# ATWin: An Improved and Detailed Startup Model of TTP/C
## 4 Formal Demonstration of the Validity of the ATWin Model
### The Ability of Contention Eliminating
: Since the sending node does not detect contention within ATW, it needs at least one TDMA round to determine the contention according to clique detection. For node \(i\) and node \(j\), \(i<j\) in set \(\mathrm{S}_{l}(n)\), they confirm themselves as a startup success and insist on sending the cold-start frame. According to AS6003 and the ATwin model, node \(i\) and node  
Figure 8: The contention elimination. In subfigure (**a**), node \(i\) and node \(j\) perform the cold-start concurrently. The contention will be eliminated in the next startup round where the node \(i\) successfully sends a cold start frame. In subfigure (**b**), node \(j\) receives invalid cold-start frames, so it detects a concurrent start-up in this startup round. Node \(j\) will re-listen in the next startup round, while node \(i\) will successfully send a cold start frame. \(j\) need to enter the PRP stage after the ATW, as shown in Figure 8a. After successfully sending, the nodes need to perform a TDMA. In the next sending slot, they enter the PSP phase to perform clique detection. This process takes \((n-1)\Delta_{slot}\) time, and the clique detection takes \(\Delta_{PSP}\). Then, the nodes receive the CB (communication blindness) error and wait for \(\Delta_{startup}^{k}\) to try the cold-start again. According to Theorems 3 and 4, all nodes in \(\text{S}_{l}(n)\) and \(\text{S}_{z}(n)\) are timed-out after nodes in \(\text{S}_{t}(n)\), that is, all sending nodes do not receive confirmation from the receiving nodes during this period. After waiting for a \(\Delta_{TDMA}\), the sending nodes detect a CB error and then lose the sending authentication and wait for the start timer timeout. If the start timeout timer of node \(i\) expires earlier \((N_{i}<N_{j})\) and the sending authentication is obtained, then Formula (19) holds (see Appendix D for the detailed derivation). \[T_{listen}^{j}(n+1)-TR_{j}^{i}(n+1)\geq\Delta_{PSP}+\Delta_{frame} \tag{19}\]  
It can be known from Formula (19) that if node \(i\) obtains the sending authentication, that is that the node \(i\) belongs to set \(\text{S}_{t}(n+1)\), then the node \(j\) belonging to set \(\text{S}_{t}(n)\) necessarily loses the sending authentication in the _(n+1)-th_ round of startup. Node \(j\) belongs to \(\text{S}_{l}(n+1)\). As shown in Figure 8a, node \(j\) receives the cold-startup frame from node \(i\) before \(T_{startup}^{j}(n+1)\). Because in the _(n+1)-th_ round, only the head node \(hd(n+1)\) can obtain the sending authentication. Only the node with the lowest node ID in set \(\text{S}_{t}(n)\) belongs to set \(\text{S}_{t}(n+1)\) in this startup round. In summary, when the core nodes do not start concurrently during the _n-th_ round of startup, all the initial startup nodes regard the received frame as the first cold-start frame, which is the case of pseudo-contention. This round of startup is still considered as a contention startup. When it comes to true contention, which means that the core nodes start concurrently in the _n-th_ round, the ATWin startup model eliminates contention in a way that only one node obtains sending authentication in the next round. Our ATWin startup model can eliminate contention effectively.