## 3 Preliminaries  
In this section, we describe the application scenario of the DMRF protocol. Various variables used in the DMRF protocol will be defined. Here, we consider a scenario where the WSN is formed by stationary sensors in a two-dimension sensing field. Figure 1 shows the scenario. The source node collects relevant data, and transmits through intermediate nodes to the sink node. The nodes deployed in the network conform to standards or random distribution. The initial energy of each node is homogeneous and the batteries are not rechargeable. The communication radius of the node is \(r\). In the network topology, there exist failure nodes, empty area and congestion area. Data packets are required to be sent from the source node to the sink node in the specified time limit. For each node, any other node in its communication range could become its neighbor. The DMRF protocol aims to reduce the effect of failure nodes, empty area and congestion area in the network to guarantee real-time performance. We first give the definitions of some variables used in the DMRF protocol, as listed in Table 1.  
\begin{table}
\begin{tabular}{l l} \hline
**Variable** & **Description** \\ \hline FAULTY \& JFAULTY & Faulty \& JFAulty state \\ CONG \& JCONG \& NORMAL & Congestion \& JCongestion \& Normal state \\ VOID & Void region \\ \(Suc_{j}\) & Successful transmission ratio to node \(j\) \\ \(p_{i}\) & Jumping probabilities to node \(i\) \\ \(\lambda\) & Remaining transmission time factor \\ \hline \end{tabular}
\end{table}
Table 1: Definitions of variables and notations used in the DMRF protocol.  
**Definition 1:** Node Failure State **FAULTY.****FAULTY** indicates that the node is in failure state. If the states of the **FCS** nodes of the current node are all **FAULTY**, then the state of current node is set to **JFAULTY**, indicating the jumping transmission mode should be used to transmit data packet due to the congestion.  
**Definition 2:** Node Congestion State **CONG.****CONG** indicates that the node is in congestion state. If the states of the **FCS** nodes of the current node are all **CONG**, then the state of current node is set to **JCONG**, indicating the jumping transmission mode should be used to transmit data packet due to the congestion.  
**Definition 3:** Empty Area State **VOID.****VOID** indicates that there is no node in the area. If the states of the **FCS** nodes of the current node are all **VOID,** then the state of the current node is set to **VOID**, indicating the jumping transmission mode should be used to transmit data packet due to the empty area.  
**Definition 4:** Jumping Probability \(p_{i}\). The \(p_{i}\) indicates the probability of jumping from current node to the node \(i\), the higher value of \(p_{i}\) indicates the higher jumping probability.  
\begin{table}
\begin{tabular}{l l} \hline \(\theta_{{}_{low}}\), \(\theta_{{}_{high}}\) & Lower or Upper threshold of remaining transmission time factor \\ \(\theta_{{}_{jump}}\) & Jumping threshold \\ \(delay_{{}_{i}\,j}\) & Delay between node \(i\) and node \(j\) \\ LOW \& MEDIUM \& HIGH \\ \(T\) & Estimated transmission time \\ \(L\) & Remaining transmission time of data packet \\ \(v\) & Average transmission rate \\ \(t\) & Maximum remaining transmission time of data packet \\ \(h\) & Average length of one hop \\ \(e\) & Average energy consumption of each transmission \\ \(d\) & Node density \\ \(r\) & Sensing radius of node \\ \(c\) & Confidence variable of node \\ \hline \end{tabular}
\end{table}
Table 1: _Cont._  
Figure 1: System scenario.  
**Definition 5:** Confidence variable \(c\). The \(c\) indicates the confidence level of reliability of the node, the higher value of \(c\) indicates the higher reliability of the node.