## 4 Performance Analysis and Evaluation
### Analysis of Signaling Costs  
The state transition diagram in Figure 7 shows that there are no transient sets in the model, but only a single ergodic set with only one cyclic class. Hence, the properties of regular Markov chains can be exploited to analyze the behavior of the proposed model [22].  
Let P be the regular transition probability matrix; then the steady state probability vector \(\pi\) can be solved using the following equations:  
\[\pi P=\pi\;\text{and}\;\sum_{i-1}^{m}\pi_{i}=1\]  
Here, m is the number of states and P is the fundamental matrix for the regular Markov chain. This is given by:  
\[Z=[Z_{{}_{\psi}}]=(I-P+A)^{{}^{-1}}\]  
where  
1. \(A\) is a limiting matrix determined by \(P\), and the powers of \(P^{n}\) approach the probability matrix \(A\);
2. Each row of \(A\) consists of the same probability vector \(\pi=\{\pi_{1},\pi_{2},.....\pi_{n}\}\), _i.e._, \(A=\xi\pi\), where \(\xi\) is the column vector with all entries equal to 1; and
3. \(I\) is the identity matrix.  
The matrix \(Z\) can be used to study the behavior of the regular Markov chain, and, using this matrix, one can compute the mean number of times that the process is in a particular state.  
Let \(Y_{j}(k)\) is the number of times that a process is in state \(S_{j}\) in the first \(k\) steps; then \(M_{i}[y(k)]\), the mean number of times the process is in state \(S_{j}\) starting from state \(S_{i}\), is given by:  
\[M_{i}[y_{{}_{j}}^{{}^{(k)}}]\rightarrow(Z_{{}_{\psi}}-\pi_{i})+k\pi_{j}\]  
The total number of boundary updates in \(k\) steps, starting from state \(S_{i}\), can be computed from the total number of times that the process is in the boundary states starting from state \(S_{j}\), the initial state. The average number of location updates (\(U_{bu}\)) in the analytical model, is given by:  
\[U_{bu}=M_{i}[y_{{}_{1}}^{{}^{(1)}}]+M_{i}[y_{{}_{2}}^{{}^{(k)}}]+M_{i}[y_{{}_ {3}}^{{}^{(k)}}]+M_{i}[y_{{}_{4}}^{{}^{(k)}}]\]  
So: \(U_{bu}=\sum_{s=1}^{4}M_{i}[y_{{}_{n}}^{{}^{(k)}}]\)  
where 1, 2, 3 and 4 are the boundary states in the model:  
Generalizing:  
\[U_{bu}=\sum_{s=1}^{u}M_{i}[y_{{}_{n}}^{{}^{(k)}}]\]We can use the above equation to determine the number of binding update messages. Because we need to send a binding update message whenever the sensor node moves between IP-WSNs, each time a node enters a boundary state, a binding update message is generated. Therefore, we need to determine the expected number of times that the process enters into a boundary state within k steps.  
Thus, sensor nodes need to send \(U_{bu}\) binding update messages, given that the sensor node experiences a total of k transitions between SMAGs. Therefore, the ratio of intra-IP-WSN mobility is denoted as \(M_{\text{int}\,n_{\text{int}\,n_{\text{int}\,n_{\text{int}\,n_{\text{NN}}}}}}\) and is expressed by:  
\[M_{\text{int}\,n_{\text{int}\,n_{\text{int}\,n_{\text{int}\,n_{\text{NN}}}}}}= \left(k-U_{\text{\emph{bus}}}\right)/\,k\]  
The ratio of inter-IP-WSN mobility is denoted as \(M_{\text{int}\,n_{\text{int}\,n_{\text{int}\,n_{\text{int}\,n_{\text{NN}}}}}}\) and is expressed by:  
\[M_{\text{int}\,n_{\text{int}\,n_{\text{int}\,n_{\text{NN}}}}}=U_{\text{\emph{bus }}}\,/k\]