# Throughput Analysis of IEEE 802.11 WLANs with Inter-Network Interference
## 3 General Concept of Throughput Analysis with Inter-Network Interference  
This paper proposes analytical expressions of network throughput for WLANs using the airtime concept with the consideration of inter-network interference. We aim to analytically obtain each network throughput by expressing transmission time, carrier-sensing duration, idle time, and frame-existence probabilities concerning each ED. Our analysis is based on the following assumptions:  
1. ED in each network generates user datagram protocol (UDP) frames to the AP (uplink) following the Poisson distribution. 2. The condition of the physical layer is ideal. Therefore, transmission failures occur due to only frame collisions in the medium access control (MAC) layer. For better understanding, we show the mathematical symbols of this paper in Table 1. Each network node is in one of three states (transmission, carrier-sensing, or idle). The transmission airtime is defined as the time-share of frame transmissions, which includes the duration of both the successful and failed transmissions. Then the transmission airtime of Node \(i\) denoted as \(X_{i}\) is as follows:  
\[X_{i}=\lim_{Time\rightarrow\infty}\frac{S_{i}}{Time}, \tag{1}\]  
where \(S_{i}\) is the parameter related to the airtime concept for IEEE 802.11, initially introduced in [9]. \(S_{i}\) represents the total duration during which Network \(i\) is in the state of DATA frame or ACK frame transmission, SIFS, or DIFS in the time interval of \([0,Time]\). Because of the carrier-sensing, any transmission within Network \(i\)'s carrier-sensing range leads to channel busy. This paper considers the interaction of carrier sensing at each network. Specifically, we examine the overlaps of the multiple transmissions among other networks, which are within the carrier-sensing range of Network \(i\). We use a new function to define \(Y_{i}\), which is the carrier-sensing airtime value (number) of Network \(i\). \(Y_{i}\) can be expressed as  
\[Y_{i} = f(X_{h},\gamma_{i,h}),h\in v(i) \tag{2}\] \[= \sum_{h\in v(i)}X_{h}(1-\gamma_{i,h})-\sum_{h_{1}<h_{2}}X_{h_{1}} (1-\gamma_{i,h_{1}})X_{h_{2}}(1-\gamma_{i,h_{2}})\] \[+ \sum_{h_{1}<h_{2}<h_{3}}X_{h_{1}}(1-\gamma_{i,h_{1}})X_{h_{2}}(1- \gamma_{i,h_{2}})X_{h_{3}}(1-\gamma_{i,h_{3}})-...\] \[+ (-1)^{|v(i)|-1}\prod_{h\in v(i)}X_{h}(1-\gamma_{i,h}),\]  
where \(v(i)\) is the set of networks within the carrier-sensing of Network \(i\). \(h\) is one of the elements in set \(v(i)\). In other word, Network \(h\) is in the carrier-sensing range of Network \(i\). \(h_{s}\) in \(h_{1},h_{2}...h_{s}...h_{|v(i)|}\) is the s-th smallest element of \(v(i)\). \(|v(i)|\) is the number of elements of set \(v(i)\). When the back-off timers of both Network \(i\) and its adjacent one become zero in the idle state, the two networks will transmit frame simultaneously. The simultaneous transmission probability, in this case, is calculated as \(\gamma_{i,h}\). Then the idle airtime of Network \(i\), \(Z_{i}\), is defined as  
\[Z_{i}=1-X_{i}-Y_{i}. \tag{3}\]  
Note that \(Z_{i}\) includes the backoff period in the airtime analysis. Let \(E_{i}\) be the throughput of Network \(i\), then from (1),  
\[E_{i}=X_{i}\frac{P}{T}, \tag{4}\]  
\(T=DIFS+DATA+SIFS+ACK\) in which \(DATA\), \(ACK\) are transmission time of only one \(DATA\) frame, only one ACK frame, respectively. \(P\) is the payload size of a data frame. According to [21], we also define \(G_{i}\) as the transmission probability when Network \(i\) is in the idle state under the saturation condition, then  
\[G_{i}=\frac{R_{i}}{V_{i}}=\frac{1}{W_{0}}, \tag{5}\]