## 3 Improved LEACH Protocol
### Parallel Transmission Based on Scheduling for Cluster Head Nodes and Sensor Nodes
#### 3.3.2 Scheduling for Sensor Nodes  
When a node transmits a data packet, the acoustic wave propagates outward in a spherical pattern. It is possible that the transmission of a node will affect the reception of other nodes in adjacent clusters. It is assumed that when a destination node is within a circle with a radius of \(a_{d}r\) centered on the source node, its reception will be interfered with by this source node, and vice visa, where \(r\) is the distance from the source node to the destination node and \(a_{d}\) is a threshold. There are three types of conflicts that should be considered when scheduling the transmission time of the \(j\)th node in the \(i\)th cluster. The first type is the intra-cluster conflicts, which are caused by the scheduled nodes in the same cluster. A data conflict occurs when the completion time for the \(i\)th cluster head to receive data from the \(j\)th sensor node exceeds the arrival time to receive data from the \(l\)th sensor node, and vice versa. Therefore, for the \(j\)th node in the \(i\)th cluster, the transmission time that will cause conflict with the \(l\)th node in the \(i\)th cluster at the \(i\)th cluster head node satisfies the following inequalities:  
\[t_{j}^{i}+d_{ij}^{i}+t_{guard}+t^{p}\geq t_{l}^{i}+d_{il}^{i}\text{ and }t_{j}^{i}+d_{ij}^{i}\leq t_{l}^{i}+d_{il}^{i}+t_{guard}+t^{p} \tag{15}\]  
where \(1\leq l\leq j-1\). \(t_{j}^{i}\) and \(t_{l}^{i}\) denote the transmission times of the \(j\)th and \(l\)th sensor nodes in the \(i\)th cluster. \(d_{ij}^{i}\) and \(d_{il}^{i}\) denote the propagation delays from the \(j\)th and \(l\)th sensor nodes to the \(i\)th cluster head node. \(t^{p}\) denotes the duration of the cluster head node receiving a packet. So, \(t_{j}^{i}\) that will cause conflict with the transmission of the first to the (\(j-1\))th node in the \(i\)th cluster belongs to the set of  
\[U_{i,j}^{iita}=\overset{j-1}{\cup}\Big{[}t_{l}^{i}+d_{il}^{i}-d_{ij}^{i}-t_{ guard}-t^{p},t_{l}^{i}+d_{il}^{i}-d_{j}^{i}+t_{guard}+t^{p}\Big{]} \tag{16}\]  
The second type is the inter-cluster conflicts caused by the current transmission at the adjacent cluster head nodes. It is assumed that there are \(M\) cluster head nodes that are interfered with by the transmission of the \(j\)th node in the \(i\)th cluster. A data conflict occurs at the \(m\)th adjacent cluster head node if the completion time for the \(m\)th cluster head node to receive data from the \(l\)th sensor node within the same \(m\)th cluster exceeds the arrival  
Figure 2: Scheduling for cluster head nodes. time of the data sent by the \(j\)th sensor node from the \(i\)th cluster. Similarly, a data conflict is also triggered at the \(m\)th cluster head node if the completion time for the \(m\)th cluster head node to receive data from the \(j\)th sensor node in the \(i\)th cluster exceeds the arrival time of the data from the \(l\)th sensor node in the \(m\)th cluster to its cluster head node. Then, \(t_{j}^{i}\) that will cause conflict with the transmission of the \(l\)th sensor node in the \(m\)th cluster at the \(m\)th cluster head node satisfies the following inequalities:  
\[t_{j}^{i}+d_{mj}^{i}+t_{guard}+t^{p}\geq t_{l}^{m}+d_{ml}^{m}\text{ and }t_{j}^{i}+d_{mj}^{i}\leq t_{l}^{m}+d_{ml}^{m}+t_{guard}+t^{p} \tag{17}\]  
So, \(t_{j}^{i}\) that will cause conflicts at the \(m\)th adjacent cluster head node belongs to the set of:  
\[U_{i,j}^{iten}=\mathop{\cup}\limits_{m=1}^{M}\mathop{\cup}\limits_{l=1}^{l_{ m}}\left[t_{l}^{m}+d_{ml}^{m}-d_{mj}^{i}-t_{guard}-t^{p},t_{l}^{m}+d_{ml}^{m}-d_{mj} ^{i}+t_{guard}+t^{p}\;\right] \tag{18}\]  
where \(J_{m}\) is the number of sensor nodes in the \(m\)th cluster. The third type is the inter-cluster conflicts caused by the transmission of the sensor nodes in adjacent clusters at the \(i\)th cluster head node. It is assumed that there are \(N\) adjacent cluster transmissions that will interfere with the reception of the \(i\)th cluster head nodes.