# A Joint Strategy for Fair and Efficient Energy Usage in WLANs in the Presence of Capture Effect
## 3 Analytical Model  
This section explains an analytic model which is based on the model presented in [29]. However, unlike the model presented in [29], our proposed model takes into account capture effect in addition to unsaturated traffic condition and erroneous channel as shown in Figure 2. Figure 2 shows that backoff stage increases due to two reasons (i) when two or more frames arrives at the receiver while none of them has significantly higher receiving power than that of all the other remaining frames to successfully capture the channel (ii) when a transmission is significantly impaired by the channel errors. In contrast, a node after successful transmission reaches to state 'E' if its transmission queue is empty or randomly selects a backoff counter from the initial contention window (_CW_) if there is at least one data frame waiting for transmission in its transmission queue. Table 1 shows the list of notations used in the derivation of the main equations. \begin{table}
\begin{tabular}{c c} \hline
**Notation** & **Description** \\ \hline \(P_{tx}\) & Power consumption during transmission. \\ \(P_{rx}\) & Power consumption during reception. \\ \(P_{dalle}\) & Power consumption in idle mode. \\ \(T_{s}\) & Time spent in successful transmission. \\ \(T_{c}\) & Time spent due to frame collision. \\ \(T_{E}\) & Time spent in transmission failure due to error. \\ \(\sigma\) & Idle slot time. \\ \(E_{s}\) & Energy consumption during successful transmission. \\ \(E_{c}\) & Energy consumption due to collision. \\ \(E_{E}\) & Energy consumption in transmission failure due to error. \\ \(E_{dalle}\) & Energy consumption during idle time slot. \\ \(p\) & Probability of transmission failure. \\ \(P_{tr}\) & Probability of channel being busy. \\ \(W\) & Initial size of the contention window. \\ \(\lambda\) & Frame arrival rate with Poisson distribution. \\ \(P_{Cap}\) & Frame capture probability. \\ \(P_{c}\) & Probability of frame collision. \\ \(P_{e}\) & Probability of transmission failure due to channel error. \\ \(P_{S}\) & Probability of successful transmission. \\ \(\tau\) & Probability of attempt by a node. \\ \(\eta\) & Energy efficiency of network in Mb/j. \\ \(\eta_{1}\) & Energy efficiency of nodes in Zone 1. \\ \(\eta_{2}\) & Energy efficiency of nodes in Zone 1. \\ \(N\) & Total number of nodes in network. \\ \(N_{1}\) & Total number of nodes in Zone 1. \\ \(N_{2}\) & Total number of nodes in Zone 2. \\ \(\mu_{fair}\) & Fairness index for energy consumption among different zones. \\ \hline \end{tabular}
\end{table} 
Table 1: Notations and their descriptions. Figure 2: A two dimensional Markov chain model that takes into account capture probability. There are two key probabilities. 1. A node attempts transmission in a generic time slot with probability \(\tau\) and is constant across all time slots. 2. Probability of collision, represented as \(P_{c}\) is constant and independent of the number of collisions already suffered and includes probability that the node does not capture the channel. Now \(\tau\) when the channel is unsaturated is given as: \[\tau=\frac{2(1-2P)q}{q((1-2P)(W+1)+WP(1-(2P)^{m}))+2(1-q)(1-P)(1-2P)}\] (1) where \(W\) and \(m\) represent the initial size of \(CW\) and the maximum backoff stage. \(q\) is the probability that there is at least one frame in the transmission queue of the tagged node, whereas \(P\) represents the probability of transmission failure. Parameters \(q\) and \(P\) are given as: \[q=1-e^{-\lambda T}\] (2) \[P=P_{c}+P_{e}-P_{c}P_{e},\] (3) where \(\lambda\) in Equation (2) is the frame arrival rate with Poisson distribution and \(T\) is the mean slot time.