# Evaluating the Functioning Quality of Data Transmission Networks in the Context of Cyberattacks
## 4 Method for Evaluating the Impact of Cyberattacks
### The First Stage  
Using the Mason equation for closed graphs [4; 5; 6], we shall define the equivalent function \(h(s)\) of the stochastic network fragment (Figure 3) which corresponds to the service process:  
\[h(s)\ =\ \frac{k_{f}\prod_{i=1}^{n}[1-r_{i}\ (s+b)]\left(\overline{d}+\overline {c}+s\right)\beta(s)}{\left(1-\sum_{i=1}^{n}r_{i}(s+b)\delta_{i}(s)\beta(s) \prod_{i=1}^{n-1}[1-r_{i}(s+b)]\right)\left(\overline{d}+s\right)}, \tag{5}\]  
where \(b\) is a mathematical expectation of the data packet transmission time;  
\(\overline{d}\ =\ \frac{1}{n}\ \sum\limits_{i=1}^{n}\frac{p_{i}}{t_{i}}\)--mathematical expectation of the intensity of restoring the functioning of network elements after the successful realization of the cyberattack;  
\(\overline{c}\ =\ \frac{1}{n}\sum\limits_{i=1}^{n}\frac{p_{i}}{t_{i}}\)--mathematical expectation of the intensity of the information impact against network elements, each of which is accessible for the cybercriminal intelligence system, and realization of cyberattack with probability \(P_{i}\)[3; 24];  
\(k_{f}\ =\ \frac{\overline{d}}{\overline{d}+\overline{c}}\)--coefficient that has the meaning of the probability of the successful operation of the network equipment when network traffic is absent until the time of the successful realization of the cyberattack, assuming that during the restoration of the function, the network element is not used for its intended purpose [24]. It is known [4; 5; 26] that knowledge of the equivalent function (5) allows one to determine the initial moments of the random delivery time by the means of numerical methods,  
\[M_{k}\ =\ (-1)^{k}\frac{d^{k}}{ds^{k}}\left\{\frac{h(s)}{h(0)}\right\}_{s\ =\ 0}, \tag{6}\]  
followed by using Pearson nomograms [27] to determine the desired distribution function. However, many years of experience in the use of nomograms by the authors [27] for modeling complex organizational and technical systems shows that in the absolute majority of cases, the solution converges to a type III Pearson distribution, i.e., approximation of the real distribution function by an incomplete gamma function, which is consistent  
Figure 4: Enlarged stochastic network of the packet delivery process. with [28; 29] and is confirmed by system studies [8; 30]. In this case, the parameters of the shape and scale of the gamma distribution are determined as  
\[\alpha\ =\ \frac{M_{1}^{2}}{M2-M_{1}^{2}};\ \ \mu\ =\ \frac{M_{1}}{M2-M_{1}^{2}}, \tag{7}\]  
and the distribution function of the service time (successful transmission in the context of cyberattacks) of the data packet is of the form:  
\[H_{\gamma}(t)\ =\ \frac{\mu^{\alpha}}{\Gamma(\alpha)}\int\limits_{0}^{t}x^{ \alpha-1}exp\{-\mu x\}dx, \tag{8}\]  
this form corresponds to Laplace-Stieltjes image  
\[h_{\gamma}(s)\ =\ \int\limits_{0}^{\infty}exp\{-st\}d[H_{\gamma}(t)]\ =\ \left(\frac{\mu}{\mu+s}\right)^{\alpha}\approx h(s). \tag{9}\]  
It should be noted that the mathematical expectation and service time dispersion are equal:  
\[T_{h}\ =\ M_{1};\ \ D_{h}\ =\ M_{2}-M_{1}^{2}. \tag{10}\]  
In this particular case, when we use exponential distributions as initial distributions, the image (5) can be submitted in a fractional rational form \(h(s)\ =\ f(s)\varphi(s)\), where \(f(s)\) and \(\varphi(s)\) are polynomials, respectively, of degrees \(m\) and \(n\) (\(m<n\)).