# A Lifetime Maximization Relay Selection Scheme in Wireless Body Area Networks
## 3 System Model
### Energy Consumption Model  
In WBANs, energy is consumed mainly on transmission, sensing and abnormal energy waste including idle listening, collision and overhearing. As described in the previous subsection, a TDMA MAC is implemented. Thus, it is considered that there is no energy consumed on abnormal energy waste. Moreover, the energy cost of sensing is so small that can be neglected when compared with  
Figure 1: Network model.  
transmission costs [26]. As a consequence, we focus on the energy consumed by transmission. We have chosen the transmission energy consumption model in [27, 28], which is widely used in many WBAN related works. The energy cost to guarantee reliable transmission during transmitting and receiving are described in Equations (1) and (2) as follows:  
\[E_{tx}(k,d,n)=E_{TXelec}\cdot k+E_{amp}\cdot k\cdot d^{n}, \tag{1}\]  
\[E_{rx}(k)=E_{RXelec}\cdot k. \tag{2}\]  
The model takes \(d^{n}\) as energy costs due to channel variation and the path loss in respect with distance \(d\) between sender and receiver. \(E_{tx}\) represents the transmission energy, \(E_{rx}\) the receiver energy, \(E_{TXelec}\) and \(E_{RXelec}\) the energy the radio dissipates to run the circuitry for the transmitter and receiver, respectively, and \(E_{amp}\) is the energy for the transmitter amplifier. The specific values of these parameters are hardware dependent. In addition, \(k\) represents the number of bits sent for the transmission. \(n\) is pathloss coefficient related to shadow effect.  
As introduced in Section 1, when making relay selection, the amount of energy in each node may not be always the same. Two conditions should be taken into consideration:  
* A WBAN set up with sensor nodes with different battery capacities.
* A WBAN resumes from a restart when each sensor node has already depleted energy under different consumption rates.  
The diversity of energy storage of each node in the network can bring obvious influence on the performance of WBANs. Hence, when making relay selection, the amount of energy in each node must be taken into consideration. As indicated from the reasons stated above, the energy difference among sensor nodes varies unpredictably. Therefore, according to the Central Limit Theorem and the indication, we adopt normal distribution to represent the energy storage condition of each sensor node. Meanwhile, we select an appropriate upper bound and lower bound for the energy storage values to avoid extreme large or small random values in normal distribution, which is impossible to appear in reality. In summary, the residual energy of the \(i\)th sensor node when making a relay selection in this paper follows truncated normal distribution which is represented as the following:  
\[E_{i}\sim N(\mu,\sigma^{2}),\qquad\mu-\Delta\leq E_{i}\leq\mu+\Delta,\]  
where \(\mu\) and \(\Delta\) are standard values for the residual energy and maximum energy deviation from standard value in each node, respectively, while \(\sigma^{2}\) stands for the energy difference degree among sensor nodes. Larger values of \(\sigma^{2}\) reflects more residual inequality in the network.