## 3 Mathematical Modeling Approach of the WSN and Solution  
The mathematical modeling of a WSN, utilizing the concept of the continuous-time Markov process, has been performed by considering different units' failure and repair. Once a failure or repair takes place, the system moves from one state to another state. The different state and their interconnections are shown in Figure 3, which is the transition state diagram of the WSN. Considering the defined notations and the state transition in the WSN within a given time frame, the following set of differential equations, also known as Chapman-Kolmogorov equations, is derived. The probability that the WSN is in the state \(P_{0}(t)\) (Figure 3) in the interval \((t,t+\Delta t)\) is given as  
\[P_{0}(t+\Delta t)=(1-\lambda_{m}\Delta t)(1-\lambda_{su}\Delta t )(1-\lambda_{ts}\Delta t)(1-\lambda_{p}\Delta t)P_{0}(t)+\mu_{m}P_{3}(t)\Delta t\] \[+\mu_{p}P_{5}(t)\Delta t+\sum_{i,j}\int_{0}^{\infty}\mu_{i}P_{j}(x,t)\ \Delta tdx\]  
\begin{table}
\begin{tabular}{c c} \hline  
\[P_{0}(t+\Delta t)=(1-\lambda_{m}\Delta t)(1-\lambda_{su}\Delta t)(1- \lambda_{ts}\Delta t)(1-\lambda_{p}\Delta t)P_{0}(t)+\mu_{m}P_{3}(t)\Delta t\] \[+\mu_{p}P_{5}(t)\Delta t+\sum_{i,j}\int_{0}^{\infty}\mu_{i}P_{j}(x, t)\ \Delta tdx\] \[\frac{P_{0}(t+\Delta t)-P_{0}(t)}{\Delta t}+(\lambda_{m}+\lambda_ {su}+\lambda_{ts}+\lambda_{p})P_{0}(t)=\mu_{m}P_{3}(t)\] \[+\mu_{p}P_{5}(t)+\sum_{i,j}\int_{0}^{\infty}\mu_{i}P_{j}(x,t)dx\] \[\lim_{\Delta t\to 0}\frac{P_{0}(t+\Delta t)-P_{0}(t)}{\Delta t}+( \lambda_{m}+\lambda_{su}+\lambda_{ts}+\lambda_{p})P_{0}(t)=\mu_{m}P_{3}(t)\] \[+\mu_{p}P_{5}(t)+\sum_{i,j}\int_{0}^{\infty}\mu_{i}P_{j}(x,t)dx\] \[\left(\frac{d}{dt}+\lambda_{m}+\lambda_{su}+\lambda_{ts}+\lambda_ {p}\right)P_{0}(t)=\mu_{m}P_{3}(t)+\mu_{p}P_{5}(t)+\sum_{i,j}\int_{0}^{\infty }\mu_{i}P_{j}(x,t)\ dx \tag{4}\]  
where \(i=su\), \(ts\), \(m\), \(psp\), \(tsp\), \(sup\), \(pm\), \(psm\), \(psm\), \(psm\), \(psu\), \(msu\), \(ms\), \(j=1\), \(2\), \(4\), \(6\), \(7\), \(8\), \(10\), \(11\), \(12\), \(13\), \(14\), \(15\). Similarly, the Chapman-Kolmogorov equations for other states are obtained as follows. \[\left(\frac{d}{dt}+\mu_{m}+\lambda_{m}+\lambda_{su}+\lambda_{ts}+\lambda_{p} \right)P_{3}(t)=\lambda_{m}P_{0}(t)+\mu_{p}P_{9}(t) \tag{5}\]  
\[+\mu_{p}P_{5}(t)\Delta t+\sum_{i,j}\int_{0}^{\infty}\mu_{i}P_{j}(x,t)\ \Delta tdx\]0  
\[+\mu_{p}P_{5}(t)\Delta t+\sum_{i,j}\int_{0}^{\infty}\mu_{i}P_{j}(x,t)\ \Delta tdx\]1  
\[+\mu_{p}P_{5}(t)\Delta t+\sum_{i,j}\int_{0}^{\infty}\mu_{i}P_{j}(x,t)\ \Delta tdx\]2  
where \(i=su\), \(ts\), \(m\), \(psp\), \(tsp\), \(sup\), \(pm\), \(psm\), \(psm\), \(psu\), \(ms\), \(j=1\), \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(9\), \(10\), \(11\), \(12\), \(13\), \(14\), \(15\). \[P_{i}(t)=\left\{ \begin{array}{ll}1,&\text{if $i=0,t=0$}\\ 0,&\text{otherwise}\end{array} \right. \tag{9}\]  
Figure 3: State transition diagram of the WSN. \[P_{k}(0,t)=\lambda_{su}P_{l}(t) \tag{10}\]  
where \(k=1\), \(8\), \(12\), \(14\); \(l=0\), \(5\), \(9\), \(3\). \[P_{m}(0,t)=\lambda_{ts}P_{n}\ (t) \tag{11}\]  
where \(m=2\), \(7\), \(13\), \(15\); \(n=0\), \(5\), \(9\), \(3\). \[P_{q}(0,t)=\lambda_{m}P_{r}(t) \tag{12}\]  
where \(q=4\), \(10\); \(r=3\), \(9\). \[P_{u}(0,t)=\lambda_{m}P_{v}(t) \tag{13}\]  
where \(u=4\), \(10\); \(v=3\), \(9\). Taking the Laplace transformations of Equations (4)-(13) and solving the obtained equations, the authors find out the various state probabilities of the WSN, as given below (Equations (14)-(17)).