## 4 DMRF Protocol
### Network Congestion Detection Method
The transmission time of the node \(i\) to the sink node is \(T_{i}\). The delay between node \(i\) and node \(j\) is \(delay_{i,j}\). The remaining transmission time of the data packet in node \(i\) is \(L_{i}\). When the source node collects data packet, a deadline time will be assigned to the data packet. The deadline is larger than the total transmission delay from the source node to the sink node. Therefore, \(\lambda\geq 1\). Suppose the communication delay between the node \(i\) and its **FCS** in the network conforms to normal distribution, with expected value \(\mu\), and node \(i\) has \(n\) forwarding candidate nodes. When determining \(\theta_{low}\) and \(\theta_{high}\), if the current state of data packet is LOW, then the state of the data packet in next hop node can not be HIGH, and vice versa. This requirement can make the node energy consumption smooth because dramatic transmission rate change makes the energy consumption imbalance. In most of the time, \(\theta_{low}>\lambda\geq\theta_{high}\). Therefore, Equation (2) exists. \[ \begin{array}{c}\includegraphics[width=142.26378 pt]{Fig2}\end{array}  \tag{2}\]

The first two lines in Equation (2) limit that a packet whose status is LOW cannot be converted into the HIGH status in the next transmission process. The next two lines of Equation (2) represent the transmission constraint in which a data packet owns a HIGH status. By analyzing the Equation (2), we work out the following results. \[ \begin{array}{c}\text{Select}\,\theta_{high}=\theta_{jump}+\frac{\underset{1} {max\{delay_{i,n\}\}}}{T_{i}},\theta_{low}=\frac{\theta_{high}}{\omega}+\frac{ \mu}{T_{i}}\,,\,\text{and}\,\omega=\frac{\underset{1}{max}\{delay_{i,i+1}\}}{ T_{i}}.\end{array} \]Hence, we can calculate \(\theta_{low}\) and \(\theta_{high}\) directly using local information, without need to understand the global network topology. ## 5 Feasibility Proof and Performance Analysis of DMRF

In this section, we present the feasibility proof of DMRF and analyze its performance in terms of message complexity and energy consumption complexity. ### Feasibility Analysis of DMRF Protocol

Suppose a path contains \(m\) nodes, the delay between node \(i\) and node \(i+1\) is denoted as \(delay_{i_{L}\,i+1}\). The remaining transmission time of each packet when generated from the source is denoted as \(L_{source}\). **Theorem 1:** The DMRF protocol can meet the requirement \(\mathop{\hbox{\raise 1.29 pt\hbox{$\delta$}}}\limits_{i-1}^{n-1}\)\(delay_{i_{L}+1}\)\(\xi\)\(L_{source}\). _Proof._ We prove **Theorem 1** in two cases: the best case and the worst case. **The Best case:** In the best case, every hop-by-hop transmission can satisfy \(\lambda\geq\theta_{low}\), because the remaining transmission time of the source is larger than the estimated transmission time from the source to the destination. Then we can get \(L_{source}\geq T_{source}\). Before the last transmission, it can guarantee that the remaining time \(L_{m-1}\) of node \(m-1\) is larger than the estimated transmission time \(T_{m-1}\) of node \(m\) - 1, which is no less than the delay between node \(m-1\) and node \(m\). Therefore we can get \(L_{m-1}\!>T_{m-1}\!\geq\!delay_{m-1,\,m}\), implying that the transmission can meet the real-time requirement. **The Worst case:** In the worst case, suppose the remaining transmission time of the data packet is 0 when it is sent to node \(k(k<m)\), then \(\mathop{\hbox{\raise 1.29 pt\hbox{$\delta$}}}\limits_{1}^{k-1}\)\(delay_{i_{L}+1}\)\({}^{3}\)\(L_{1}\). Each transmission satisfies \(\theta_{high}>\lambda>\theta_{jump}\). Consequently \(L_{i}\)\({}^{3}\)\(T_{i}\). Since \(T_{1}=\mathop{\hbox{\raise 1.29 pt\hbox{$\delta$}}}\limits_{1}^{n-1}\)\(delay_{i_{L}+1}\), we can get \(\mathop{\hbox{\raise 1.29 pt\hbox{$\delta$}}}\limits_{1}^{k-1}\)\(delay_{i_{L}+1}>\mathop{\hbox{\raise 1.29 pt\hbox{$\delta$}}}\limits_{1}^{m-1}\)\(delay_{i_{L}+1}\), which means \(k>m\). This is in contradiction with \(k<m\), thereby the transmission still can meet the real-time performance in the worst case.