## 4 Collaborative Distributed Scheduling Approach
### One-Step Collaborative Distributed Scheduling Approach
#### 4.2.2 Scheduling Update  
The parameters of the working state model for each node, the behavior model of its child nodes and the dynamics of the communication channel are all worked out at the SU stage and are used at the next round. At the SU stage, each node only sends its parameter of total transmission time to its parent node, which is referred to the upstream update, as is illustrated in Figure 5. Take note that there is no reverse variables exchanging here, which is the second update process in T-CDSA.  
Figure 4: State transition of child nodes.  
Figure 3: State transition diagram of a node in O-CDSA.  
After a node receives the total transmission times of all its child nodes, it calculates the parameters \(v\) and \(m\) as follows:  
\[v=\sum_{jech}t_{j}\Big{/}t_{a} \tag{1}\]  
\[m=\prod_{jech}(1-\Big{/}t_{a}\Big{/}t_{a}) \tag{2}\]  
where \(ch\) is the set of its child node, \(t_{j}\) denotes the total transmission time of node \(j\) in the last DD stage, and \(t_{a}\) represents the total active time of the node in the last DD stage.  
In this paper, in order to avoid the cases that \(v\) is bigger than 1 or \(m\) is lower than 0, we set \(v\in[0,1]\) and \(m\in[0,1]\) in case of the severe collision in packet transmission.  
The parameter \(\beta\) can be computed as below:  
\[\beta=\frac{The\;\;number\;\;of\;\;data\;\;packets\;\;transmitted\;\;successfully}{ The\;\;\;total\;\;number\;\;of\;\;data\;\;packets\;\;transmitted\;d} \tag{3}\]  
Equation (3) can be extended to other MAC protocols. For the RTS/CTS based CSMA/CA protocol, the parameter \(\beta\) can be obtained by replacing the total number of data packets transmitted and packets successfully transmitted with the number of RTS frames transmitted and the number of ACK frames received.  
Our CDSAs are implemented above the MAC layer, so we do not consider the transition probability to state R, state Ix and I\({}_{0}\), which are jointly decided through negotiations of MAC protocols between the sender and the receiver. Let P(S\({}_{i+1}\)|S\({}_{i}\)) denote the scheduling probability of SM that a node transmits from state S\({}_{i}\) to state S\({}_{i+1}\). By the analysis of Section 4.2.1, to derive the probabilities, we should take into account the ratio of a data packet successfully transmitted by a node to the next hop and the state of its child nodes. Take \(p(T|R)=m\times\beta\) as an example. In order to successfully transmit a data packet, the precondition for a node to transmit its received data packet depends on the state when all its child nodes stop transmitting and the data packet can be successfully received by its next hop. Therefore, the result of a node's SM can be calculated from Table 1. The SM will instruct operations of the node at the next DD stage, where the first column is the current state, the second column represents the next state and the third column is the state transition probability from the current state to the next state. For a sleep node, it will transfer to state I\({}_{0}\) after waking up.  
As is seen from Table 1, the child nodes have direct influence on the probability to Transmit and the probability to Sleep of its parent node. It is noted that the parent node has indirect influence on its child  
Figure 5: Parameter delivery procedure in O-CDSA.  
nodes because the data transmission times of child nodes are related to the probability of their parent node from state I\({}_{0}\) to state S. For example, if the parent node has an over probability to Sleep state, it will have much more time in Sleep state, leading to more data packets transmission time in its child nodes. Because the parameter \(v\) of the parent node has much more value, the parent node will probably reduce its time in Sleep state accordingly, this is to say that the child nodes and the parent node affect each other interactively.