# Wireless May Benefit Blockchain +
## 4 Analysis on Consensus Performance
### Scalability
It is noteworthy from the reference that we consider no queueing delay as it is less significant than the network delay [57]. We also approximate that \(\mathsf{T}_{\text{prop}}\approx 0\) as even wireless signals propagate at the speed of light. Nonetheless, as has been discussed in Section 3.1, \(\mathsf{T}_{\text{to}}\) is too significant to ignore. This justifies the approximation \(\mathsf{T}_{ij}\approx\mathsf{T}_{\text{to}}\). In Equation (3), it is also significant to notice that \(n_{\text{N}[\mathbb{S}(x)]}\) heavily depends on the type of consensus, i.e., PoW, PoS, and PoC, which is defined as the number of propagations within a consensus attempt via the gossip protocol:  
\[n_{\text{N}[\mathbb{S}(x)]}=n\big{|}_{\theta(n)\to 0} \tag{5}\]  
where \(n\) denotes the number of nodes in the blockchain network and \(\theta(n)\) gives the rate of nodes who have \"not yet received\" a block, which is formulated as [40]\[\theta(n) =\mathbb{E}[s_{t+1}(n)]\] \[\mathsf{r} =\frac{\mathsf{\#\ transactions}}{\mathsf{\#\ seconds}}\]0  
where \(t\) is each time slot where a single-hop propagation takes place. To further clarify, let us consider the variables \(s_{t}\) and \(r_{t}\), representing the proportions of nodes that have \"not yet received\" and \"already received\" the block in the \(t\)th time slot. It is evident that \(s_{t}=1-r_{t}\). In the initial time slot, denoted as \(s_{0}\), we assume the master peer has a block to propagate, leading to \(r_{0}=1/n\) and \(s_{0}=1-1/n\), indicating that only one node has received the update. Now, assuming the randomly selected node is chosen independently of other nodes and past decisions, we express Equation (6) as the expectation of \(s_{t}+1\) in terms of \(s_{t}\). In Equation (5), it is noteworthy that \(\mathbb{N}\big{[}\mathbb{S}_{\text{pos}}\big{]}<<\mathbb{N}\big{[}\mathbb{S}_ {\text{pow}}\big{]}\) as the validators in PoS are a subset of nodes. We also emphasize that \(\mathbb{N}\big{[}\mathbb{S}_{\text{pos}}\big{]}>>\mathbb{N}\big{[}\mathbb{S}_ {\text{pow}}\big{]}\) is also true; however, the identities of witnesses \(\in\mathbb{S}_{\text{pos}}\) in a PoC network are subject to change. Although it is not captured in the scalability, this rotation in the witnesses' identities will affect the decentralization, which will be detailed in the next subsection. Now, let us quantify \(p\) for the geometric random variable \(X\) in Equation (3), which is defined as the probability of a successful consensus among \(\mathbb{N}[\mathbb{S}(x)]\) nodes. We highlight that the probability \(p\) is a function of \(\mathbb{N}[\mathbb{S}(x)]\) in every round of consensus attempt, which can be formally written as  
\[\mathsf{r} =\frac{\mathsf{\#\ transactions}}{\mathsf{\#\ seconds}}\]1 \[\mathsf{r} =\frac{\mathsf{\#\ transactions}}{\mathsf{\#\ seconds}}\]2 \[\mathsf{r} =\frac{\mathsf{\#\ transactions}}{\mathsf{\#\ seconds}}\]3  
where \(f\) denotes the number of Byzantine nodes, which is designed to fluctuate due to wireless connections. It has been proved in the literature [58] that both \(\mathbb{N}[\mathbb{S}(x)]\) and \(f\) followed Poisson distributions with their own respective densities, which we denote by \(\lambda_{\mathbb{S}}\) and \(\lambda_{f}\), respectively. Here, we propose to model the impacts of wireless channel fluctuation on \(f\) as an increase in \(\lambda_{f}\). This is a plausible assumption in the sense that the number of Byzantine nodes \(\lambda_{f}\) should increase as more nodes experience communication errors due to the imperfect wireless connections. Also, notice in Equation (7) that \(F_{\mathbb{S}}\) denotes the cumulative distribution function (CDF) of random variable \(\mathbb{N}[\mathbb{S}(x)]\), which can be formally written as \(F_{\mathbb{S}}(s)=\mathbb{P}[\mathbb{S}\leq s]\).