# Joint Deployment of Sensors and Chargers in Wireless
## 3 Model and Problem Statement
### Sensor Perception Model  
We adopt a probability model for the sensor perception. In this model, the probability value of the Pols being monitored by the sensor decreases with the growth of the distance between them. The perception model is as follows [34]:  
\[P(s_{j},o_{i})=\left\{\begin{array}{ll}0&r+r_{e}\leq d\big{(}s_{j},o_{i}\big{)} \\ e^{-\lambda a\beta}r-r_{e}\leq d\big{(}s_{j},o_{i}\big{)}\leq r+r_{e}\\ 1&r-r_{e}\leq d\big{(}s_{j},o_{i}\big{)}\end{array}\right. \tag{1}\]  
where \(r_{e}(r_{e}<r)\) and \(r\) is the uncertainty in the measurement sensor detection, \(\alpha=d\big{(}s_{j},o_{i}\big{)}\)\(-(r-r_{e})\) and \(\lambda\), \(\beta\) are parameters for measuring the probability of detection. \(d\big{(}s_{j},o_{i}\big{)}\) is Euclidean distance between \(s_{j}\) and \(o_{i}\), with \(s_{j}\in\Phi,o_{i}\in\Phi\).  
We assume that the sensors are required to monitor the PoIs with a probability of no less than a certain probability \(c_{th}\). Specifically, when \(P\big{(}s_{j},o_{i}\big{)}\geq c_{th}\), the PoI \(o_{i}\) is covered by the sensor \(s_{j}\).  
In real life, the working state and the charging state of the sensor cannot occur at the same time, which means that the sensor cannot be charged during the working state, and cannot work during the charging process. Therefore, based on the many-to-one mode of sensor, it is necessary to schedule the status of sensors (working or charging) reasonably to meet the full-time domain monitoring of all PoIs. Specifically, we assume that a sensor's life cycle is divided into \(J\) identical time slots, and the time slots of all sensors are synchronized by default. The scheduling scheme of all sensors is \(S_{T}=\{s_{1t},s_{2l},\ldots s_{mt}\}\), where \(s_{ft}=\big{(}a_{1},a_{2},..,a_{J}\big{)}\). \(a_{j}=1\) indicates that the sensor is in a working state in the current time slot, and the sensor can monitor the PoIs. Otherwise, \(a_{j}=0\) means that the sensor is in a charging state.  
As shown in Figure 2, the scheduling scheme of the sensor \(s_{j}\) in one life cycle is \(s_{ft}=(1,0,0,1)\). Obviously, the sensor \(s_{j}\) only enters the working state in the time slot \(t\in\{4 kt,(4k+1)t\}\) or \(t\in\{(4k+3)t,(4k+4)t\}\), where \(k\) is an integer.  
\begin{table}
\begin{tabular}{c p{284.5 pt}} \hline \hline
**Variables** & **Description** \\ \hline \(o_{i}\) & A two-dimensional continuous variable. The location of the \(i\)-th deployed PoI, and \(O=\{o_{1},o_{2},...,o_{N}\}\)) is the set of all deployed PoIs, and \(o_{i}\) also refers to the PoIs itself for simplicity \\ \(s_{j}\) & A two-dimensional continuous variable. The location of the \(j\)-th deployed sensor, and \(S=\{s_{1},s_{2},...,s_{M}\}\) is the set of all deployed sensors, and \(s_{j}\) also refers to the sensor itself for simplicity \\ \(c_{x}\) & A two-dimensional continuous variable. The location of the \(x\)-th deployed charger, and \(C=\{c_{1},c_{2},...,c_{K}\}\) is the set of all deployed chargers, and \(c_{x}\) also refers to the charger itself for simplicity \\ \(P_{h}^{s_{j},c_{x}}\) & A two-dimensional continuous variable. Wireless charging power delivered by a charger \(c_{x}\) to the sensor \(s_{j}\) \\ \(P_{h}^{s_{j},c_{x}}\) & A two-dimensional continuous variable. Wireless charging power delivered by all chargers \(C\) to the sensor \(s_{j}\) \\ \(P_{min}^{s_{j}}\) & A two-dimensional continuous variable. The minimum charging power of \(s_{j}\) \\ \(s_{ft}\) & A binary-continuous variable. The schedule of \(s_{j}\) in one cycle \\ \(S_{T}\) & The scheduling scheme of all sensors \\ \(P\Big{(}s_{j},o_{i}\Big{)}\) & A two-dimensional continuous variable. The probability of sensing a point \(o_{i}\) by a sensor \(s_{j}\). \\ \hline \hline \end{tabular}
\end{table}
Table 2: _Cont._