# ATWin: An Improved and Detailed Startup Model of TTP/C
## 3 The Proposed Startup Model ATWin
### The Lower Time Bound of ATW
\begin{table}
\begin{tabular}{l l} \hline \hline
**Symbols** & **Annotations** \\ \hline \(S\) & Set of core nodes allowed for cold-start \\ \hline \(S_{t}(n)\) & Set of sending nodes in the \(n\)-th round of startup \\ \hline \(S_{t}(n)\) & Set of receiving nodes in the \(n\)-th round of startup \\ \hline \(S_{z}(n)\) & Set of unstarted nodes in the \(n\)-th round of startup \\ \hline \(T^{i}_{listen}(n)\) & Listening timeout time of node \(i\) during the \(n\)-th round of startup \\ \hline \(prop^{i}_{j}\) & Propagation delay from node \(i\) to node \(k\) \\ \hline \(\Delta_{prop}\) & The max propagation delay in a TTP/C cluster \\ \hline \(\Delta_{window}\) & Duration of ATW \\ \hline \(\Delta_{frame}\) & Duration of transmitting a complete cold-start frame \\ \hline \(N_{k}\) & Specified slot number of node \(k\) in designed phase \\ \hline \(TR^{k}_{i}(n)\) & The arrival time of the cold-start frame from node \(k\) to node \(i\) \\ \hline \(hd(n)\) & The head node in the \(n\)-th round of startup \\ \hline \(T^{i}_{start}(n)\) & Time when node \(i\) starts the PSP phase in the \(n\)-th round of startup \\ \hline \(T^{i}_{start}(n)\) & Activity time for node \(i\) in the \(n\)-th round of startup \\ \hline \(EPO_{i}(n)\) & The epoch node of node \(i\) in the \(n\)-th round of startup \\ \hline \(\delta\) & The compensating time \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The definitions of symbols proposed in this paper. where \(TR_{k}^{i}(n)\) is related to the listening timeout time of node \(i\) and the propagation delay between node \(i\) and node \(k\). According to the definition and Formula (9), \(\text{max}\Big{\{}TR_{k}^{i}(n)-TR_{k}^{j}(n)\Big{\}}\) can be computed as:  
\[ \begin{split}&\text{max}\Big{\{}TR_{k}^{i}(n)-TR_{k}^{j}(n)\Big{\}}\\ &=\text{max}\Big{\{}T_{listen}^{i}(n)+prop_{k}^{i}-T_{listen}^{j} (n)-prop_{k}^{j}\Big{\}}\\ &=\text{max}\Big{\{}T_{listen}^{i}(n)-T_{listen}^{j}(n)\Big{\}}+ \text{max}\{prop_{k}^{i}-prop_{k}^{j}\Big{\}}\\ &=\Delta_{prop}+\text{max}\Big{\{}prop_{k}^{i}-prop_{k}^{j}\Big{\}} \end{split}  \tag{12}\]  
According to Formula (8), the propagation delay must be greater than zero. Combined with Formula (11), when \(prop_{k}^{i}\) obtains the maximum value, \(prop_{k}^{i}\) could obtain the minimum value. Therefore, \(\Big{(}prop_{k}^{i}-prop_{k}^{j}\Big{)}\) can also obtain the maximum value under the condition. The following formula must be satisfied. \[\text{min}(\Delta_{window})\geq 2\Delta_{prop}+\Delta_{frame} \tag{13}\]  
\(\Delta_{window}\) supplements one of the missing details of AS6003, marked as Mark 1 in Figure 4.  
Formula (14) is derived from the above analysis. \[\text{max}\Big{\{}prop_{k}^{i}-prop_{k}^{j}\Big{\}}=\Delta_{prop} \tag{14}\]  
Since the nodes are not allowed to perform other operations in the protocol within ATW, there is a compensation time \(\delta\) reserved for nodes to complete the remaining operations for synchronization. The required value for \(\delta\) should not exceed the slot exhaustion time of the sending node, as shown in Figure 7, where \(\Delta_{PRP}\) is the duration time of the PRP in the TTP/C three-phase cycle. To maintain compatibility with the AS6003, the duration time of the \(\Delta_{TP}\) phase in its time slot can be set to the length of \(\Delta_{window}\) without a dedicated timer if the dedicated MEDL is used in the startup phase. In the MEDL design phase, \(\Delta_{TP}\) should satisfy Formula (15). \[\Delta_{TP}\geq 2\Delta_{prop}+\Delta_{frame} \tag{15}\]  
At this time, the compensation time should meet the constraint condition defined in Formula (16). The compensation time supplements one of the missing details of AS6003, marked Mark 5 in Figure 4.  
\[\delta\leq\Delta_{PRP}-\Delta_{prop} \tag{16}\]  
To sum up, in order to start correctly, the lower bound time of ATW is (\(2\Delta_{prop}+\Delta_{frame}\)). Figure 7: The sequence diagram of the Compensating time.