# A Hybrid Scheme for Disaster-Monitoring Applications in Wireless Sensor Networks
## 5 Theoretical Analysis
When the token ring system reaches a stable state, the instant at which ordinary node \(i\) begins to transmit frames, the probability of ordinary node \(k\) with \(j_{k}\) frames waiting is \(g_{i}(j_{1}\), \(j_{2}\),..., \(j_{P1}\), \(j_{P2}\)). In ordinary nodes, assume that the instant when an ordinary node begins to transmit the data is..., \(t_{n}\), \(t_{n+1}\),..., so that \(\ldots<t_{n}<t_{n+1}<\ldots\). According to the random process, random variables may be defined as follows: \(\varepsilon_{n}(i)\) is the frame number in ordinary node \(I\) when the very moment is \(t_{n}\); \(\varepsilon_{n}\) is the ordinary node identifier when ordinary node \(i\) starts transmitting packets in the period from \(t_{n}\) to \(t_{n+1}\); in ordinary node \(i\), \(h_{i}\) is the mean frame transmission time; and \(h_{P1}\) and \(h_{P2}\) are the mean frame transmission times of superior nodes \(P1\) and \(P2\), respectively. Subsequently, the state of the system at \(t_{n}\) is described by (\(\varepsilon_{n}\), \(\varepsilon_{n}(1)\), \(\varepsilon_{n}(2)\),..., \(\varepsilon_{n}(N)\), \(\varepsilon_{n}(P1)\), \(\varepsilon_{n}(P2)\)); meanwhile, the state space of the system becomes \(I=\{(i,k_{1},k_{2},\ldots,k_{j},\ldots,k_{N},k_{P1},k_{P2})\): \(i=1\), \(2\),..., \(N\); \(k_{j}=0\), \(1\), \(2\),..., \(j=1\), \(2\),..., \(N\), \(P1\), \(P2\)). Hence, the transition probabilities with respect to the state (\(\varepsilon_{n}\), \(\varepsilon_{n}(1)\), \(\varepsilon_{n}(2)\),..., \(\varepsilon_{n}(N)\), \(\varepsilon_{n}(P1)\), \(\varepsilon_{n}(P2)\)) construct an irreducible, aperiodic Markov chain. The Markov chain is ergodic, and the system can be assumed to be in statistical equilibrium. In this case, the limiting probabilities of the states (\(\varepsilon_{n}\), \(\varepsilon_{n}(1)\), \(\varepsilon_{n}(2)\),..., \(\varepsilon_{n}(N)\), \(\varepsilon_{n}(P1)\), \(\varepsilon_{n}(P2)\)) are obtained as follows:  
\[\lim_{n\to\infty}\Pr\{\varepsilon_{n}=i,\varepsilon_{n}(k)=j_{k};k=1,2, \ldots,N,P1,P2\}=g_{i}(j_{1},j_{2},\ldots,j_{N},j_{P1},j_{P2}). \tag{5}\]  
Equation (5) indicates that the number of frames in a node remains steady after a long time. For the existence of an equilibrium state, the necessary and sufficient conditions are as follows:  
\[\sum_{i=1}^{N}\rho_{i}+\rho_{P1}+\rho_{P2}<1\rho_{i}=\lambda_{i}h_{i},\rho_{ P1}=\lambda_{P1}h_{P1},\rho_{P2}=\lambda_{P2}h_{P2}, \tag{6}\]  
where \(\rho_{i}\) is the traffic intensity for ordinary node \(i\), and \(\rho_{P1}\) and \(\rho_{P2}\) are the traffic intensities for superior nodes \(P1\) and \(P2\), respectively. The above equation shows that the system exists in an equilibrium state when the total traffic intensity of the system is less than \(1\). \(F_{P1}(x)\) and \(F_{P2}(x)\) are the frame transmission times for superior nodes \(P1\) and \(P2\) at any instant \(t_{i}\) with \(x\) frames, respectively. Between \(t_{n}\) and \(t_{n+1}\), in superior nodes \(P1\) and \(P2\), the frame transmission times are \(F_{P1}[\varepsilon_{n}(P1)]\) and \(F_{P2}[\varepsilon_{n}(P2)]\), respectively. Consequently, the superior and ordinary nodes have the following number of frames at instant \(t_{n+1}\):  
\[ \begin{cases}&\varepsilon_{n+1}(P1)=v_{P1}(\tau_{i}+\mu_{i}+s_{i})+v_{P1}\{F_ {P1}[\varepsilon_{n}(P1)]\}+v_{P1}\{F_{P2}[\varepsilon_{n}(P2)]\}\\ &\varepsilon_{n+1}(P2)=v_{P2}(\tau_{i}+\mu_{i}+s_{i})+v_{P2}\{F_{P2}[ \varepsilon_{n}(P2)]\}\\ &\varepsilon_{n+1}(G_{i})=v_{G_{i}}(\tau_{i}+\mu_{i}+s_{i})\\ &\varepsilon_{n+1}(G_{j})=\varepsilon_{n}(G_{j})+v_{G_{j}}(\tau_{i}+\mu_{i}+s_ {i})+v_{G_{j}}\{F_{P1}[\varepsilon_{n}(P1)]\}+v_{G_{j}}\{F_{P2}[\varepsilon_ {n}(P2)]\}\end{cases} .