## 2 Materials and Methods
### Elements of the Research of a TCP Window Size Dynamic Properties in Continuous Time  
Let us examine the stochastic process \(\left\{\overset{\rightarrow}{F}(t)\right\}_{t>0}\) in continuous time. For this process, a Markov chain \(\left\{\overset{\rightarrow}{F}_{k}\right\}_{k>0}\) is needed; accordingly, the process \(\left\{\overset{\rightarrow}{F}(t)\right\}_{t>0}\) is semi-Markov. Let \(\left\{t_{i}\right\}\) be a sequence of moments during which the value of the TCP Window Size parameter changed. For all \(T_{sls}\) values, we introduce the notation \(\gamma_{S_{w}}^{\delta}=\Phi\left[t_{i+1}-_{i}t_{i}\right]\overset{\rightarrow} {F}_{i}=\left(T_{sls},S_{w},\alpha\right)\).  
If \(T_{rt}\) has a finite, non-zero mathematical expectation, then the stochastic process \(\left\{\overset{\rightarrow}{F}(t)\right\}_{t>0}\) is ergodic and its distribution converges to the stationary one:  
\[P_{\overset{\rightarrow}{F}}=\lim_{t\rightarrow\infty}P_{\overset{ \rightarrow}{F}}(t)=\gamma_{S_{w}}^{\delta}q_{T_{sls},S_{w}}^{\delta}/\sum_{S _{w}=1}^{S_{\max}}\gamma_{S_{w}}^{\delta}\sum_{j=2}^{L}q_{j,S_{w}}^{\delta}. \tag{16}\]  
The statement enclosed in Equation (16) is derived as per an ergodic theorem [33] for semi-Markov processes. It is important to mention that \(t_{i+1}-t_{i}>0\), \(t_{i+1}-t_{i}\geq S_{w}/T_{cc}\); \(t_{i+1}-t_{i}=T_{rt}\) if \(T_{rt}>S_{w}/T_{cc}\). Accordingly, the stochastic process \(\left\{\overset{\rightarrow}{F}(t)\right\}_{t>0}\) is regular, so \(0<\gamma_{S_{w}}^{\delta}<\infty\) if \(\Phi[T_{rt}]\) is finite. To establish the stationary distribution (16), it is enough to calculate \(\gamma_{S_{w}}^{\delta}\).  
If the process of information interaction between the SNEIoTD environment and a hub is controlled through the AI/MD mechanism, the value of the TCP Window Size parameter changes when the number of sent data blocks matches a value of \(S_{w}\) at the current time of sending data to a hub. The described procedure for sending data at a fixed TCP Window Size takes \(S_{w}t_{0}\). If at the end of this time interval, the reverse communication signal from a hub is not received, the SNEIoTD waits for it, subsequently updating the value \(S_{w}\). The waiting interval is equal to \(T_{rt}\); therefore,  
\[\gamma_{S_{w}}^{1}=\int\limits_{S_{w}t_{0}}^{\infty}xdT_{rt}(x)+S_{w}t_{0}T_{rt }(S_{w}t_{0}), \tag{17}\]  
where \(t_{0}=1/T_{cc}\).  
If the process of information interaction between the SNEIoTD environment and a hub is controlled through the SIS mechanism, then the value of the TCP Window Size parameter is doubled in the period \(T_{rt}\). Accordingly, the value of the parameter \(S_{w}\) manages to change several times during one implementation \(T_{rt}\). A direct description of this variant of the studied process of information interaction within the framework of this research is not rational. It is more productive to use the verified approximate scheme of the structure \(T_{rt}\) described in [34]: \(\gamma_{2S_{w}}^{0}=\int\limits_{2S_{w}t_{0}}^{\infty}(x-2(S_{w}-1)t_{0})dT_{ rt}(x)+t_{0}T_{rt}(2S_{w}t_{0})\). In all other cases, \(\gamma_{i}^{0}=t_{0}\), except for \(\gamma_{1}^{0}\), the value of which is regulated by the postponement procedure. As we noted above, in the absence of a signal \(s^{+}\), the delay procedure involves doubling the waiting period \(k\) times in a row. Taking this into account, we write:  
\[\gamma_{1}^{0}=\left(p_{1}^{+}\sum_{i=1}^{k-1}i\big{(}1-p_{1}^{+}\big{)}^{i-1}+ k\big{(}1-p_{1}^{+}\big{)}^{k-1}\right)\Phi[T_{rt}] \tag{18}\]To calculate the characteristics in (16)-(18) based on the \(P_{\vec{F}}(t)\) distribution, we suggest using the method described in [35].