# Efficient Actor Recovery Paradigm for Wireless Sensor and Actor Networks
## 4 Problem Statement and Optimized Deterministic Actor Recovery System Model
### Problem Formulation
: _Let \(a_{i}\) be an actor node located at \((y_{i},z_{i})\) that sends information to a sensor using RSSI model with standard deviation \({}^{\prime}\sigma^{\prime}\) and path loss \({}^{\prime}\beta^{\prime}\)._  
_Let \(r_{i}\) be the calculated distance from the actor node \(a_{i}\) at the sensor node. The probability density function for correct location \((y,z)\) of the sensor node is obtained by Equation (9):_  
\[\left\{ \begin{aligned} & E_{Y,Z}^{(i)}(Y,Z)\frac{10\beta\exp\left(- \left(\log\beta\log\frac{r_{i}}{\sqrt{(y-y_{i})^{2}+(z-z_{i})}}\right)^{2}/2 \sigma^{2}\right)}{2\sigma^{2}\sqrt{2\pi\,\,\log(10)\left(\left(y-y_{i} \right)^{2}+(z-z_{i})\,\,\right)^{2}}}\\ \end{aligned} \right\} \tag{9}\]  
_Probability distribution can be simplified due to an actor \(a_{i}\) with Equation (10):_  
\[E_{Y,Z}^{(i)}(Y,Z)=\Psi_{a_{i}}(y,z) \tag{10}\]  
This can further be extended by using finite set of actors \(A=\{a_{1},\,a_{2},\,a_{3},\,\ldots,\,a_{k}\}\) that produces Definition 2. **Definition 2**. : _Let \(A=\{a_{1},\,a_{2},\,a_{3},\,\ldots,\,a_{k}\}\) be the set of the actors sending information to the set of sensor nodes using RSSI model with path loss exponent \({}^{\prime}\beta^{\prime}\)._  
_If the calculated distance from the actor node \(a_{i}\) at the sensor nodes \(S=\{s_{1},\,s_{12},\,\ldots,\,s_{n}\}\), then the probability density function of correct location \((y,z)\) of the sensor nodes can be obtained by:_  
\[\left\{ \begin{aligned} &\Psi(A)(y,z)=\frac{\sum_{i=0}^{k}\Psi_{a_{i}} \left(y,z\right)}{\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\left( \sum_{i=0}^{k}\Psi_{a_{i}}(y,z)dzdy\right)}\\ \end{aligned} \right\} \tag{11}\]_where \(\Psi a_{i}(y,z)\): probability distribution because of an actor \(a_{i}\)._  
**Definition 3**. : _Assume an actor node \(\rm a_{i}\) reads a sample distance \(R=\{r_{1}\), \(r_{2}\), \(r_{3}\), \(\ldots\), \(r_{n}\}\) using beacon messages \(B=\{b_{1}\), \(b_{2}\), \(b_{3}\), \(\ldots\), \(b_{n}\}\) that is modeled with RSSI with path loss \(\beta\) and standard deviation \({}^{\prime}\)\({}^{\prime}\)\({}^{\prime}\)._  
_If 'R' provides the mean sample distances and \(\sigma_{i}\) is the mean standard deviation, then the square of the actual distance from actor to sensor using beacons can be determined as:_  
\[R=\frac{R^{4}}{R^{2}+\sigma_{i}^{2}} \tag{12}\]  
_Furthermore, square standard deviation can be found as:_  
\[\sigma_{i}^{2}=\frac{100\beta^{2}}{\log^{2}(10)}\ \log\Bigl{(}1+\frac{\sigma}{R} \Bigr{)}^{2} \tag{13}\]  
The definition shows that the actual distance is greatly dependent on the distribution of the measured ranges. Hence, our proposed formulas for RSSI-based wireless node location are optimized and modified. They are different from the original RSSI-based formulas. We focused particularly on the energy consumed for transmission and receiving the data including determining the distance between actor-sensor nodes and error rate for finding location of the node that helps identifying the accurate position of the deployed actor nodes for events. Thus, the previous model is used by our proposed algorithm in order to identify the node locations during deployment in addition to during the network lifetime.