## 2 System Model
### Network Model  
\(M\) sensor nodes and one base station node form a wireless sensor network with a known topology. As shown in Figure 1, BS is the base station or the destination node, while others are sensor nodes. The sensor nodes periodically generate data with different strict deadlines and send it to the base station node through data transmission between nodes. The entire network topology can be represented by a directed graph \(G(V,E,E^{\prime})\). \(V\) represents the set of vertices of all sensor nodes. \(E\) represents the set of all communication links, \(\overline{a}\overline{a}\) represents node \(e\) sends data to node \(a\), which is called the parent node of \(e\). \(E^{\prime}\) represents the set of all interference edges. Interference edge \(\overline{ab}\) means that the transmission of node \(a\) will interfere with the transmission of \(b\), and similarly, the transmission of node \(b\) will also interfere with the transmission of \(a\).  
Sensor nodes periodically generate data, and data transmission is carried out according to a time slot. The data packet \(P_{i}\) generated by \(i\)-th node \(v_{i}\) can be expressed as  
\[P_{i}=(T_{i},H_{i},D_{i},\phi_{i}) \tag{1}\]  
where \(T_{i}\) is the packet generation cycle, \(D_{i}\) is the packet transfer cutoff time, \(\phi_{i}\) is the routing path of the packet, \(H_{i}\) represents the total hops from the source node to the sink node. The unit of \(T_{i}\) and \(D_{i}\) are represented by the number of time slots. Generally, \(T_{i}\) is greater than \(H_{i}\) to ensure the transmission time required for packets. At any time slot \(t\), the packet contains three attributes, which are \(C_{i}\), \(h_{i}\), \(t_{i}\). \(C_{i}\) indicates the node where the data packet resides (\(C_{i}\in\phi_{i}\)), \(h_{i}\) indicates the number of hops remaining when data are transmitted from the current node to the destination node (\(0<h_{i}\leq H_{i}\)), \(t_{i}\) indicates the number of time slots contained in the remaining cutoff time of the packet (\(0<t_{i}\leq T_{i}\)).  
For the data in the process of transmission in WSNs, if the data state is \(t_{i}>h_{i}\), it means that the data can be transmitted to the destination theoretically. If the next time slot is allocated by the system for data transfer, the remaining time of the current data transfer status and the remaining hops are reduced by one and the data are transmitted to the next node. Otherwise, the remaining time of the data transfer status is reduced by one. When \(t_{i}=0\), node \(v_{i}\) will generate new data and start to wait for transmission scheduling. The initial remaining time \(t_{i}=T_{i}\) and the initial remaining hop number \(h_{i}=H_{i}\).  
In order to consider the practical applications, the following assumptions are assumed:  
1. The sensor node cannot transmit and receive data at the same time or receive data from more than one node at the same time;
2. The node that receives multiple data selects at most one data packet for data transmission in each time slot;  
Figure 1: Communication topology of WSNs.  
3. The data generated periodically by the source node has a strict cut-off time limit, and the cut-off time is equal to the data generation cycle;
4. In the process of data transmission, if the deadline is exceeded, the data will be directly discarded because it has become invalid;
5. The probability of success of wireless communication transmission will be affected by physical factors such as transmission power, encoding mode, and modulation scheme. This paper assumes that if the data are arranged for transmission, the probability of success is 1.