# An Overview on Denial-of-Service Attacks in Control Systems: Attack Models and Security Analyses
## 4 Probabilistic Approaches to Models of DoS Attacks and Non-Malicious Transmission Failures
The following result from [25] shows that DoS attack indicator processes \(\{l_{\mathrm{A}}(i)\in\{0,1\}\}_{i\in\mathbb{N}_{0}}\) satisfying Equation (21) belong to the class \(\Lambda_{\rho}\) for \(\rho\in(\rho_{\mathrm{A}},1]\). **Proposition 5** ([25]). : _Suppose \(\{l_{\mathrm{A}}(i)\in\{0,1\}\}_{i\in\mathbb{N}_{0}}\) satisfies Assumption 4 with \(\rho_{\mathrm{A}}\in[0,1)\). Then, \(l_{\mathrm{A}}\in\Lambda_{\rho}\) for any \(\rho\in(\rho_{\mathrm{A}},1]\)._  
Interestingly, the combination of non-malicious transmission failures and malicious DoS attacks can also be modeled with processes that belong to the class \(\Lambda_{\rho}\) for certain values of \(\rho\). To see this, first let the failure indicator process \(\{l(i)\in\{0,1\}\}_{i\in\mathbb{N}_{0}}\) be given by  
\[l(i)= \begin{cases}1,&l_{\mathrm{R}}(i)=1\text{ or }l_{\mathrm{A}}(i)=1,\\ 0,&\text{otherwise},\end{cases} \qquad i\in\mathbb{N}_{0}. \tag{22}\]  
The following result covers two cases: (1) non-malicious failures and DoS attacks occur independently, i.e., \(l_{\mathrm{R}}(\cdot)\) and \(l_{\mathrm{A}}(\cdot)\) are independent processes and (2) the DoS attack strategy of the attacker may depend on non-malicious failures in the network, i.e., \(l_{\mathrm{R}}(\cdot)\) and \(l_{\mathrm{A}}(\cdot)\) are _not_ independent. **Proposition 6** ([25]). : _Consider a time-inhomogeneous Markov chain \(\{l_{\mathrm{R}}(i)\in\{0,1\}\}_{i\in\mathbb{N}_{0}}\) with transition probabilities that satisfy \(p_{\mathrm{A},0}(i)\leq p_{0}\) and \(p_{\mathrm{A},1}(i)\leq p_{1}\), for \(i\in\mathbb{N}_{0}\), where \(p_{0},p_{1}\in(0,1)\). Moreover, consider the process \(\{l_{\mathrm{A}}(i)\in\{0,1\}\}_{i\in\mathbb{N}_{0}}\) satisfying Assumption 4 with \(\rho_{\mathrm{A}}\in[0,1)\). If \(l_{\mathrm{R}}(\cdot)\) and \(l_{\mathrm{A}}(\cdot)\) are independent processes and \(p_{1}+p_{0}\rho_{\mathrm{A}}<1\), then \(l(\cdot)\) given by Equation (22) satisfies \(l\in\Lambda_{\rho}\) for \(\rho\in(p_{1}+p_{0}\rho_{\mathrm{A}},1]\). If, on the other hand, \(l_{\mathrm{R}}(\cdot)\) and \(l_{\mathrm{A}}(\cdot)\) are not independent processes and \(p_{1}+\rho_{\mathrm{A}}<1\), then \(l(\cdot)\) given by Equation (22) satisfies \(l\in\Lambda_{\rho}\) for \(\rho\in(p_{1}+\rho_{\mathrm{A}},1]\)._  
The proof of this result utilizes a key lemma (see Lemma A.1 in [25]), where Markov's inequality is used for obtaining Chernoff-type tail probability bounds for the term \(\mathbb{P}\left[\sum_{i=0}^{t-1}l(i)>\rho t\right]\). In the literature, Chernoff-type bounds are essential in obtaining concentration inequalities for sums of random variables (see Section 1.9 in [89], Chapter 27 in [90], and [91]). Notice that in the case where the DoS attack strategy of the attacker may depend on non-malicious failures in the network, \(l\in\Lambda_{\rho}\) holds for larger values of \(\rho\) indicating that the average number of failures in the long run can be larger in that case. Ranges of \(\rho\) values associated with different transmission issues are illustrated in Figure 5.