# A Hybrid Scheme for Disaster-Monitoring Applications in Wireless Sensor Networks
## 5 Theoretical Analysis
\tag{7}\]From the assumption of Poisson arrivals of node data, the following relation holds if \(\tau_{i}\), \(\mu_{i}\) and \(s_{i}\) do not overlap:  
\[v(\tau_{i}+\mu_{i}+s_{i})=v(\tau_{i})+v(\mu_{i})+v(s_{i}). \tag{8}\]  
Therefore, the generating function of the transition probability Pr(\(i+1\), \(\varepsilon_{n+1}\)(1), \(\varepsilon_{n+1}\)(2),..., \(\varepsilon_{n+1}\)(N), \(\varepsilon_{n+1}\)(P1), \(\varepsilon_{n+1}\)(P2)) is as follows:  
\[E[\prod\limits_{k=1}^{N}x_{G_{k}}^{\varepsilon_{n+1}(G_{k})}\cdot x_{P1}^{ \varepsilon_{n+1}(P1)}\cdot x_{P2}^{\varepsilon_{n+1}(P2)}]=U_{i}^{*}(A)\cdot S _{i}^{*}(A)\cdot E[\prod\limits_{k\
eq i}x_{G_{k}}^{\varepsilon_{n}(G_{k})} \cdot[H_{i}^{*}(A)]^{\varepsilon_{n}(i)}\cdot[H_{P1}^{*}(B)]^{\varepsilon_{n} (P1)}\cdot[H_{P2}^{*}(C)]^{\varepsilon_{n}(P2)}], \tag{9}\]  
where \(A\), \(B\), and \(C\) are given by the following:  
\[\left\{ \begin{array}{l}A=\sum\limits_{k=1}^{N}\lambda_{G_{k}}(1-x_{G_{k}}) +\lambda_{P1}(1-x_{P1})+\lambda_{P2}(1-x_{P2})\\ B=\sum\limits_{k\
eq i}\lambda_{G_{k}}(1-x_{G_{k}})+\lambda_{P1}(1-x_{P1})\\ C=\sum\limits_{k\
eq i}\lambda_{G_{k}}(1-x_{G_{k}})+\lambda_{P1}(1-x_{P1})+ \lambda_{P2}(1-x_{P2})\end{array} \right.. \tag{10}\]  
Letting \(n\)\(\rightarrow\)\(\infty\) on both sides of Equation (9), we obtain the generating function of the transition probability Pr(\(i+1\), \(\varepsilon_{n+1}\)(1), \(\varepsilon_{n+1}\)(2),..., \(\varepsilon_{n+1}\)(N), \(\varepsilon_{n+1}\)(P1), \(\varepsilon_{n+1}\)(P2)) is \(G_{i}\)(\(x_{1}\), \(x_{2}\)..., \(x_{N}\), \(x_{P1}\), \(x_{P2}\)) as follows:  
\[G_{i+1}(x_{1},x_{2},\ldots,x_{N},x_{P1},x_{P2})=U_{i}^{*}(A)\cdot S_{i}^{*}(A) \cdot G_{i}(x_{1},x_{2},\ldots,H_{i}^{*}(A),\ldots,x_{N},H_{P1}^{*}(B),H_{P2}^ {*}(C)). \tag{11}\]  
In the above equation, when the polling token moves forward during the period from \(t_{n}\) to \(t_{n+1}\), \(U_{i}^{*}\)(\(s\))(\(i\) = 1,2,..., \(N\)) is the Laplace-Stieltjes transform of the walking time probability distribution; \(S_{i}^{*}\)(\(s\)) is the Laplace-Stieltjes transform of the sleeping time probability distribution when the polling token moves forward from \(t_{n}\) to \(t_{n+1}\). In ordinary node \(i\) with a Poisson input, \(H_{i}\)(\(x\)) represents the busy period probability distribution, and \(H_{i}^{*}\)(\(s\)) is the Laplace-Stieltjes transform of \(H_{i}\)(\(x\)). In superior nodes \(P1\) and \(P2\) with a Poisson input, \(H_{P1}\)(\(x\)) and \(H_{P2}\)(\(x\)) represent the busy period probability distributions for \(P1\) and \(P2\), respectively, and \(H_{P1}^{*}\)(\(s\)) and \(H_{P2}^{*}\)(\(s\)) are the Laplace-Stieltjes transforms of \(H_{P1}\)(\(x\)) and \(H_{P2}\)(\(x\)), respectively. Using the above recursive formula, \(G_{i}\)(\(x_{1}\), \(x_{2}\)..., \(x_{N}\), \(x_{P1}\), \(x_{P2}\)) can be described as a functional equation; however, an explicit representation cannot be derived. We denote \(g_{i}\)(\(j\)) as the mean node queue length in queue \(j\) (\(j\) = 1, 2,..., \(N\), P1, \(P2\)), when ordinary node \(i\) begins to transmit data, where \(i\) = 1, 2,..., \(N\):  
\[g_{i}(j)=\lim\limits_{x_{1},x_{2},\ldots,x_{N},x_{P1},x_{P2}\to 1}\frac{ \left\{\partial G_{i}(x_{1},x_{2},\ldots,x_{N},x_{P1},x_{P2})/\partial x_{j} \right\}}{G_{i}(1,1,\ldots,1,1,1)}. \tag{12}\]  
From Equation (11) for \(i\) = 1, 2,..., \(N\), we set:  
\[G_{i}(1,1,\ldots,1,1,1)=K\ (\text{constant})\ i=1,2,\ldots,N. \tag{13}\]  
In Equation (11), the differential operation of \(x_{j}\) and \(x_{l}\)\(\rightarrow\)1 (\(l\) = 1, 2,..., \(N\), \(P1\), \(P2\)) yields  
\[\left\{