## 2 Materials and Methods
### Statement of the Research
The control of the flow of incoming requests is regulated by the above-described logic of application of the SIS and AI/MD mechanisms built into the TCP. Let us assume that at the time of receiving the signal \(s\) from the population of incoming service requests, \(n>0\) blocks of data are transmitted, where the value \(n\) depends on the parameter \(S_{w}\) and several data blocks being sent to a hub without confirming the fact of their receipt. Accepted incoming requests are served in order of arrival (FIFO). We denote with \(p_{n}^{+}\) the probability of obtaining a series of signals \(s^{+}\) of length \(n\). We denote with \(p_{n}^{+-}\) the probability of obtaining an arbitrary series of signals \(s^{+},s^{-}\) of length \(n\). We denote with \(p_{n}^{o}\) the probability of obtaining a signal \(s^{o}\) for a series of data blocks of length \(n\) sent to a hub at \(S_{w}=n\) (if the studied process is controlled via the AI/MD mechanism) or \(S_{w}\geq n\) (if the studied process is controlled via the SIS mechanism). All further initiatives regarding the formalization and research of the relevant queuing system are based on the following initial postulates:  
* the probabilities \(p=\{p_{n}^{+},p_{n}^{+-},p_{n}^{o}\}\) are determined exclusively by an existing value of the parameter \(S_{w}\) and do not depend on the previous values of the parameters \(S_{w}\), \(T_{sls}\);
* for the probabilities \(p\), the equality \(p_{n}^{+}+p_{n}^{+-}+p_{n}^{o}=1\) is fulfilled. At the same time, when \(n=1\), the values of probabilities \(p\) represent the functioning of the SIS mechanism, and when \(n>1\) the values of probabilities \(p\) represent the functioning of the AI/MD mechanism;
* SNEIoTD always has relevant data for sending to a hub;
* the initiating value \(S_{w}^{\max}\) is at the same time a deterministic upper limit of the value of the parameter \(S_{w}\) for any moment of implementation of the studied process;
* parameter \(T_{rt}\) is a stochastic value with distribution function \(T_{rt}(x)\). A visual interpretation of the above-described logic of interaction between the SIS and AI/MD mechanisms is presented in Figure 2.  
Research of the Distribution Function of the Stochastic Parameter TCP Window Size for the Studied Process  
In Section 2.1, we noted that the controlled parameter \(S_{w}\) is significant for the target connection between the SNEIoTD environment and a hub. Let us investigate the stochastic nature of this parameter, taking into account that the studied process is controlled via SIS & AI/MD mechanisms, the interaction between which is characterized by the controlled parameter \(T_{sls}\). Let us introduce the vector \(\overset{\rightarrow}{F}=(S_{w},T_{sls},\alpha)\), where \(\alpha\) is a binary parameter whose value \(\alpha=0\) corresponds to the SIS mechanism, and \(\alpha=1\) corresponds to the AI/MD mechanism. Next, we will investigate the stochastic process \(\left\{\overset{\rightarrow}{F}(t)\right\}_{t>0}\). Let the value of the parameter \(S_{w}\) be measured at discrete moments represented by the sequence \(\left\{t_{i}\right\}_{i>0}\). Subject to the postulates defined in Section 2.1, the sequence \(\left\{\overset{\rightarrow}{F}_{k}=\overset{\rightarrow}{F}(t_{k})\right\}_ {k>0}\) ordered by the value of the argument is the aperiodic, irreducible Markov chain with a bounded set of states and with a stationary distribution \(q_{\overset{\rightarrow}{F}}\). Considering that \(T_{sls}:=\left[S_{w}/2\right]\) after each fact of receiving a signal \(s^{-}\) or \(s^{o}\), then (under the condition \(\exists n:p_{n}^{+-}>0\) or \(p_{n}^{o}>0\)) all \(\overset{\rightarrow}{F}\) of the inequalities \(T_{sls}>S_{w}^{\max}\), \(q_{\overset{\rightarrow}{F}}=0\) are true, because all such states will be irreversible. Let us introduce the notation \(q_{\overset{\rightarrow}{F}}^{\rightarrow}=q_{T_{sls}S_{w}}^{\pm}\).