# Evaluating the Functioning Quality of Data Transmission Networks in the Context of Cyberattacks
## 3 Problem Statement  
Let us suppose that there is a data transmission network. In this data transmission network, data transmission routes are organized for information exchange in accordance with the required quality of user service (Figure 2). Let us suppose that the designated route for data delivery contains (\(n-1\)) sections and \(n\) nodes (including terminal network elements), each of which can be exposed to cyberattacks. In a general way, the data packet transmission time \(t_{b}\) is a random value that is characterized by the distribution function \(B(t)\) and is specified by the volume \(V\), the transmission speed \(S\) and the number of sections on this route \(n\), i.e., \(t_{b}~{}=~{}\frac{(n-1)V}{S}\). Let us assume that a malefactor realizes a destructive information impact (cyberattack) on the data transmission network; in this case, the functioning of network elements is disrupted in a random time \(t_{ri}\) with the distribution function \(R_{i}(t)\), where \(i\) is a number of the network element, \(i~{}=~{}\frac{1}{1,n}\). The availability of network elements for the implementation of a cyberattack by a malefactor is characterized by the probability \(P_{i}\) that is determined using the known model of the cybercriminal intelligence system [8; 24; 25]. If the malefactor's actions are successful, then in a random time \(t_{di}\) with the distribution function \(\Delta_{i}(t)\), \(i~{}=~{}\frac{1}{1,n}\), the network  
Figure 2: The route plan of the data packets’ transmission in a network, functioning in the context of cyberattacks. administrator detects the impact and realizes measures to restore the functioning of the damaged network element because of the cyberattack, and the data packet received for transmission is transmitted again. Data packets arrive at the input of the transmission route with intensity \(\lambda_{input}\). The value of this intensity is determined in accordance with a specified network traffic model. In this case, a data packet received for transmission waits for operation during some time \(t_{w}\) with distribution function \(W(t)\), which depends on the time of the successful transmission of the previously received data packet, the storage capacity \(K\) at the input of the network element and the characteristics of the incoming traffic. Assumptions are as follows: operation is carried out according to the rule \"first come, first served\" and the coefficient of operational reliability of the network equipment is equal to one. We need to determine the distribution function of the delivery time of data packets to the final user. Let us present the process of data packet flow transmission, described in a task statement, in the form of a stochastic network (Figure 3). It is denoted in Figure 3 that:  
\[\beta(s)\ =\ \int_{0}^{\infty}e^{-st}d[B(t)] \tag{1}\]  
* Laplace-Stieltjes transformation of the distribution function of packet transmission time \(B(t)\) without taking into account the malefactor's information impact with a distribution parameter equal to \(b\ =\ 1/\overline{t}_{b}\); \[\delta_{i}(s)\ =\ \int_{0}^{\infty}e^{-st}d[\Delta_{i}(t)],\ i\ =\ \overline{1,n}\] (2)
* Laplace-Stieltjes transformation of the distribution function of restoration time of the \(i\)-th network element functioning after a malefactor's cyberattack with a distribution parameter equal to \(d_{i}\ =\ 1/\overline{t}_{di}\); \[r_{\ i}(s)\ =\ \int_{0}^{\infty}e^{-st}d[R_{i}(t)];\ i\ =\ \overline{1,n}\] (3)
* Laplace-Stieltjes transformation of the distribution function of the successful realization of the cyberattack against the \(i\)-th network element; \[\omega(s)\ =\ \int_{0}^{\infty}e^{-st}d[W(t)]\] (4)  
Figure 3: Stochastic network of data packet delivery process in the context of cyberattacks. Laplace-Stieltjes transformation of the distribution function of service waiting time by packet.