# A Power Planning Algorithm Based on RPL for AMI Wireless Sensor Networks
## 3 ETX Theoretical Model  
RPL supports a flexible set of metrics and constraints [21]. One of the preferred metrics is ETX, which represents the expected number of transmissions required to successfully transmit and acknowledge a packet on a wireless link. The ETX metric was introduced in [4], and is defined by the following expression  
\[ETX=\frac{1}{d_{f}*d_{r}}, \tag{1}\]  
where \(d_{f}\) is the measured probability that a packet is received by the neighbor and \(d_{r}\) is the measured probability that the acknowledgment packet is successfully received. ETX is sensitive to packet loss but not to delay. If the transmission of a packet is delayed by contention, but the packet is transmitted, the metric is not affected. Thus, the ETX metric is affected by two main components: the quality of the wireless channel and collisions. RPL does not assume a probe mechanism based on broadcast messages, as in [4], to compute the value for ETX. In the following, we present a theoretical model to estimate a value for ETX that is used as the initial value during the DODAG construction.  
A channel outage probability model gives a prediction for the packet error rate (PER) caused by radio channel impairments, such as path loss and fading. The Nakagami-\(m\) distribution [22] is employed in this work to model the multi-path fading effects of the propagation environment. The fading severity is adjusted through the parameter \(m\). Supposing the use of capacity achieving codes, an outage occurs at the receiver when the instantaneous signal-to-noise ratio (SNR) \(\gamma\) is below a threshold \(\beta=\left(2^{\Delta}-1\right)\) that allows error free decoding. The parameter \(\Delta\) is the system spectral efficiency. The Nakagami-\(m\) fading outage probability for frames transmitted from a node with power \(w\) to a receiver at a distance \(d\) is  
\[\mathcal{O}=\mathbb{P}\left[\gamma<\beta\right]=\frac{\Psi\left(m,\frac{m \cdot\beta}{\overline{\gamma}}\right)}{\Gamma(m)}, \tag{2}\]  
where \(\overline{\gamma}=\frac{\eta_{d}\cdot w}{N_{0}\cdot\overline{B}}\) is the average received SNR, \(B\) is the system bandwidth in Hertz, \(\eta_{d}=\frac{G\cdot\lambda^{2}}{(4\pi)^{2}d^{2}}\) is the path loss of the channel and \(N_{0}\) is the thermal noise power spectral density in Watts/Hertz. We assume a path loss model with attenuation factor \(\frac{1}{d^{2}}\), where \(\alpha\) is the path loss exponent used to represent different environment propagation conditions. \(G\) is the total gain of the transmission and reception  
Figure 2: Power planning.  
antennas, and \(\lambda\) is the wavelength. The \(\Gamma(a)\) and \(\Psi(a,b)\) are the complete and lower incomplete gamma functions. The ETX metric of a wireless link between nodes \(a\) and \(b\) can be calculated by Equation (3), considering the outage probability of transmissions from \(a\) to \(b\) (\(\mathcal{O}_{ab}\)) and \(b\) to \(a\) (\(\mathcal{O}_{ba}\)):  
\[ETX=\frac{1}{(1-\mathcal{O}_{ab})*(1-\mathcal{O}_{ba})}. \tag{3}\]  
The ETX of a wireless link can be reduced by increasing the transmission power of the adjacent nodes, but this also increases the interference range caused by this link over other links in the same channel frequency. The interference range can be defined as the distance where the received power equals the receiver carrier-sensing sensitivity.