# An Overview on Denial-of-Service Attacks in Control Systems: Attack Models and Security Analyses
## 3 Deterministic and Game-Theoretical Approaches to Denial-of-Service Attack Modeling
### Deterministic Attack Models with Average Duration and Frequency Constraints  
In [47], the authors proposed a model that allows DoS attacks to happen in an arbitrary fashion as long as the total duration of attacks in a given time interval is upper-bounded by a deterministic function of the length of that interval. In the continuous-time setting of this model, the timing of the attacks can be characterized as follows. First, two sequences \(\{a_{k}\geq 0\}_{k\in\mathbb{N}_{0}}\) and \(\{\tau_{k}\geq 0\}_{k\in\mathbb{N}_{0}}\) are used for denoting the starting times and durations of attack intervals. As we illustrate in Figure 4, the \(k\)th attack interval starts at time \(a_{k}\) and lasts for \(\tau_{k}\) units of time. It is assumed that \(a_{k+1}>a_{k}+\tau_{k}\) so that different attack intervals are not overlapping. This model fits well into scenarios with jamming, where the attacker may be emitting very strong jamming signals during the intervals \([a_{k},a_{k}+\tau_{k}]\). Note that, when a packet transmission time \(t_{j}\) overlaps with any attack interval (i.e., \(t_{j}\in\cup_{k\in\mathbb{N}_{0}}[a_{k},a_{k}+\tau_{k}]\)), then there will be a transmission failure at that time. To model capabilities of the attacks, the notion of total attack duration is used. Specifically, for any time interval \([\tau,t]\subset[0,\infty)\), the set \(\mathcal{A}(\tau,t)\subset[\tau,t]\) is used for denoting the times that the network faces DoS attacks, i.e.,  
\[\mathcal{A}(\tau,t)\triangleq\cup_{k\in\mathbb{N}_{0}}[a_{k},a_{k}+\tau_{k}] \cap[\tau,t], \tag{12}\]  
and \(|\mathcal{A}(\tau,t)|\) is used for denoting the total duration of the attacks in the same interval. Figure 4: Sequence of DoS attack intervals. Transmission attempts that occur in any of the DoS attack intervals (represented with pink regions) fail. Note that, in the case where DoS attacks cover the entire time span, we have \(|\mathcal{A}(0,t)|=t\) for all \(t\geq 0\). In such cases, communication on a network is not possible, and hence, control, estimation, and consensus goals cannot be achieved over such networks. Typically, attackers may have constraints that prevent them from attacking at all times. For instance, in the case of jamming attacks, emitting powerful interference signals is costly and attackers with limited energy resources cannot conduct attacks at all time instants. Note also that the constraints may be imposed by the attacker who wants to avoid being detected. Strategic attackers may want to keep away from attacking continuously at all times to avoid detection. The model in [47] takes into account such constraints by considering the following assumption. **Assumption 1**. : _There exist scalars \(\kappa_{\mathrm{D}}\geq 0\) and \(\rho_{\mathrm{D}}\in[0,1)\) such that_  
\[|\mathcal{A}(0,t)|\leq\kappa_{\mathrm{D}}+\rho_{\mathrm{D}}t,\quad t\geq 0. \tag{13}\]  
It follows from Equation (13) that \(\limsup_{t\to\infty}|\mathcal{A}(0,t)|/t\leq\rho_{\mathrm{D}}\). This indicates that the scalar \(\rho_{\mathrm{D}}\) represents an upper-bound on the average ratio of attack durations in long intervals. If, for instance, \(\rho_{\mathrm{D}}=0.5\), then the total attack duration in the long run does not exceed \(50\%\) of the total time. In the case where the first attack interval starts at time \(a_{0}=0\), Equation (13) implies a bound on the attack interval length \(\tau_{0}\) as \(\tau_{0}\leq\kappa_{\mathrm{D}}/(1-\rho_{\mathrm{D}})\). Thus, the scalar \(\kappa_{\mathrm{D}}\) in Equation (13) can be selected for modeling initial capabilities of the attacker. In [47; 59], Assumption 1 is utilized in analyzing stability properties of networked control systems. In [48], attackers with additional attack frequency constraints are considered. In particular, the following assumption characterizes attacks that have frequency constraints. **Assumption 2**.