# A Joint Strategy for Fair and Efficient Energy Usage in WLANs in the Presence of Capture Effect
## 5 PHY Capture Induced Unfairness in WLAN and Proposed Solutions Based on Joint Strategy  
In previous sections we showed that capture effect improves the overall network energy efficiency. In this section, we show that although capture effect increases the overall energy efficiency of a WLAN, it it results in unfairness, i.e., the group of nodes located closer to the AP is twofold more energy efficient than the group of nodes located far away from AP.  
To understand the unfairness caused by the capture effect, we expanded the scenario given in Figure 1 by increasing the number of nodes in the network as shown in Figure 7 and broadly dividing the network geographically into two zones. We call the area closer to the AP Zone 1 and the area at farther distance from AP is called Zone 2. When two nodes, one located in Zone 1 and other located in Zone 2, transmit packets simultaneously, the former being closer to AP successfully captures the channel, while the latter's packet is dropped causing node from Zone 2 to increase its \(CW\). Hence, long run nodes located in Zone 1 transmit more data to AP consuming almost the same or less amounts of energy than the nodes located in Zone 2. In addition, Zone 2 nodes spend more energy in contention and overhearing transmissions between Zone 1 nodes and AP. Zone 1 nodes successfully transmit only if one of them wins contention and none of the Zone 2 nodes transmit in the same time slot. We demonstrate the effect of capture on unfairness in energy efficiency by performing some simulations.  
the simulation scenario consists of WLAN as shown in Figure 7 having AP in the centre of the network. Let the total number of nodes in network be \(N_{total}=20\) and number of nodes in Zone 1 and Zone 2 are \(N_{1}=10\) and \(N_{2}=10\), respectively. Groups of nodes in Zone 1 and Zone 2 surround AP at distances of 40\(\sim\)50 m and 70\(\sim\)140 m from AP, respectively. Traffic in both zones is saturated, transmission failure due to erroneous channel is negligible and \(L=2000\) bytes.  
To measure fairness, we used [PERSON]'s fairness index proposed in [PERSON] (2014). [PERSON]'s fairness index has been widely used as a representative fairness index and the resulting value is bounded between 0 and 1  
Figure 6: Energy efficiency in the presence and absence of capture effect with varying number of nodes in network.  
and denoted as \(\mu_{fair}\). Let \(\eta_{1}\) and \(\eta_{2}\) are the energy efficiencies of group of nodes located in Zone 1 and Zone 2, respectively, then \(\mu_{fair}\) according to [PERSON]'s fairness index among the two groups of nodes is  
\[\mu_{fair}=\frac{(\eta_{1}+\eta_{2})^{2}}{2(\eta_{1}^{2}+\eta_{2}^{2})}. \tag{19}\]  
Simulations were performed with and without the capture effect as shown in Figure 8. At first, Figure 8 shows that in the absence of capture effect, i.e., \(Z_{0}=\infty\), energy efficiency of both zones is almost equal, i.e., \(\mu_{fair}=1\) but this fairness comes at the cost of lower aggregate energy efficiency, i.e., \(\eta=1.30\). In the presence of the capture effect, when \(Z_{0}=6\) dB, although the aggregate energy efficiency was improved, i.e., \(\eta=1.51\), but this caused unfairness, i.e., energy efficiency of the group of nodes in zone 2 was only \(\eta_{1}=0.7\) Mb/J as compared to energy efficiency of the group of nodes in zone 1 which was \(\eta_{1}=2.2\) Mb/J.  
In the presence of the capture effect, according to (19) \(\mu_{fair}=0.81\) which shows significant unfairness. We call the results obtained from the simple capture effect as baseline and will compare them with the results obtained using various joint strategies in the following sections.  
Figure 8: Energy efficiency comparison for different values of capture threshold.  
Figure 7: Wireless local area networks (WLANs) division in various zones depending on their respective distances from AP.