# Efficient Actor Recovery Paradigm for Wireless Sensor and Actor Networks
## 4 Problem Statement and Optimized Deterministic Actor Recovery System Model
### Problem Formulation
The RSSI model is used to calculate the distance. It has also been combined with further techniques for better accuracy and to find the relative error. Equations (1)-(5) illustrate applying RSSI in actor nodes [55]. RSSI can also be used to determine the link quality measurement in wireless sensor networks [56].The RSSI shows the relationship between the received energy of the wireless signals and transmitted energy and the required distance among the actor-sensor nodes. This process helps determine failure node recovery process given in Definition 1. The relationship is given by Equation (1):  
\[E_{r}=E_{t}\times\left(\frac{1}{r}\right)^{\beta} \tag{1}\]  
where \(E_{r}\): Received energy of wireless signals, \(E_{t}\): transmitted energy of wireless signals, \(r\): Distance between forwarding and receiving node, and \(\beta\): Path loss transmission factor whose value depends on the environment. Taking the logarithm of Equation (1) provides:  
\[10\log E_{r}=10\log E_{t}-\ 10\beta\log r \tag{2}\]  
where \(10\log E\): Description of energy that could be converted into \(dBm\). Therefore, Equation (2) can be converted to its \(dBm\) form as:  
\[E_{r}(dBm)=\gamma-\ 10\beta\log r \tag{3}\]  
where \(\gamma\): transmission parameter. Here, \(\gamma\) and \(\beta\) represent the relationship between the strength of the received signals and the distance of the signal transmission among sensor-to-sensor, actor-to-sensor or actor-to-actor. RSSI propagation models cover free-space model, log-normal shadow model and ground bidirectional [20]. In this study free space model is used due to following conditions. * The transmission distance is larger than carrier wavelength and antenna size. * There is no obstacle between forwarding actor and either receiving actor or sensor. The transmission energy of the wireless signals and the energy of the received signals of sensor nodes located at distance of \(\tau^{\prime}\) can be obtained by Equations (4) and (5):  
\[\frac{E_{t}\ A_{gt}A_{gr}\lambda^{2}}{(4\pi)^{2}r^{2}\omega} \tag{4}\]  
where \(\lambda\): \(1/\)Frequency of the actor node, \(A_{gt}\) & \(A_{gr}\): An antenna gains, \(\omega\): failure factor of the actor and \(r\): distance of the node:  
\[E\omega(dB)=10\log\frac{A_{gt}}{A_{gr}}\ =-10\ \log\Biggl{[}\frac{\lambda^{2}}{ (4\pi)^{2}r^{2}}\Biggr{]} \tag{5}\]Equation (5) represents the signal attenuation using a logarithmic expression. Assume a field with k actor nodes a\({}_{1}\), a\({}_{2}\), a\({}_{3}\),..., a\({}_{k}\). The coordinates of the actor nodes are \((p_{i},\,q_{i})\) for \(i=1,\,2,\,\ldots,\,k\). The actor nodes transmit the information regarding their location with their signal strength to the sensor nodes \(\{\text{s}_{1},\,\text{s}_{2},\,\ldots,\,\text{s}_{n}\}\). The locations of the sensor nodes are unknown. The estimated distances of the actor nodes are calculated from the received signals. In the proposed model, the actor nodes broadcast signals to all sensors. The actor nodes are also responsible for estimating the distances between them and sensor nodes. Let \(a_{i}\) be an actor node located at \((y_{i},z_{i})\) and sensor node is located at \((y,\,\,\text{z})\). Focusing on the relative error \({}^{\prime}r_{e}^{\prime}\) relating to \(a_{i}\), suppose that the actor node reads a distance \(r_{i}\), but the correct distance is \(r_{j}\). Therefore, the relative error can be obtained by Equation (6):  
\[r_{e}=\frac{r_{i}}{r_{j}}-1\in(-1,\,\,+\infty) \tag{6}\]  
The relation between actual distance and the measured distance can be obtained by Equation (7):  
\[r_{i}=r_{j}10\frac{\pi}{10\beta}-1 \tag{7}\]  
which can be reduced to Equation (8):  
\[r_{e}=10\frac{\pi}{10\beta}-1 \tag{8}\]  
The probability distribution of the location of the actor node based on beacon messages is described in the following definition. **Definition 1**.