# A Joint Strategy for Fair and Efficient Energy Usage in WLANs in the Presence of Capture Effect
## 3 Analytical Model
In Equation (3) \(P_{e}\) and \(P_{c}\) denote the probability that a transmission is unsuccessful due to erroneous channel and collision respectively, and can be expressed as: \[P_{e}=1-(1-BER)^{D}\] (4) \[P_{c}=1-(1-\tau)^{(N-1)}-P_{cap}.\] (5)  
BER and D in Equation (4) are bit error rate and size of payload in bits, respectively, whereas \(P_{cap}\) is the frame capture probability and is obtained by summing the contributions from all combinations of the rest of the network nodes transmitting or not. \(P_{cap}\) can be obtained as follows:  
\[P_{cap}=\sum_{i=1}^{N-1}\left( \begin{array}{c}N\\ i+1\end{array} \right)\tau^{i+1}(1-\tau)^{N-1-i}P_{cp}(SIR>Z_{Q}g(S_{f})|i). \tag{6}\]  
\(P_{cp}(\text{SIR}{>}Z_{Q}g(S_{f})|\text{i})=\frac{1}{[1+Z_{Q}g(S_{f})]^{i}}\) in Equation (6) is the conditional capture probability where SIR is the signal to interference ratio of the strongest signal represented as \(P_{r}\) to the \(i\) interfering signals such that \(\text{SIR}=\frac{P_{r}}{\sum_{n=1}^{i}P_{r(n)}}\). To find \(P_{cp}\), we used the capture model that is derived under the hypothesis of power-controlled stations in infrastructure mode and Rayleigh fading [30; 31]. The function \(g(s_{f})\) is the inverse processing gain where \(s_{f}=11\) for using the direct sequence spread spectrum (DSSS) and \(Z_{0}\) is the capture threshold. For detailed derivation of \(P_{cp}\) readers are refered to [30]. The mean slot time, \(T\), is expressed as:  
\[T=P_{tr}P_{S}(1-P_{e})T_{S}+P_{tr}(1-P_{S})T_{C}+P_{tr}P_{S}P_{e}T_{E}+(1-P_{ tr})\sigma, \tag{7}\]  
where \(P_{tr}=1-(1-\tau)^{N}\) is the probability that there is at least one transmission in the network in a randomly chosen time slot. \(P_{S}\)=\(\frac{N\tau(1-\tau)^{(N-1)}+P_{cap}}{P_{tr}}\) is the successful conditional probability that a transmission is successful and corresponds to the case when exactly one frame is transmitted, or when more than one frames are transmitted simultaneously that result in successful frame capture. \(T_{S}\), \(T_{C}\) and \(T_{E}\) are the time durations for the successful frame transmission, transmission failure due to collision and channel interference, respectively, and are given as:  
\[T_{S}=DIFS+T_{D}+SIFS+T_{A} \tag{8}\]\[T_{C} =T_{D}+EIFS \tag{9}\] \[T_{E} =T_{D}+EIFS \tag{10}\]  
where \(T_{D}\) and \(T_{A}\) are data frame transmission time including the PHY and MAC headers, and ACK transmission time, whereas EIFS=SIFS+\(T_{A}\)+DIFS. Now that we have derived some basic expressions, energy efficiency, denoted as \(\eta\), of the considered WLAN in CAM can be expressed as:  
\[\eta=\frac{E[L]}{E} \tag{11}\]  
where \(E[L]\)= \(P_{tr}P_{S}\)(1-\(P_{e}\))L is the amount of payload successfully transmitted in the network, and E is the total energy consumed by the network and is the sum of the energies spent in the four instances, namely successful transmission, collision, failed transmission due to channel interference and the time when nodes were idle, and is given as:  
\[E=P_{tr}P_{S}(1-P_{e})E_{S}+P_{tr}(1-P_{S})E_{C}+P_{tr}P_{S}P_{e}E_{E}+(1-P_{ tr})E_{I}. \tag{12}\]  
Below are the derived expressions for \(E_{S}\),\(E_{C}\),\(E_{E}\),and \(E_{I}\)  
\[E_{S}=T_{D}[N_{cap}P_{tx}+(N-N_{cap})P_{rx}]+NT_{A}P_{rx}+NP_{idle}[SIFS+DIFS] \tag{13}\]  
\[E_{C}=T_{D}[N_{col}P_{tx}+(N-N_{col})P_{rx}]+NT_{A}P_{rx}+NP_{idle}[EIFS] \tag{14}\]  
\[E_{E}=T_{D}[P_{tx}+(N-1)P_{rx}]+NP_{idle}[EIFS] \tag{15}\]  
\[E_{idle}=NP_{idle}\sigma, \tag{16}\]  
where \(P_{tx}\), \(P_{rx}\), \(P_{idle}\) are the transmission, reception and idle mode power consumption (in Watts), respectively, and \(N_{cap}\) is the average number of simultaneously transmitted frames resulting in atleast one successful transmission due to capture with probability \(P_{cap}\) and \(N_{col}\) is the average number of frames transmitted simultaneously resulting in collision. \(N_{cap}\) and \(N_{col}\) can be calculated as:  
\[N_{cap}=\sum_{i=1}^{N-1}i\left( \begin{array}{c}N\\ i+1\end{array}