## 4 DMRF Protocol
### Network Congestion Detection Method
Therefore, **Theorem 1** is proved to be correct. ### Message Complexity Analysis

**Theorem 2**: The message complexity of DMRF is \(O(N)\) in the whole network, where \(N\) is the number of nodes in the network. _Proof._ During the network initialization, each node monitors the notice information of its **FCS** to obtain the information of **FCS**. The monitoring process requires each node to send message once. The complexity is \(O(N)\) for this kind of control information. In the feedback procedure, when the states of all nodes in **FCS** are FAULTY or JFAULTY, or congestion and empty area exist, the current node will send feedback message to its upper stream node. In the jumping transmission mode, if the transmission failed, the current node will also send feedback message to its upper stream to inform them to adjust the jumping probabilities. In the worst case, in the feedback procedure within a path with the maximum \(S\) hops (\(S<N\)), each node sends four kinds of feedback messages. Therefore, the control complexity is \(O(S)\). It can be seen that the total message complexity of DMRF is \(O(N)+O(S)=O(N)\). Therefore, the **Theorem 2** is proved to be correct. ### Energy Consumption Complexity Analysis

**Theorem 3**: \(O\left(\sqrt{N}\right)\) is the energy consumption complexity of the DMRF Protocol. Proof. : Suppose there are \(N\) nodes deployed in the network, the average data packet transmission rate of a node is \(v\), the maximum remaining transmission time of data packet is \(t\), and the average length of one hop is \(h\), consequently a node needs to transmit \(t/(vh)\) hops. The energy consumption of each hop transmission is \(e\). The density of nodes is \(d\). Accordingly the average energy consumption complexity of the transmission in a path for SPEED, SPEED-S, SPEED-T, and FTSPEED is

\[O\big{(}\frac{t}{vh}\cdot\ e\big{)}=O\big{(}\sqrt{\frac{4N}{pd}}\big{)}=O\big{(} \sqrt{N}\big{)}\]

As DMRF establishes constant paths in the initial phase, the energy consumption is also \(O(1)\cdot\ O(\sqrt{N})=O(\sqrt{N})\). MMSPEED establishes a tree-like path structure in the initial transmission process. With the distance between the subsequent node and destination node decreasing, the transmission will converge into one path in the latter transmission process. Therefore, its energy consumption complexity is \(\frac{1}{2}O\big{(}2^{\frac{1}{2}\frac{NN}{md}}\big{)}+\frac{1}{2}O\big{(} \sqrt{N}\big{)}=O\big{(}2^{\sqrt{N}}\big{)}\), which is higher than DMRF, SPEED, and FTSPEED. ### Time Complexity of Faulty Nodes, Congested Nodes and VOID Region Detection

**Theorem 4**: \(O(N)\) is the time complexity of detecting faulty nodes, congested nodes and VOID region. Proof. : Suppose \(N\) nodes are deployed in the network uniformly. In the initialization phase, each node will establish its **FCS** and initialize the status of the nodes in **FCS**. Since all nodes are uniformly distributed, the average size of each **FCS** can be considered as a constant \(f_{cs}\). Therefore the complexity of detecting faulty nodes, congestion nodes and VOID region in the initial phase is \(O(f_{cs}\times N)=O(N)\). When faulty node, congested nodes or VOID region occur in the network, the upper stream nodes will receive the feedback message, which means the detecting complexity is the identical with the complexity of the control messages. Therefore, the time complexity of detecting faulty nodes, congested nodes and VOID region is \(O(N)\). ## 6 Performance Evaluation

In this section, we present the results of several simulations to evaluate the performance of the DMRF protocol. We use the JProwler simulation platform [30], which is an event-driven WSN-based simulation platform. Consider a typical wireless sensor network in a real-time transmission scenario. Nodes conform to uniform or random distribution. During the experiments, the source node periodically sends data to the sink node. A certain amount of failure nodes are deployed.