure 2: PoW, PoS, and PoC—scalability R versus the number of nodes (\(\text{p}_{\text{fail}}=5\%\), \(10\) transactions per block).  
Figure 2 compares the three consensus mechanisms, i.e., PoW, PoS, and PoC-on the scalability, R. One can easily find that PoC is the most scalable among the three since it is designed to select a fixed number of witnesses, which also explains the \"flatness\" of its scalability versus the number of nodes. Due to such selectivity, PoC should pose a lower level of decentralization, as shall be presented in Section 5.2.2.  
In Figure 2, one can find it interesting that the scalability of PoS varies in accordance with parameter \(R_{v}\), indicating the rate of validators. We set the parameter as a Poisson random variable with a justification that there must be a \"certain amount\" of tokens that the largest number of participants hold. Suppose an Ethereum network. The number of ETHs will vary by holders, but the validators are strictly required to hold more than 32 ETHs. It means that the rate of validators can be formally written as  
\[r_{v}=1-\mathbb{P}[Z\geq 32] \tag{9}\]  
Figure 4: PoS—scalability R versus the number of nodes (\(p_{\text{fail}}=\{1,5,10\}\%\), 10 transactions per block).  
Figure 3: PoW—scalability R versus the number of nodes (\(p_{\text{fail}}=\{1,5,10\}\%\), 10 transactions per block).  
where \(Z\sim(\lambda_{\text{eth}})\) denotes a Poisson random variable indicating the number of ETHs held by a holder. The Poisson random variable was chosen to represent the distribution of ETHs across the globe; this modeling refers to a recent study that proved that the distribution of wealth converges to a Poisson distribution [63].  
Through Figures 3 and 5, we focus on visualizing the scalability R for each of the three consensus mechanisms, i.e., PoW, PoS, and PoC, respectively, with the probability of Byzantine failure at each node (due to the uncertain wireless connection), \(\mathsf{p_{fail}}\), varied. One can notice a straightforward but clear inversely proportional relationship between \(\mathsf{p_{fail}}\) and \(\mathsf{R}\): a higher \(\mathsf{p_{fail}}\) yields a lower R. The rationale behind this relationship is as follows. A consensus protocol adopts a gossip protocol to propagate the given block. If achieved among only \(n<3f-1\) nodes, the consensus is regarded to have a Byzantine fault, which calls for another round of consensus attempt. This is where a significant increase in the consensus delay \(\mathsf{T}_{c}\) occurs, which negatively affects the scalability R. (See Equations (2) and (3) for the formulation of the two parameters.)  
Figure 5 reveals a particularly interesting phenomenon in regard to R for PoC. As has already been discussed earlier in this subsection, R remains flat regardless of the number of nodes participating in consensus because PoC employees exactly 10 witnesses for each consensus. However, the same principle applies to PoC as well: a higher \(\mathsf{p_{fail}}\) degrades R due to the need for multiple block propagation attempts for a single consensus.