# 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
### Multi-Hop Network Models  
In addition to SINR-based jamming models, the probabilistic characterization through the class \(\Lambda_{\rho}\) can be employed for modeling transmission failures in multi-hop networks. A multi-hop network (similar to those considered on the right diagram in Figure 1) can be represented by a directed acyclic graph \(\mathcal{G}\triangleq(\mathcal{V},\mathcal{E})\), where \(\mathcal{V}\) is the set of nodes, and \(\mathcal{E}\subset\mathcal{V}\times\mathcal{V}\) is the set of edges. Note that \(\mathcal{V}\) and \(\mathcal{E}\), respectively, represent the set of communication devices and the set of communication links. On a multi-hop network, data packets are transmitted over paths, which are sequences of non-repeating edges. Specifically, a _path_\(\mathcal{P}\) from a node \(\
u_{1}\in\mathcal{V}\) to another node \(\
u_{h}\in\mathcal{V}\) is given as  
Figure 6: The transmission failure probability function \(p\) and a concave upper-bounding function \(\hat{p}\) for an example wireless channel under jamming attacks. \(\mathcal{P}=\big{(}(v_{1},v_{2}),(v_{2},v_{3}),\ldots,(v_{h-1},v_{h})\big{)}\). Notice that there may be multiple paths between two nodes, and, moreover, those paths may be utilized in transmission of the same data. It is shown in our previous work [27] that the class \(\Lambda_{\rho}\) can be utilized in modeling failures on individual links as well as paths and entire graphs. Specifically, in [27], a network \(\mathcal{G}\) with source \(v_{\mathrm{P}}\) (corresponding to the plant) and sink \(v_{\mathrm{C}}\) (corresponding to the controller) ia considered (see Figure 7). The paths between nodes \(v_{\mathrm{P}}\) and \(v_{\mathrm{C}}\) are identified as \(\mathcal{P}_{1},\mathcal{P}_{2},\ldots,\mathcal{P}_{|\mathcal{G}|}\), where \(|\mathcal{G}|\) denotes the total number of paths. Moreover, the individual links on the \(i\)th path are denoted as \(\mathcal{P}_{i,1},\mathcal{P}_{i,2},\ldots,\mathcal{P}_{i,|\mathcal{P}_{i}|}\) where \(|\mathcal{P}_{i}|\) is the number of links on path \(\mathcal{P}_{i}\). Transmission failures on those links can be represented by binary-valued failure indicator processes \(l_{\mathcal{P}_{i}}^{\mathcal{P}_{i,j}}(\cdot)\). Similarly, overall failures on each path \(\mathcal{P}_{i}\) can be represented with an indicator process \(l_{\mathcal{P}_{i}}(\cdot)\), and, moreover, the failures of transmission between nodes \(v_{\mathrm{P}}\) and \(v_{\mathrm{C}}\) on network \(\mathcal{G}\) can be represented with a binary-values indicator process \(l_{\mathcal{G}}\). Notice that \(l_{\mathcal{G}}(t)=l_{\mathcal{P}_{1}}(t)\wedge l_{\mathcal{P}_{2}}(t)\wedge \cdots\wedge l_{\mathcal{P}_{|\mathcal{G}|}}(t)\) and \(l_{\mathcal{P}_{i}}(t)=l_{\mathcal{P}_{i}}^{\mathcal{P}_{i,1}}(t)\lor l_{ \mathcal{P}_{i}}^{\mathcal{P}_{i,2}}(t)\vee\cdots\lor l_{\mathcal{P}_{i}}^{ \mathcal{P}_{i,|\mathcal{P}_{i}|}}(t)\). The following result is obtained in [27] to show that the class \(\Lambda_{\rho}\) can be used for characterizing the failure indicators for each individual link as well as the paths that include those links. **Proposition 9** ([27]). : _Suppose \(l_{\mathcal{P}_{i}}^{\mathcal{P}_{i,j}}\in\Lambda_{\mathcal{P}_{i,j}}\), \(j\in\{1,\ldots,|\mathcal{P}_{i}|\}\), where \(\rho_{\mathcal{P}_{i}}^{\mathcal{P}_{i,j}}\in[0,1]\) and \(j\in\{1,\ldots,|\mathcal{P}_{i}|\}\) satisfy \(\sum_{j=1}^{|\mathcal{P}_{i}|}\rho_{\mathcal{P}_{i}}^{\mathcal{P}_{i,j}}\leq 1\). Then, \(l_{\mathcal{P}_{i}}\in\Lambda_{\rho_{\mathcal{P}_{i}}}\) with \(\rho_{\mathcal{P}_{i}}\triangleq\sum_{j=1}^{|\mathcal{P}_{i}|}\rho_{\mathcal{ P}_{i}}^{\mathcal{P}_{i,j}}\)._  
The next result shows that the overall failures on a network \(\mathcal{G}\) can be characterized with the class \(\Lambda_{\rho}\) where the \(\rho\) value is identified as the minimum of \(\rho_{\mathcal{P}_{i}}\) obtained for each path \(\mathcal{P}_{i}\) on the network \(\mathcal{G}\). **Proposition 10** ([27]).