## 2 Problem Formulation and Model
### Communication Model  
We use a packet-erasure channel model to describe the realistic wireless communication links, in which the receiver drops all the packets with a received SNR below a predefined threshold. The received signal-to-noise ratio (SNR) \(\Gamma_{ij}\) at node \(j\) from a transmitting node \(i\) is given by:  
\[\Gamma_{ij}=\frac{P_{i}G_{ij}}{N_{j}} \tag{9}\]  
where \(P_{i}>0\) is the power delivered to the transmitting antenna of node \(i\); \(N_{j}>0\) is the average noise power when node \(j\) operating as a receiver, and \(G_{ij}\) is the channel gain. Using the Rayleigh model, \(G_{ij}\) can be expressed as:  
\[G_{ij}=\frac{C_{ij}|h_{ij}|^{2}}{d_{ij}^{\alpha}} \tag{10}\]  
where \(C_{ij}\) is a constant in which the antenna gains and shadowing are considered. \(h_{ij}\) represents the multi-paths fading. The distance between nodes \(i\) and \(j\) is \(d_{ij}\). \(\alpha\) is the propagation loss factor. Maintaining a successful transmission with an acceptably small packet loss requires a sufficiently high received SNR. In this paper, the required SNR threshold that to obtain minimal link quality is \(\gamma\). Therefore, if we assume that the channel has fast Rayleigh fading [24], the probability of a successful transmission between nodes \(i\) and \(j\) is given by:  
\[P_{r}^{ij}(\Gamma_{ij}\geq\gamma)=\exp\left(-\frac{N_{j}\gamma d_{ij}^{\alpha} }{C_{ij}P_{i}}\right) \tag{11}\]  
Equation (11) describes a realistic wireless communication model. Please note that a larger distance between nodes \(i\) and \(j\) results in a lower received SNR. If \(\Gamma_{ij}<\gamma\), the packet from node \(i\) will be dropped by node \(j\), and thus nodes \(i\) and \(j\) are regarded as disconnected. If \(\Gamma_{ij}\geq\gamma\), the packet from node \(i\) will be successfully received by node \(j\), and thus nodes \(i\) and \(j\) are regarded as connected. The relationship between \(P_{r}^{ij}\) and the distance \(d_{ij}\) is shown in Figure 2; for the parameters listed in Table 1.  
Figure 2: The probability of a successful transmission vs. distance, as determined by Equation (11) for the parameters listed in Table 1.  
To compute the highest successful transmission probability between the source and desired destination, we should examine all possible multi-hop routes. If the successful transmission of the packet in each hop is independent, the successful transmission probability in a given multi-hop route (route\({}_{k}\)) is given by:  
\[P_{r}(\text{route}_{k})=\prod_{(i,j)\in\text{route}_{k}}P_{r}^{ij} \tag{12}\]  
The specific route will be chosen by the routing algorithm. In this paper, the packet routing algorithm will always choose the route with the highest successful transmission probability.  
In addition to the Rayleigh fading model used in our algorithm, other fading models can also be employed in a similar way. In practice, the distribution of the channels can be collected and employed within the framework of this paper in a straightforward way. In future work, we will design other communication fading models, such as Rician fading, and integrate them into the framework of co-optimization of communication and sensing.