# Evaluating the Functioning Quality of Data Transmission Networks in the Context of Cyberattacks
## 4 Method for Evaluating the Impact of Cyberattacks
### The Second Stage  
Let us determine the function \(W(t)\) by means of the results [20; 21; 22], which make it possible to calculate the values of the average time and dispersion of the service waiting time for arbitrary distribution laws of arrival and service, but with the difference that the values included in them are determined by means of data obtained at the first stage.  
Therefore, the average service waiting time for a packet flow is determined by the formula [20; 21]:  
\[T_{w}\ =\ \frac{T_{h}C_{\wp}\rho\big{[}1+(K+1)\rho+K\rho^{K+1}\big{]}}{(1-\rho)(1- \rho^{K+2}\ )} \tag{16}\]  
and dispersion [21; 22]:  
\[D_{w}\ =\ \frac{T_{h}{}^{2}C_{\wp}^{2}\rho\big{[}2\big{(}1-\rho^{K}\big{)}+K \rho^{K}(1-\rho)[\rho(K+1)-K-3]\big{]}}{(1-\rho)(1-\rho^{K+2}\ )}-T_{w}^{2}, \tag{17}\]  
where \(\rho\ =\ \lambda T_{h}<1\); \(C_{\wp}^{2}\ =\ \frac{C_{\wp}^{2}+C_{h}^{2}}{2}\)--root mean square coefficient of distribution variation of the integrated time between packet arrival and service.  
Depending on the type of network traffic, the value \(C_{w}^{2}\) is equal to:  
* Pareto distribution [27]: \[C_{wp}^{2}\ =\ \frac{1}{(2H-1)(2H-3)};\] (18)
* Weibull distribution [27]: \[C_{w\wp}^{\ 2}\ =\ \frac{\Gamma\Big{(}1+\frac{2}{a_{\wp}}\Big{)}}{\Gamma \Big{(}1+\frac{2}{a_{\wp}}\Big{)}^{2}}-1,\] (19) where \(a_{\wp}>0\)--shape parameter of the Weibull distribution; \(\Gamma\)(*)--gamma-function [27]; \(H\)--Hurst exponent; \(\alpha\)--distribution shape parameter, which is numerically equal to:
* for Weibull distribution [14; 16]: \[\alpha_{v}\ =\ 2-2H;\] (20)
* for Pareto distribution [16]: \[\alpha_{p}\ =\ H^{\frac{-5}{4}}.\] (21)  
On the basis of the results [22] of the application of the diffusive approximation method [29; 30], which in the context of a heavy load achieves the solution for the queueing systems with arbitrary distribution laws of the arrival and service of claims, based on the solution for an M/M/1/K type system and \(K<\infty\), as well as the results [30], we assume that the waiting time for service is described by the gamma distribution [27] with parameters:  
\[\theta\ =\ \frac{T_{w}}{D_{w}};\,\eta\ =\ \frac{T_{w}^{2}}{D_{w}} \tag{22}\]  
The utilized assumption is in good agreement with the data described in fundamental works [28; 29; 32].  
In this case,  
\[\omega(s)\ =\ \frac{\theta^{\eta}}{\Gamma(\eta)}\int_{0}^{\infty}t^{\eta-1}exp\{-(s +\theta)t\}dt\ =\ \bigg{(}\frac{\theta}{\theta+s}\bigg{)}^{\eta}. \tag{23}\]  
Then, the equivalent function of the stochastic network (Figure 4) is of the form:  
\[Q(s)\ =\ \omega(s)*h(s)\ =\ \bigg{(}\frac{\mu}{\mu+s}\bigg{)}^{\alpha}\bigg{(} \frac{\theta}{\theta+s}\bigg{)}^{\eta}\ =\ \bigg{(}\frac{\beta_{n}}{\beta_{n}+s}\bigg{)}^{\alpha_{n}}, \tag{24}\]  
where \(\alpha_{n}\ =\ \frac{(T_{h}+T_{w})^{2}}{D_{w}+D_{h}}\); \(\beta_{n}\ =\ \frac{T_{h}+T_{w}}{D_{w}+D_{h}}\).  
Hence, the distribution function of the delivery time of data packets in the context of cyberattacks can be defined as:  
\[F(t)\ =\ \frac{(\beta_{n})^{\alpha_{n}}}{\Gamma(\alpha_{n})}\int_{0}^{t}x^{ \alpha_{n}-1}exp\{-t\beta_{n}\}\ dx. \tag{25}\]  
Thus, the formulated task has been solved, and the solution obtained in general terms allows one to define:  
1. Average delivery time of data packets: \[T_{f}\ =\ T_{h}+T_{w}.\] (26)
2. Jitter of packet delay time: \[\sigma\ =\ \sqrt{D_{h}+D_{w}}.\] (27)
3. Loss probability of packets [16; 21]: \[P_{Loss}\ =\ \frac{1-\rho}{1-\rho^{\frac{K}{C_{s}}+2}}\rho^{\frac{K}{C_{s}}+1}.\] (28)  
The novelty of the obtained solution lies in the parameter definition for the quality of service of the network in the context of cyberattacks in the absence of restrictions on the type of distribution functions of the transmission time \(B(t)\), the time of the successful realization of cyberattacks \(R_{i}(t)\), the recovery time of the network element function \(\Delta_{i}(t)\) and the type of network traffic.