## 5 Analysis of CDSAs
### Adaptability  
Each node should sum up \(t_{a}\) - the time in active phrase, \(t\) - the transmission time of data packets and \(\beta\) - the ratio of data packets successfully transmitted at the DD stage. Each node should also calculate the parameter \(g\) -the scheduling probability from state F to state S in the T-CDSA. Thus we classify the tendency of \(t\) into three situations and illustrate the adaptability through analyzing the tendency of \(t\) and \(t_{a}\) as below:  
1) \(t^{n}>t^{n\cdot 1}\)  
Since the nodes generate fixed-size packets at a regular interval, then \(t^{n}>t^{n\cdot 1}\) means either \(\beta^{n}<\beta^{n\cdot 1}\) for the same workload as before or more data packets have been transmitted. For the former case, because the parameter \(\beta\) reflects the conflict status of the shared channel surrounding the node, the child nodes should also be characterized as a bigger transmission time than before. For the latter case, the child nodes must have transmitted more data packets than before due to a better state of the channel. Therefore, the variable \(v\) of the node in any case will increase at the next stage by Equation (1), resulting in the decrease of its transmission probability and its probability from I\({}_{0}\) to S in Table 1 and Table 2. The reduction of the transmission probability will make a lower conflict probability surrounding the node, leading to the decrease of its transmission time \(t^{n+1}\). The decrease of the probability from I\({}_{0}\) to S will cause the node a lower time in state Sleep and lead to more active time at the next DD stage. For the T-CDSA, \(t^{n}>t^{n\cdot 1}\) also increases the variable \(v\) of the next-hop. Subsequently the probability of the next-hop from F to S will reduce accordingly. After downstream update, the node itself will have a lower probability from F to S in Table 2. Therefore, the active time at the next DD stage \(t_{a}^{n+1}\) will also increase. 2) \(t^{n}<t^{n\cdot 1}\)  
If \(t^{n}<t^{n\cdot 1}\), we must have either \(\beta^{n}>\beta^{n\cdot 1}\) for the same workload as before or less data packets have been transmitted. For the former case, this implies that the successful transmission ratio of data packets for its child nodes is bigger than the previous one. For the latter case, the child nodes must have  
Figure 8: The sequences of SM and SP. transmitted less data packets than the last DD stage. Thus, in any case, the random variable \(v\) will decrease after the scheduling update by Equation (1). This will increase the transmission probability and conflict probability, resulting in an increase in the transmission time \(t^{n+1}\). An increase of variable \(v\) will also increase the probability from I\({}_{0}\) to S. Similarly, the probability from state F to state S in T-CDSA will increase. Hence, the node will possess a less active time \(t_{a}^{n+1}\) at the next stage. 3) \(t^{n}=t^{n\text{-}1}\)  
This situation means that the successful transmission ratio of data packets remains stable, which also implies that the successful transmission ratio of data packets for child nodes remains stable. After the scheduling update, the variable \(v\) will equal to that of the previous one. Therefore, the transmission probability and state scheduling probability of this node from I\({}_{0}\) to S will not change. Similarly, the variable \(v\) of the next-hop will keep stable, resulting in a stable probability from state F to state S. That is to say, the probability of state S will not change. Hence, the active time at the next data delivery state \(t_{a}^{n+1}\) will keep stable. Figure 9 illustrates the transition relationship. From this figure, we can see that the situation 3 is in an equilibrium state while the situations 1 and 2 above are not stable. Nodes will adapt to the workload and the status of the communication channel, and finally run to the situation 3 after a long time running. When in situation 3, the active time and the data packets transmission time will be stable.