## 2 Materials and Methods
### Statement of the Research
Based on the value of the probabilities \(q_{i}^{0}\) and \(q_{i}^{1}\), we can analytically express the dynamics of the value of the parameter \(S_{w}\) for AI/MD and SIS mechanisms, respectively. We characterize the dynamics of the value of the TCP Window Size parameter for the case when the information interaction between a SNEIoTD and a hub is controlled via the AI/MD mechanism through probabilities \(q_{i}^{1}\). For \(i<L\), we write: \(q_{n,i}^{1}=p_{i-1}^{+}q_{n,i-1}^{1}\), \(n=\overline{2,i-1}\), \(i=\overline{n+1,L}\). Let us sum the obtained equations in terms of \(n\):  
\[q_{n,n}^{1}=p_{2n}^{+-}\sum\limits_{i=2}^{2n}q_{i,2n}^{1}+p_{2n+1}^{+-}\sum \limits_{i=2}^{2n+1}q_{i,2n+1}^{1}+p_{1}^{+}q_{n,n-1}^{0}+p_{1}^{+-}\left(\sum \limits_{i=2n+1}^{L}q_{i,2n}^{0}+\sum\limits_{i=2n+2}^{L}q_{i,2n+1}^{0}\right), \tag{1}\]1  
Based on the previously introduced notation, Equation (10), we can write \(q_{i}^{1}-q_{i,i}^{1}=p_{i-1}^{+}q_{i-1}^{q}\) and, in turn:  
\[q_{n,n}^{1}=p_{2n}^{+-}\sum\limits_{i=2}^{2n}q_{i,2n}^{1}+p_{2n+1}^{+-}\sum \limits_{i=2}^{2n+1}q_{i,2n+1}^{1}+p_{1}^{+}q_{n,n-1}^{0}+p_{1}^{+-}\left(\sum \limits_{i=2n+1}^{L}q_{i,2n}^{0}+\sum\limits_{i=2n+2}^{L}q_{i,2n+1}^{0}\right), \tag{1}\]2\[q_{n,n}^{1}=p_{2n}^{+-}\sum\limits_{i=2}^{2n}q_{i,2n}^{1}+p_{2n+1}^{+-}\sum \limits_{i=2}^{2n+1}q_{i,2n+1}^{1}+p_{1}^{+}q_{n,n-1}^{0}+p_{1}^{+-}\left(\sum \limits_{i=2n+1}^{L}q_{i,2n}^{0}+\sum\limits_{i=2n+2}^{L}q_{i,2n+1}^{0}\right), \tag{1}\]3  
Now, let us characterize the dynamics of the value of the TCP Window Size parameter for the case in which the information interaction between the SNEIoT and a hub is controlled via the SIS mechanism through probabilities \(q_{i}^{0}\). This mechanism functions as long as the condition \(S_{w}<T_{sls}\) is fulfilled. In terms of \(q_{i}^{0}\), this means that  
\[q_{n,n}^{1}=p_{2n}^{+-}\sum\limits_{i=2}^{2n}q_{i,2n}^{1}+p_{2n+1}^{+-}\sum \limits_{i=2}^{2n+1}q_{i,2n+1}^{1}+p_{1}^{+}q_{n,n-1}^{0}+p_{1}^{+-}\left(\sum \limits_{i=2n+1}^{L}q_{i,2n}^{0}+\sum\limits_{i=2n+2}^{L}q_{i,2n+1}^{0}\right), \tag{1}\]4  
Let us reduce the value of the iterator in Equation (13) by one: \(q_{i-1}^{0}=\sum\limits_{k=i}^{L}q_{k,i-1}^{0}=\left(p_{1}^{+}\right)^{i-2} \sum\limits_{k=i}^{L}q_{k,1}^{0}\), from which \(q_{i-1}^{0}=\left(p_{1}^{+}\right)^{i-2}\left(\left(p_{1}^{+}\right)^{1-i}q_{ i}^{0}+q_{i,1}^{0}\right)\). Let us present the last equation in its completed form:  
\[q_{n,n}^{1}=p_{2n}^{+-}\sum\limits_{i=2}^{2n}q_{i,2n}^{1}+p_{2n+1}^{+-}\sum \limits_{i=2}^{2n+1}q_{i,2n+1}^{1}+p_{1}^{+}q_{n,n-1}^{0}+p_{1}^{+-}\left(\sum \limits_{i=2n+1}^{L}q_{i,2n}^{0}+\sum\limits_{i=2n+2}^{L}q_{i,2n+1}^{0}\right), \tag{1}\]5  
Therefore, Equations (11), (12) and (14) characterize the dynamics of the value of the TCP Window Size parameter for AI/MD and SIS mechanisms (in terms of probabilities \(q_{i}^{0}\) and \(q_{i}^{1}\), respectively). Let us summarize the process of solving the system of Equations (1)-(4) in the context of the obtained recurrent expressions (8) and (9). As a result of the substitution, we find:  
\[q_{n,n}^{1}=p_{2n}^{+-}\sum\limits_{i=2}^{2n}q_{i,2n}^{1}+p_{2n+1}^{+-}\sum \limits_{i=2}^{2n+1}q_{i,2n+1}^{1}+p_{1}^{+}q_{n,n-1}^{0}+p_{1}^{+-}\left(\sum \limits_{i=2n+1}^{L}q_{i,2n}^{0}+\sum\limits_{i=2n+2}^{L}q_{i,2n+1}^{0}\right), \tag{1}\]6  
where \(\forall\lfloor L/2\rfloor\leq n\leq L\): \(C_{k,1}^{0}=p_{2n}^{0}H_{2n}+p_{2n+1}^{0}H_{2n+1}\), \(C_{n,n}^{1}=p_{2n}^{+-}H_{2n}+p_{2n+1}^{+-}H_{2n+1}+\left(p_{1}^{+}\right)^{n- 1}\)C\({}_{n,1}^{0}\); and \(\forall 2<n<\lfloor L/2\rfloor\): \(C_{n,1}^{0}=p_{1}^{0}(B_{2n}+B_{2n+1})+p_{2n}^{0}A_{2n}+p_{2n+1}^{0}A_{2n+1}\), \(C_{n,n}^{1}=p_{2n}^{+-}A_{2n}+p_{2n+1}^{+-}A_{2n+1}+\left(p_{1}^{+}\right)^{n- 1}\)B\({}_{n,1}+p_{1}^{+-}(B_{2n}+B_{2n+1})\), where \(A_{i}=q_{i}^{1}/q_{L}^{1}\), \(i<L\); \(A_{n,j}=q_{n,i}^{1}/q_{L}\), \(B_{n,i}=q_{n,i}^{0}/q_{L}\), \(B_{i}=q_{i}^{0}/q_{L}^{1}\).