## 2 Materials and Methods
### Statement of the Research
In this case, starting from the logic of interactions of SIS and AI/MD mechanisms described in the previous section, we can write: \(q_{T_{sls}S_{w}}^{0}=0\)\(\forall S_{w}\geq T_{sls},q_{T_{sls}S_{w}}^{1}=0\)\(\forall S_{w}<T_{sls}\). We accept that \(L=\left\lfloor S_{w}^{\max}/2\right\rfloor\). Taking into account the introduced notations, the stationary distribution of the stochastic parameter \(S_{w}\) for SIS and AI/MD mechanisms will be described by the expressions \(q_{i}^{0}=\sum\limits_{n=i+1}^{L}q_{n,i}^{0}\), \(q_{i}^{1}=\sum\limits_{n=2}^{\min(L)}q_{n,i}^{1}\), respectively. For all \(2<n\leq\left\lfloor L/2\right\rfloor\), the probability \(q_{\overset{\rightarrow}{F}}^{+}\) satisfies the following equations:  
\[q_{n,n}^{1}=p_{2n}^{+-}\sum\limits_{i=2}^{2n}q_{i,2n}^{1}+p_{2n+1}^{+-}\sum \limits_{i=2}^{2n+1}q_{i,2n+1}^{1}+p_{1}^{+}q_{n,n-1}^{0}+p_{1}^{+-}\left(\sum \limits_{i=2n+1}^{L}q_{i,2n}^{0}+\sum\limits_{i=2n+2}^{L}q_{i,2n+1}^{0}\right), \tag{1}\]  
\[q_{n,1}^{0}=p_{2n}^{o}\sum\limits_{i=2}^{2n}q_{i,2n}^{1}+p_{2n+1}^{o}\sum \limits_{i=2}^{2n+1}q_{i,2n+1}^{1}+p_{1}^{o}\left(\sum\limits_{i=2n+1}^{L}q_{ i,2n}^{0}+\sum\limits_{i=2n+2}^{L}q_{i,2n+1}^{0}\right). \tag{2}\]  
Equations (1) and (2) match with the first quarter of the index space and characterize the system's transitions between the SIS and AI/MD mechanisms. At the same time, Equation (1) characterizes the transitions for the AI/MD mechanism after the arrival of the signal \(s^{-}\) and the transitions from the SIS mechanism to the AI/MD mechanism, which  
Figure 2: Graphic interpretation of the interaction of SIS and AI/MD mechanisms in the context of the research object. can occur if, as a result of the operation of the SIS mechanism, \(S_{w}\) becomes equal to \(T_{sls}\), or if the SIS mechanism detects a pass in the data sequence. Equation (2) is interpreted so that a transition to a state \(q_{n,1}^{0}\) can take place from any other state of a Markov chain if the SNEIoTD receives a signal \(s^{o}\). The equations for \(q_{n,n}^{1}\), \(q_{n,1}^{0}\) for indices \(\lfloor L/2\rfloor<n\leq L\) are obtained as a result of the reduction of Equations (1) and (2), with the exclusion of the aforementioned irreversible states:  
\[q_{n,n}^{1}=p_{1}^{+}q_{k,k-1}^{0}+p_{2n}^{+-}\sum\limits_{i=2}^{2n}q_{i,2n}^{ 1}+p_{2n+1}^{+-}\sum\limits_{i=2}^{2n+1}q_{i,2n+1}^{1}, \tag{3}\]  
\[q_{n,1}^{0}=p_{2n}^{0}\sum\limits_{i=2}^{2n}q_{i,2n}^{1}+p_{2n+1}^{o}\sum \limits_{i=2}^{2n+1}q_{i,2n+1}^{1}. \tag{4}\]  
Finally, let us characterize the traffic control on the SNEIoTD side under the conditions of the arrival of signals \(s^{+}\), which leads to an increase in the value of the parameter \(S_{w}\). For all, \(2<n\leq S_{w}^{\max}\) we write:  
\[q_{n,i}^{1}=p_{i-1}^{+}q_{n,i-1}^{1},\ n<i; \tag{5}\]  
\[q_{n,S_{w}^{\max}}^{1}=p_{S_{w}^{\max}-1}^{+}q_{n,S_{w}^{\max}-1}^{1}+p_{S_{w} ^{\min}}^{+}q_{n,S_{w}^{\max}}^{1}. \tag{6}\]  
After the value of the parameter \(S_{w}\) reaches \(S_{w}^{\max}\), its growth stops, even if new signals \(s^{+}\) arrive. For the SIS mechanism, this means that  
\[q_{n,i}^{0}=p_{1}^{+}q_{n,i-1}^{0},\ n<L,\ n>i. \tag{7}\]  
Based on Equation (7), we write:  
\[q_{n,i}^{0}=\left(p_{1}^{+}\right)^{i-1}q_{n,1}^{0}. \tag{8}\]  
Let us select implementations of Equation (5) for which the condition \(L<i<S_{w}^{\max}\) is fulfilled and sum them for \(\forall n=\overline{2,L}\):  
\[q_{n,n}^{1}=p_{2n}^{+-}\sum\limits_{i=2}^{2n}q_{i,2n}^{1}+p_{2n+1}^{+-}\sum \limits_{i=2}^{2n+1}q_{i,2n+1}^{1}+p_{1}^{+}q_{n,n-1}^{0}+p_{1}^{+-}\left(\sum \limits_{i=2n+1}^{L}q_{i,2n}^{0}+\sum\limits_{i=2n+2}^{L}q_{i,2n+1}^{0}\right), \tag{1}\]0  
Based on Equation (9), we write: \(q_{i}^{1}=H_{i}q_{L}^{1}\ \forall i>L\), where \(H_{i}=\prod\limits_{k=L}^{i-1}p_{k}^{+}\). Note that \(q_{i}^{1}=\sum\limits_{n=2}^{\min(i,L)}q_{n,i}^{1}\). If \(i>L\), then \(q_{i,i}^{1}=0\); otherwise, \(q_{i,i}^{1}>0\). Accordingly, the system of Equations (1)-(7) can be reduced to the form of a system of equations concerning the variables \(q_{n,1}^{0}\) and \(q_{n,n}^{1}\).