# Robot Delay-Tolerant Sensor Network for Overhead Transmission Line Monitoring
## 3 RDTSN Model and Data Delivery Strategy
### Robot Delay-Tolerant Sensor Network (RDTSN) Model
#### 3.1.2 Discrete Time DTSN Mathematical Model  
This section solves a problem of how to determine the best delivery path from one or more sources to a destination. The solution is performed based on the statistical methods of delivery probability and energy loss between nodes. For RDTSN, as robots walk randomly along transmission lines, the network topology is unreliable and constantly changing. Furthermore, as the robots have high mobility, the entire network is locally strongly interconnected at discrete times. In order to meet the needs of the nodes (Inspection Robots, SNs or MMS) in RDTSN to transmit information to the CMP via the WCNs connected with OPGW, this paper draws on references [14; 32], to introduce the graph theory to mathematically describe the above network. In the mathematical model, each link has an availability weight, which is jointly determined by link delivery probability and energy loss. The availability weights of links can change momentarily. Furthermore, the node will determine whether to send or store the message according to the weight of the link. Therefore, this paper proposes a discrete-time multi-path selection model with maximum link weights and maximum information at different time, as shown in Figure 6.  
Figure 5: BMI movement models of robot. (**a**) State one; (**b**) state two. In the network mathematical model, it is assumed that the current moment is t, and the information of the network is represented by graph \(G\) (\(t\)) = (\(N\), \(E\)). Elements of N represent the corresponding node in the RDTSN, which are a set of nodes; Elements of E reveal the physical link between those nodes, which is an edge set. At the next moment \(t+1\), graph \(G\) changes. The network may be represented by an extension \(G\) (\(t+1\)) = (\(N\)', \(E\)') of \(G\) (\(t\)), as shown in Figure 7. The graph \(G\) contains all the information of the network. In addition, Equation (1) is used to calculate the weight function \(w_{ij}(t)\) of each edge. It shows the change rule of the links' availability weight over time. In each new extension \(G\) (\(t+n\)) of the network, a node makes decisions to forward or store the messages based on the elements and relationships of the current graph. \(P_{ij}(t)\) is the best path for the message to be transmitted from the source node to the target node. One path can be jointly created by the physical links of different nodes at several discrete moments. \[w_{ij}(t)=\alpha d_{ij}(t)+\beta f_{ij}(t) \tag{1}\]  
\[d_{ij}(t)=a_{ik}(t)\prod_{kj\in V}d_{kj}(t) \tag{2}\]  
\[f_{ij}(t)=1-\frac{\sum\limits_{j\in P}e_{ij}(t)}{E_{i}(t)} \tag{3}\]  
At each moment, an \(a_{ik}(t)\) is given, which represents the link availability probability between node \(i\) and node \(k\). Equation (2) is used to obtain the variable \(d_{ij}(t)\) denoting the delivery probability of  
Figure 6: Network representation. (**a**) Two-node graph; (**b**) five-node graph. Figure 7: Network extended graph with the availability weights through time represented at once. the source node \(i\) forwarding to node \(j\), through node \(k\). The variable \(d_{ki}(t)\), in Equation (2), denotes the delivery probability from node \(k\) to node \(j\). At time \(t\), \(E_{i}(t)\) denotes the total energy of node \(i\), \(e_{ij}(t)\) denotes the energy consumption of message delivery by path \(P_{ij}(t)\), and the variable \(f_{ij}(t)\) denotes the node survivability, which is given in Equation (3). In Equation (4), the objective function of the network mathematical model is constructed. As the description of the objective function, messages are forwarded by nodes at different discrete moments. It ensures that the appropriate message delivery probability, energy consumption, and message delay are obtained. The constraints on network traffic and energy consumption are given in Equations (5)-(8). Each node has a certain storage capacity. \(bi\) denotes the buffer size of node \(i\), \(b_{ij}\) denotes the information capacity of link \(P_{ij}(t)\).