# A Hybrid Scheme for Disaster-Monitoring Applications in Wireless Sensor Networks
## 5 Theoretical Analysis  
The details of the HT-MAC scheme in the steady-state stage are shown in Figure 7. HT-MAC is based on the duty cycle. After clustering, all nodes in one cluster begin to listen. A polling token is created and owned by the cluster head node. Subsequently, the token owner node asks the first superior node whether there are data to transmit. If the superior node has data to transmit, it simultaneously transmits the data directly to the destination node. The token owner node then asks the second superior node to transmit the data. This polling among the superior nodes is performed until the last superior node, superior node \(M\). Subsequently, an ordinary node with a polling token determines whether there are data to transmit. If there are data, they are transmitted to the destination node directly. The node with the polling token then transfers the polling token to the next ordinary node in the virtual ring, and the algorithm for handling abnormalities when a node is unavailable is performed. The other nodes receive polling tokens and determine whether they are successor nodes. If there is a successor node, it obtains the polling token and checks if the token is corrupted. If the token is corrupted, the successor node will discard this token and deal with the abnormality using Algorithm 1; otherwise, the token reply will be sent. Then, the successor node performs abnormity handling of multiple tokens and enters the sleeping state. After a superior node receives a token reply from an ordinary node and there are alert data in this superior node, the superior node transmits the alert data at once under the low-power listening and shortened preamble approach. Otherwise, the superior node enters a sleeping state. During the sleeping state, if an alert appears in one superior node, the alert awakens the superior node and is transmitted at once, using the low-power listening and shortened preamble approach. The next period starts at the end of the sleeping state. Therefore, the polling token is transferred in a virtual token ring network. In the following sections, to facilitate theoretical analysis, we assume that there are two superior nodes within one cluster. Figure 7: Flow chart of the HT-MAC scheme at the steady-state stage. We assume that the ring network includes \(N+2\) nodes, and that there are two superior nodes and \(N\) ordinary nodes. Every node follows the first come, first served queue principle, and assumes that the queue capacity is unlimited. All the nodes provide a gated service; in other words, they send the arrived frames only when it is their turn to transmit. In ordinary node \(i\) (\(i=1\), \(2\),..., \(N\)), queue \(i\) (\(i=1\), \(2\),..., \(N\)) represents the queue of node \(i\). Node \(i\) has a Poisson process input, with an arrival rate of \(\lambda_{i}\). The service time for the frames transmitted by an ordinary node is independent. The distribution function of the service time in an ordinary node \(i\) is \(H_{i}(x)\). \(P1\) and \(P2\) represent the queues of superior Nodes \(1\) and \(2\), respectively. The input of the superior nodes is a Poisson process, and in superior Nodes \(1\) and \(2\), the arrival rates of the Poisson process are \(\lambda_{P1}\) and \(\lambda_{P2}\), respectively. The service time for the frames transmitted by a superior node is independent. The probability distribution of the frame transmission time is a general distribution in superior Nodes \(1\) and \(2\), and the distribution functions are \(H_{P1}(x)\) and \(H_{P2}(x)\), respectively. In the period from \(t_{n}\) to \(t_{n+1}\), as shown in Figure 2, the data frame transmission time of ordinary node \(i\) is \(\tau_{i}\), the sleep time is \(s_{i}\), and the polling token walking time is \(\mu_{i}\). Within time \(t\), the number of frames arriving in an ordinary node queue \(i\) is \(v_{i}(t)\).