# 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  
In Section 3, we looked at deterministic and game-theoretic DoS attack models and analysis techniques. Our goal in this section is to discuss _probabilistic_ approaches for modeling and analyzing the effects of DoS attacks. In particular, we focus on networks with wireless channels and multi-hop transmissions. For such networks, DoS attacks may not be the only source of failures, as transmissions may also fail due to non-malicious reasons [17; 18]. For instance, wireless channels face unintentional interference, channel noise, and fading issues. Moreover, non-malicious issues such as link errors and congestion cause failures in multi-hop networks. In networked control systems, the transmission failures due to non-malicious issues are typically modeled through the use of stochastic processes such as Bernoulli processes [30; 84] and Markov processes [31; 32]. In [25], we observed that both non-malicious transmission failures and malicious DoS attacks as well as their combinations can be modeled through a _probabilistic_ approach. The probabilistic approach in [25] is developed upon tail-probability bounds for the binary-valued processes that describe the occurrences of transmission failures on a network. In what follows, we first explain this probabilistic approach and illustrate its generality by considering non-malicious failures and DoS attacks on a network. Then, in Sections 4.1 and 4.2, we consider its utility in wireless channel and multi-hop network modeling. Consider the binary-valued process \(\{l(i)\in\{0,1\}\}_{i\in\mathbb{N}_{0}}\) discussed in Section 2.1.1. This process is used for indicating the status of packet transmissions, where \(l(i)=1\) represents a failure and \(l(i)=0\) represents a successful transmission at time \(t_{i}\). To describe the effects of certain non-malicious and malicious failure models in a unified manner, in [25; 27], we investigated classes of binary-valued transmission failure indicator processes that describe networks where the number of failures in the long run is bounded in a probabilistic sense. First, for a given scalar \(\rho\in[0,1]\), consider the class \(\Lambda_{\rho}\) of binary-valued processes given by  
\[\Lambda_{\rho}\triangleq\left\{l\colon l(i)\in\{0,1\},i\in\mathbb{N}_{0};\sum_{t =1}^{\infty}\mathbb{P}\left[\sum_{i=0}^{t-1}l(i)>\rho t\right]<\infty\right\}. \tag{19}\]  
Notice that the inequality \(\sum_{t=1}^{\infty}\mathbb{P}\left[\sum_{i=0}^{t-1}l(i)>\rho t\right]<\infty\) describes a condition on the tail probability \(\mathbb{P}[\frac{1}{t}\sum_{i=0}^{t-1}l(i)>\rho]\) (i.e., the probability of the tail event where the ratio \(\frac{1}{t}\sum_{i=0}^{t-1}l(i)\) of transmission failures exceeds \(\rho\)). Under this condition, the average number of failures is guaranteed to be bounded in the long run. In particular, a consequence of the Borel-Cantelli lemma [85] is that \(\limsup_{t\to\infty}\frac{1}{t}\sum_{i=0}^{t-1}l(i)\leq\rho\), almost surely, for every \(l\in\Lambda_{\rho}\). For the network control system discussed in Section 2.1.1, it is shown in [25] that the closed-loop system becomes stable if \(l\in\Lambda_{\rho}\) for a sufficiently small \(\rho\). It was further noted by the authors of [25] that classes \(\Lambda_{\rho}\) with different \(\rho\) values can be used to characterize binary-valued transmission failure indicators that are associated with networks that face: (1) non-malicious transmission failures; (2) malicious DoS attacks; and (3) combination of non-malicious and malicious issues.