# ATWin: An Improved and Detailed Startup Model of TTP/C
## 4 Formal Demonstration of the Validity of the ATWin Model
### The upper bound of the clique-detecting time  
A successful startup of the system means that there exist at least two nodes that have passed clique detecting and then enter into the SYNC state.  
The result of whether clique detecting succeeds or fails depends on set \(S_{l}(n+1)\). All nodes in set \(S_{l}(n+1)\) will be synchronized with node \(hd(n+1)\) in the _(n+1)-th_ round startup before the decision time of node \(hd(n+1)\). If the number of nodes in set \(S_{l}(n+1)\) is only one, the system is not judged into the SYNC state until node \(hd(n+1)\) succeeds in clique detecting; if the number of nodes in set \(S_{l}(n+1)\) are greater than two, all the receiving nodes reach the SYNC state before the decision time of the head node \(hd(n+1)\).  
Figure 9 illustrates the case when the clique-detecting time can reach its maximum. Suppose that node \(i\) is the head node in the _n-th_ round startup, and node \(j\), \(k\) are the receiving nodes in the Figure. Node \(i\) cannot ensure its SYNC state before its PSP phase in the next TDMA. According to AS6003, if a receiving node that receives a valid cold-start frame is in the majority clique, it reports a startup success in its own PSP phase. Node \(k\) is the same. Any receiving node can complete clique detection before the sending nodes; thus, Equation (29) can be entailed.  
\[\text{max}(\Delta_{note})=(n-1)\Delta_{slot}+\Delta_{PSP} \tag{29}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Cases** & **Contention** & \(\text{max}(\Delta_{ contention})\) \\ \hline \(\#S_{t}(n)=1\) & pseudo-contention & \(2\Delta_{TDMA}+\Delta_{slot}-\Delta_{PSP}\) \\ \hline \(\#S_{t}(n)\geq 2\) & true contention & \(2\Delta_{TDMA}+\Delta_{prop}-\Delta_{PSP}\) \\ \hline \hline \end{tabular}
\end{table}
Table 7: The situations of the contention-removing time.  
**C. The upper bound of the startup time**  
From the analysis aforementioned, when there exists only one node that becomes the sending node and the node has the maximum ID number, \(\Delta_{\textit{contention}}\) obtains the max value. If \(\Delta_{\textit{pote}}\) obtains the max value, then there is only one receiving node in a startup cycle. Hence, when there is only one sending node in the startup cycle \(n\) and there is only one receiving node in this cycle, the system consumes the maximal startup time.  
As to a system consisting of n core nodes that are allowed to perform startup and using the contention-removing strategy in the ATWin model based on AS6003, the startup time upper bound under fault-free circumstances is deduced as Equation (30).  
\[TS=\max(\Delta_{\textit{contention}})+\max(\Delta_{\textit{ voice}}) \tag{30}\] \[=2\Delta_{\textit{TDMA}}+\Delta_{\textit{slot}}-\Delta_{\textit{ PSP}}+(n-1)\Delta_{\textit{slot}}+\Delta_{\textit{PSP}}\] \[=2\Delta_{\textit{TDMA}}+\Delta_{\textit{slot}}+(n-1)\Delta_{ \textit{slot}}\] \[=3n\Delta_{\textit{slot}}\]  
The condition that the system consumes the maximal startup time is that there are two nodes with the highest ID number performing a concurrent startup and the other nodes are in off-line or fail-silent mode.  
From the above analysis, if the core nodes start concurrently in the \(n\)-\(th\) round of startup, our ATWin startup model can eliminate the contention in the (\(n\)+1)-\(th\) round. It takes at most \(3n\Delta_{\textit{slot}}\) time to succeed in startup. Therefore, the proposed startup model is effective both in the logical domain and the time domain.