## 6 Teletraffic Analysis of the WSN  
In this section, we develop a mathematical analysis based _On_ Continuous Time Markov Chains (CTMC) that captures the main dynamics of the system, such as the nodes that become active (inactive). according to the statistical characteristics of the particular birds. As we described in the previous section, the CoV of all the selected bird's sounds are higher than 1. Hence, we use a Hyper-Exponential distribution to approximate to the _On/Off_ periods of the nodes in the WSN. However, we first develop the mathematical analysis considering the exponential case since the Hyper-Exponential case is an extension of this exponential model. From this, the Markov Chain corresponds to an irreductible chain with valid states \(\{\Omega_{N_{On},N_{Off}}:0\leq N_{On}\leq N,0\leq N_{Off}\leq(N-N_{On})\}\) where \(N\) is the total number of nodes in the system, \(N_{On}\) (\(N_{Off}\)) is the number of active (inactive) nodes, i.e., nodes in the _On_ (_Off_) mode. Note that \(N=N_{On}+N_{Off}\). From state \((N_{On},N_{Off})\) the valid transitions are as follows (See Figure 7, and Table 2):  
* To state \((N_{On}+1,N_{Off}-1)\) when an inactive node becomes active and \(N_{Off}>0\). This occurs with rate \(N_{Off}\times\lambda_{On}\). As such, the mean time of the node in the _Off_ state is considered to be \(1/\lambda_{On}\). * To state \((N_{On}-1,N_{Off}+1)\) when an active node becomes inactive and \(N_{On}>0\). This occurs with rate \(N_{On}\times\lambda_{Off}\). As such, the mean time of the node in the _On_ state is considered to be \(1/\lambda_{Off}\). This Markov chain is numerically solved to find the probability that there are \(N_{On}\) and \(N_{Off}\) nodes in the WSN, i.e., the steady-state probabilities, \(\pi_{N_{On},N_{Off}}\). For the Hyper-exponential case, when the \(CoV>1\), recall that this distribution is formed by two exponential distributions, with parameters \(\lambda_{On}\) and \(\lambda_{Off}\), with probabilities \(P_{On}\) and \(P_{Off}\) respectively. Then, a node in phase 1 (2) of the _On_ (_Off_) state can transit to phase 1 (2) of the _Off_ (_On_) state and vice versa. Building on this, the Markov Chain also corresponds to an irreducible chain with valid states \(\{\Omega_{N_{On}^{(1)},N_{On}^{(2)},N_{Off}^{(1)},N_{Off}^{(2)}}:0\leq N_{On}^ {(1)}+N_{On}^{(2)}\leq N,0\leq N_{Off}^{(1)}+N_{Off}^{(2)}\leq(N-(N_{On}^{(1)} +N_{On}^{(2)}))\}\) where \(N\) is the total number of nodes in the system, \(N_{On}=N_{On}^{(1)}+N_{On}^{(2)}\) (\(N_{Off}=N_{Off}^{(1)}+N_{Off}^{(2)}\)), and is the number of active (inactive) nodes, i.e., nodes in the _On_ (_Off_) mode. Note that \(N=N_{On}^{(1)}+N_{On}^{(2)}+N_{Off}^{(1)}+N_{Off}^{(2)}\). From state \((N_{On}^{(1)},N_{On}^{(2)},N_{Off}^{(1)},N_{Off}^{(2)})\) the valid transitions are as follows (See Figure 8, and Table 3):  
* To state \((N_{On}^{(1)}-1,N_{On}^{(2)},N_{Off}^{(1)}+1,N_{Off}^{(2)})\) when a node is active in phase 1 and turns off in phase 1 of the _Off_ mode. This occurs with rate of \(N_{On}^{(1)}\times\lambda_{Off}^{(1)}\times P_{Off}\), this is the reason why there is one node less in state _On_ with phase 1 and one more node in the _Off_ mode with phase 1. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multicolumn{5}{c}{**Description of Node State Transitions**} \\ \hline
**Next State** & **Transition Rate** & **A Node in State** & **Turns** & **Turns to a State of** \\ \hline \((N_{On}+1,N_{Off}-1)\) & \(N_{Off}\times\lambda_{On}\) & \(Off\) & \(On\) & \(On\) \\ \((N_{On}-1,N_{Off}+1)\) & \(N_{On}\times\lambda_{Off}\) & \(On\) & \(Off\) & \(Off\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Description of the possible transitions in the proposed network model. Figure 7: Markov chain for the case when the _On/Off_ periods are exponentially distributed. * To state \((N_{On}^{(1)}-1,N_{On}^{(2)},N_{Off}^{(1)},N_{Off}^{(2)}+1)\) when a node is active in phase 1 and turns off in phase 2 of the _Off_ mode.