## 6 Teletraffic Analysis of the WSN
Then, the possible states in the system are {(0,0,0,2), (0,0,2,0), (0,2,0,0), (2,0,0,0), (0,0,1,1), (0,1,0,1), (1,0,0,1), (0,1,1,0), (1,0,1,0), (1,1,0,0)} which corresponds to all valid configurations. It is important to mention that configurations such as (2,-1,1,0) are not possible as they are considering negative nodes. And for the general case, the valid configurations follow Equation (25):  
\[|valid|=4+6(N-1)+2(N-1)(N-2)+\frac{1}{6}(N-1)(N-2)(N-3),N\in\mathbb{Z}^{+} \tag{25}\]  
where 4 belongs to the base case of all the nodes are in the same state with the same phase, \(6(N-1)\) when there are only 2 states and phases that hold all the nodes in the system, \(2(N-1)(N-2)\) when all the nodes are distributed in 3 of the states and phases, and \(\frac{1}{6}(N-1)(N-2)(N-3)\) when all the nodes are distributed in the all the four possible states and phases. And finally, for completeness reasons, since no bird song has either _On_ or _Off_ modes determined by a _CoV_\(<1\), we also develop the Erlang (Hypo-exponential) model considering that there may be bird songs that follow this Erlang distribution for the _On_/_Off_ periods that can be used in future works. This model also corresponds to an irreducible, Markov Chain with valid states \(\{\Omega_{N_{On}^{(1)}N_{On}^{(2)}\ldots N_{On}^{(k)}N_{Off}^{(k)},N_{Off}^{( 2)}\ldots N_{Off}^{(k)}}:0\leq N_{On}^{(1)}+N_{On}^{(2)}+\ldots+N_{On}^{(k)} \leq N,0\leq N_{Off}^{(1)}+N_{Off}^{(2)}+\ldots+N_{Off}^{(k)}\leq(N-(N_{On}^{ (1)}+N_{On}^{(2)}+\ldots+N_{On}^{(k)}))\}\) where \(N\) is the total number of nodes in the system, \(N_{On}=N_{On}^{(1)}+\ldots+N_{On}^{(k)}\) (\(N_{Off}=N_{Off}^{(1)}+\ldots+N_{Off}^{(k)}\)), and is the number of active (inactive) nodes, i.e., nodes in the _On_ (_Off_) mode. Note that \(N=N_{On}^{(1)}+\ldots+N_{On}^{(k)}+N_{Off}^{(1)}+\ldots+N_{Off}^{(k)}\). Note that the Erlang model corresponds to the sum of \(k_{On}\) (\(k_{Off}\)) exponential random variables with parameter \(\lambda_{On}\) (\(\lambda_{Off}\)). As such, in the _On_ (_Off_) state, nodes have to transit \(k_{On}\) (\(k_{Off}\)) exponentially distributed random times before going to the _Off_ (_On_) state. Hence, valid transitions from state \(((N_{On}^{(1)},\ldots,N_{On}^{(k)},N_{Off}^{(1)},\ldots,N_{Off}^{k}))\) are as follows (See Figure 9, and Table 4):  
* To state \((N_{On}^{(1)}-1,N_{On}^{(2)}+1,\ldots,N_{On}^{(k)},N_{Off}^{(1)}\ldots,N_{Off }^{(k)})\) when a node is active in phase 1 and becomes active in phase 2 of the _On_ mode. This occurs with rate of \(N_{On}^{(1)}\times\lambda_{On}\), this is the reason why there is one node less in state _On_ with phase 1 and one more node in the _On_ mode with phase 2. * To state \((N_{On}^{(1)},\ldots,N_{On}^{(k-1)}-1,N_{On}^{(k)}+1,N_{Off}^{(1)},\ldots,N_{ Off}^{(k)})\) when a node is active in phase \(k-1\) and becomes active in phase \(k\) of the _On_ mode. This occurs with rate of \(N_{On}^{(k-1)}\times\lambda_{On}\), this is the reason why there is one node less in state _On_ with phase \(k-1\) and one more node in the _On_ mode with phase \(k\). Figure 8: Markov chain of the transition process of a node which depends on the birdsongs behavior, Hyper-exponential case. * To state \((N_{On}^{(1)},\ldots,N_{On}^{(k)}-1,N_{Off}^{(1)}+1,\ldots,N_{Off}^{(k)})\) when a node is active in phase \(k\) and becomes inactive in phase \(1\) of the \(\mathit{Off}\) mode. This occurs with rate of \(N_{On}^{(k)}\times\lambda_{Off}\), this is the reason why there is one node less in state \(\mathit{On}\) with phase \(k\) and one more node in the \(\mathit{Off}\) mode with phase \(1\). * To state \((N_{On}^{(1)},\ldots,N_{On}^{(k)},N_{Off}^{(1)}-1,N_{Off}^{(2)}+1,\ldots,N_{ Off}^{(k)})\) when a node is inactive in phase \(1\) and becomes inactive in phase \(2\) of the \(\mathit{Off}\) mode. This occurs with rate of \(N_{Off}^{(1)}\times\lambda_{Off}\), this is the reason why there is one node less in state \(\mathit{Off}\) with phase \(1\) and one more node in the \(\mathit{Off}\) mode with phase \(2\).