## 6 Teletraffic Analysis of the WSN
This occurs with rate of \(N_{On}^{(1)}\times\lambda_{Off}^{(1)}\times(1-P_{Off})\), 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 2. * To state \((N_{On}^{(1)},N_{On}^{(2)}-1,N_{Off}^{(1)}+1,N_{Off}^{(2)})\) when a node is active in phase 2 and turns off in phase 1 of the _Off_ mode. This occurs with rate of \(N_{On}^{(2)}\times\lambda_{Off}^{(2)}\times P_{Off}\), this is the reason why there is one node less in state _On_ with phase 2 and one more node in the _Off_ mode with phase 1. * To state \((N_{On}^{(1)},N_{On}^{(2)}-1,N_{Off}^{(1)},N_{Off}^{(2)}+1)\) when a node is active in phase 2 and turns off in phase 2 of the _Off_ mode. This occurs with rate of \(N_{On}^{(2)}\times\lambda_{Off}^{(2)}\times(1-P_{Off})\), this is the reason why there is one node less in state _On_ with phase 2 and one more node in the _Off_ mode with phase 2. * To state \((N_{On}^{(1)}+1,N_{On}^{(2)},N_{Off}^{(1)}-1,N_{Off}^{(2)})\) when a node is turned off in phase 1 and turns on in phase 1 of the _On_ mode. This occurs with rate of \(N_{Off}^{(1)}\times\lambda_{On}^{(1)}\times P_{On}\), this is the reason why there is one node less in state _Off_ with phase 1 and one more node in the _On_ mode with phase 1. * To state \((N_{On}^{(1)}+1,N_{On}^{(2)},N_{Off}^{(1)},N_{Off}^{(2)}-1)\) when a node is turned off in phase 2 and turns on in phase 1 of the _On_ mode. This occurs with rate of \(N_{Off}^{(2)}\times\lambda_{On}^{(2)}\times P_{On}\), this is the reason why there is one node less in state _Off_ with phase 2 and one more node in the _On_ mode with phase 1. * To state \((N_{On}^{(1)},N_{On}^{(2)}+1,N_{Off}^{(1)}-1,N_{Off}^{(2)})\) when a node is turned off in phase 1 and turns on in phase 2 of the _On_ mode. This occurs with rate of \(N_{Off}^{(1)}\times\lambda_{On}^{(1)}\times(1-P_{On})\), this is the reason why there is one node less in state _Off_ with phase 1 and one more node in the _On_ mode with phase 2. * To state \((N_{On}^{(1)},N_{On}^{(2)}+1,N_{Off}^{(1)},N_{Off}^{(2)}-1)\) when a node is turned off in phase 2 and turns on in phase 2 of the _On_ mode. This occurs with rate of \(N_{Off}^{(2)}\times\lambda_{On}^{(2)}\times(1-P_{On})\), this is the reason why there is one node less in state _Off_ with phase 2 and one more node in the _On_ mode with phase 2. The Markov chain is numerically solved to find the steady-state probabilities, \(\mathcal{T}_{N_{On}^{(1)},N_{On}^{(2)},N_{Off}^{(1)},N_{Off}^{(2)}}\). The total nodes in the system are distributed over four possible existing states as they follow a Hyper-exponential distribution in the Singing and Silence times. To calculate the number of possible states of this Markov Chain, consider for instance the case when \(N=2\) \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Next State** & **Transition Rate** & **A Node in State** & **Turns** & **To a State of** \\ \hline \((N_{On}^{(1)}-1,N_{On}^{(2)},N_{Off}^{(1)}+1,N_{Off}^{(2)})\) & \(N_{On}^{(1)}\lambda_{Off}^{(1)}P_{Off}\) & \(On^{(1)}\) & _Off_ & _Off_(1) \\ \((N_{On}^{(1)}-1,N_{On}^{(2)},N_{Off}^{(1)},N_{Off}^{(2)}+1)\) & \(N_{On}^{(1)}\lambda_{Off}^{(1)}(1-P_{Off})\) & \(On^{(1)}\) & _Off_ & _Off_(2) \\ \((N_{On}^{(1)},N_{On}^{(2)}-1,N_{Off}^{(1)}+1,N_{Off}^{(2)})\) & \(N_{On}^{(2)}\lambda_{Off}^{(2)}P_{Off}\) & \(On^{(2)}\) & _Off_ & _Off_(1) \\ \((N_{On}^{(1)},N_{On}^{(2)}-1,N_{Off}^{(1)},N_{Off}^{(2)}+1)\) & \(N_{On}^{(2)}\lambda_{Off}^{(2)}(1-P_{Off})\) & \(On^{(2)}\) & _Off_ & _Off_(2) \\ \((N_{On}^{(1)}+1,N_{On}^{(2)},N_{Off}^{(1)}-1,N_{Off}^{(2)})\) & \(N_{Off}^{(1)}\lambda_{On}^{(1)}P_{On}\) & \(Off^{(1)}\) & _On_ & _On_(1) \\ \((N_{On}^{(1)},N_{On}^{(2)}+1,N_{Off}^{(1)}-1,N_{Off}^{(2)})\) & \(N_{Off}^{(1)}\lambda_{On}^{(1)}(1-P_{On})\) & \(Off^{(1)}\) & _On_ & _On_(1) \\ \((N_{On}^{(1)}+1,N_{On}^{(2)},N_{Off}^{(1)},N_{Off}^{(2)}-1)\) & \(N_{Off}^{(2)}\lambda_{On}^{(2)}(1-P_{On})\) & \(Off^{(2)}\) & _On_ & _On_(1) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Description of the possible transitions on the chain proposed.