# ATWin: An Improved and Detailed Startup Model of TTP/C
## 3 The Proposed Startup Model ATWin
### The Lower Time Bound of ATW
The decision time point \(T^{i}_{start}(n)\) refers to the instant that the node \(i\) starts the post-cold-start process, which is specified by the instant of the end of the current slot for cold-start sending nodes and the instant of the end of the ATW for cold-start receiving nodes. The epoch node \(EPO_{i}(n)\) refers to the node that satisfies conditions below in startup cycle n: if node \(i\in\mathsf{S}_{t}(n)\), then the \(EPO_{i}(n)\) for node \(i\) in startup cycle \(n\) is the node \(i\); if the node \(i\in\mathsf{S}_{t}(n)\), then the node \(EPO_{i}(n)\) for node \(i\) in this startup cycle is the earliest node of all nodes in the ATWin of the node \(i\). If there exist multiple nodes satisfying the conditions aforementioned, the smallest node id is \(EPO_{i}(n)\). In the startup cycle \(n\), there must be a node \(EPO_{i}(n)\) for every node \(i\). Activity time for node \(i\) refers to the instance for node \(i\) that opens its ATWin in startup cycle n. As it can be concluded from the definition, if node \(i\in\mathsf{S}_{t}(n)\), \(T^{i}_{atstart}(n)=T^{i}_{listen}(n)\); if \(i\in\mathsf{S}_{t}(n)\), \(T^{i}_{start}(n)=TR^{EPO_{i}}_{i}(n)\), where \(EPO_{i}\) is the epoch of node \(i\). Compensating time \(\delta\) refers to the longest time interval from the instant when a node ends its ATW to the instant when the node completes the synchronizing. The definitions of symbols proposed in this paper are summarized in Table 5 for easy reference. These definitions are explained in detail above. Figure 6: Node classification during cluster startup. Let the duration of the ATW be \(\Delta_{window}\). Let the time duration of sending time for a cold-start frame be \(\Delta_{frame}\). For the reason of physical contention and logic contention in the bus topology, the \(\Delta_{window}\) should be set with enough time within which the sending node can completely send a cold-start frame, and the receiving nodes can receive cold-start frames from all sending nodes. The relationship between the listening timeout values of the nodes in \(S_{t}(n)\) can be described by Theorem 1. Theorem 1: _During the \(n\)-th round of startup, for any node \(i\) and node \(j\) in the union set of \(S_{t}(n)\) and \(S_{t}(n)\), the absolute difference value between their active time points must be less than or equal to the maximum propagation delay. Its formal expression is Inequality (7) (see Appendix A for the proof)._  
\[\left|T^{i}_{start}(n)-T^{j}_{start}(n)\right|\leq\Delta_{prop} \tag{7}\]  
The meaning of Theorem 1 is that when the start time interval between node \(i\) and node \(j\) is less than \(\Delta_{prop}\), node \(i\) and node \(j\) start simultaneously. It supplements one of the missing details of AS6003, marked in Mark 2 in Figure 4.  
Let the number of the sending nodes in set \(S_{t}(n)\) be m and the number of the receiving nodes in set \(S_{t}(n)\) be g. To ensure that any receiving node \(k\) (\(k\in S_{t}(n)\)) can receive all the cold-start frames within ATW, the constraint is given in Formula (8) for any node \(i\) and node \(j\) in \(S_{t}(n)\). \[\min(\Delta_{window})\geq\max\Bigl{\{}TR^{i}_{k}(n)-TR^{j}_{k}(n)\Bigr{\}}+ \Delta_{frame} \tag{8}\]  
According to Theorem 1, the constraints are listed as follows:  
\[0\leq prop^{i}_{j}\leq\Delta_{prop},\ \forall i,j\in\{1,2,\ldots,m+g\} \tag{9}\]  
\[\left|T^{i}_{listen}(n)-T^{j}_{listen}(n)\right|\leq\Delta_{prop},\forall i,j \in\{1,2,\ldots,m\} \tag{10}\]  
\[prop^{i}_{j}=prop^{i}_{k}+prop^{j}_{k^{\prime}}\forall i,j\in\{1,2,\ldots,m+g\},i\leq k\leq j \tag{11}\]