# ATWin: An Improved and Detailed Startup Model of TTP/C
## 3 The Proposed Startup Model ATWin
### The Lower Time Bound of ATW  
This section focuses on the lower bound of ATW time to meet the requirements of a normal startup of the system. Several definitions and corresponding symbols are introduced to explain the derivation and the proof. A startup cycle, denoted as n, is a process that starts from the time when any node in the system attempts to access the bus and ends with the next accessing attempt after the contention elimination. If multiple nodes start almost simultaneously, the first transmission time is considered as the start of a startup cycle. In the _n-th_ round startup, the node abandons the cold-start if it detects channel activity before the listening timeout timer expires. Let \(S\) denote the node set; \(S_{c}(n)\) denotes the nodes that are allowed to perform cold-start; \(S_{I}(n)\) denotes the nodes that give up the cold-start operation during the _n-th_ round startup; \(S_{t}(n)\) denotes the nodes that normally perform the cold-start operation; and \(S_{z}(n)\) denotes the nodes that have not been started during the _n-th_ round startup. Thus, (4) holds, as seen in Figure 6.  
\[S_{c}(n)=S_{I}(n)+S_{t}(n)+S_{z}(n) \tag{4}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Type**} & \multicolumn{2}{c}{**Standard Model**} & \multicolumn{2}{c}{**ATWin Model**} \\ \cline{2-5}  & **Detect** & **Eliminate** & **Detect** & **Eliminate** \\ \hline \multirow{2}{*}{C1} & \multirow{2}{*}{No} & Discarding the first cold-start frame and then successfully starting in the next round. & Pseudo-contention & Discarding the first cold-start frame and then successfully starting in the next round. \\ \hline \multirow{2}{*}{C2} & \multirow{2}{*}{No} & Discarding the first cold-start frame and then successfully starting in the next round. & Contention occurs & Discarding the first cold-start frame and then successfully starting in the next round. \\ \hline \multirow{2}{*}{C3} & \multirow{2}{*}{No} & Discarding the invalid frames and then re-entering the listening state and waiting for a valid cold-start frame. & Contention occurs & Discarding the invalid frames and then successfully starting in the next round. \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The differences in detecting and eliminating contention between the standard startup model and our ATWin startup model. The listening timeout \(T^{i}_{listen}(n)\) refers to the listening timeout time point of node \(i\) in startup cycle \(n\). Channel delay \(prop^{k}_{i}\) denotes the communication delay from node \(i\) to node \(k\), including propagation delay and digitization error. For bus deployment, the channel delay between node \(i\) and node \(k\) is symmetrical. Its maximum one-way transmission delay is abbreviated as \(\Delta_{prop}\). That is to say, there is a formula, which is \(\Delta_{prop}=\max\{prop^{k}_{i}\}\). Concurrent startup refers to a phenomenon that there exists at least two nodes, named node \(i\) and node \(j\), satisfying the condition \(\left|T^{i}_{listen}(n)-T^{i}_{listen}(n)\right|\leq prop^{j}_{i}\). It is also called the true-contention phenomenon. Arrival Time \(TR^{k}_{i}(n)\) refers to the arrival time point when the cold-start frame sent by node \(i\) is received by node \(j\) at the \(n\)-_th_ round of startup. In the startup model, the sending node sends a cold-start frame immediately at the timeout time of the listening timeout timer; thus, Equation (5) can be entailed. \[TR^{k}_{i}(n)=T^{k}_{listen}(n)+prop^{k}_{i} \tag{5}\]  
For any two nodes in set \(\mathsf{S}_{t}\), they should satisfy the constraint below. \[TR^{k}_{i}(n)>T^{i}_{listen}(n) \tag{6}\]  
Head node \(hd(n)\) refers to the earliest start node that sends the cold-start frame firstly at the \(n\)-_th_ round of startup. If there exist multiple nodes whose listening timeout timers expire at the same time in a startup cycle, the head node in this startup cycle is the node with the smallest number id.