# Throughput Analysis of IEEE 802.11 WLANs with Inter-Network Interference
## 3 General Concept of Throughput Analysis with Inter-Network Interference
\begin{table}
\begin{tabular}{c c} \hline \(\lim_{Time\rightarrow\infty}\) & The limit value as \(Time\) approaches \(\infty\) \\ \hline \(S_{i}\) & The transmission duration of Network \(i\) in \([0,Time]\) \\ \hline \(X_{i}\) & The transmission airtime of Network \(i\) \\ \hline \(Y_{i}\) & The carrier-sensing airtime of Network \(i\) \\ \hline \(f(0)\) & The function to represent \(Y_{i}\) \\ ine \(v(i)\) & The set of networks within the carrier-sensing of Network \(i\) \\ \hline \(|v(i)|\) & The number of elements of set \(v(i)\) \\ \hline \(h_{1},h_{2},-h_{5},-h_{1}|_{v(i)}|\) & \(h_{5}\) is the s-th smallest element of \(v(i)\) \\ \hline \(\sum\limits_{h\in v(i)}\) & The sum over all elements \(h\) in the set \(v(i)\) \\ \hline \(\prod\limits_{h\in v(i)}\) & The product over all elements \(h\) in the set \(v(i)\) \\ \hline \(Z_{i}\) & The idle airtime of Network \(i\) \\ \hline \(E_{i}\) & The throughput of Network \(i\) \\ \hline \(P\) & The payload size of a data frame \\ \hline \(T\) & The period of successful transmission of a data packet \\ \hline \(V_{i}\) & The BT-decrement for one frame-transmission success of Network \(i\) \\ \hline \(G_{i}\) & The transmission probability when Network \(i\) is in the idle state under the saturation condition \\ \hline \(CW_{min}\) & The minimum contention-window value \\ \hline \(\sigma\) & The slot time \\ \hline \(q_{i}^{\prime}\) & The frame-existence probability of Network \(i\) in the non-saturation condition \\ \hline \(q_{i}\) & The frame-existence probability of Network \(i\) \\ \hline \(O_{i}\) & The offered load of Network \(i\) \\ \hline \(\lambda_{i}\) & The frame occurrence rate in Network \(i\) \\ \hline \(\tau_{i}\) & The frame transmission probability at the idle state of Network \(i\) \\ \hline \(\sum\limits_{k\in v(a)\cap v(b)}\) & The sum over all elements \(k\) in the set \(v(a)\cap v(b)\) \\ \hline \(\prod\limits_{k\in v(a)\cap v(b)}\) & The product over all elements \(k\) in the set \(v(a)\cap v(b)\) \\ \hline \(U_{i,h}\) & The probability that Network \(i\) is in the idle state when its neighbor Network \(h\) is in the idle state \\ \hline \(\gamma_{i,h}\) & The simultaneous transmission probability of Network \(i\) and Network \(h\) \\ \hline \(v(a)\cap v(b)=\{\$\}\) & Only Network 5 is within the carrier-sensing range of Network \(a\) and \(b\) \\ \hline \end{tabular}
\end{table} 
Table 1: Mathematical Symbols. where \(R_{i}\) and \(V_{i}\) are the mean number of frame transmission attempts and the average slot number of BT-decrement for one frame-transmission success, respectively. Additionally, \(W_{0}\) is  
\[W_{0}=CW_{min}, \tag{6}\]  
where \(CW_{min}\) is the minimum contention-window value. It is known from [19] that at the saturation condition the relationship between the transmission and idle airtime is  
\[X_{i}=\frac{G_{i}TZ_{i}}{\sigma}, \tag{7}\]  
where \(\sigma\) is the slot time. In the non-saturation condition, it is necessary to derive the frame-existence probability \(q_{i}^{\prime}\), which is the probability that Network \(i\) having at least one frame is in the idle state. Because the BT-decrement is carried out only when the node in the idle state, Network \(i\) decreases the BT in whole time is  
\[q_{i}^{\prime}Z_{i}=\lambda_{i}\sigma V_{i}=\frac{\sigma O_{i}V_{i}}{P}, \tag{8}\]  
where \(q_{i}^{\prime}\) is the frame-existence probability in the non-saturation condition; \(\lambda_{i}\) is the frame occurrence rate; \(O_{i}=\lambda_{i}P\) is the offered load. Additionally, \(\sigma V_{i}\) means the average time of BT-decrement for one successful transmission of a frame. (8) is based on the fact that the frame-occurrence number same as the frame-transmission number in the non-saturation condition. From (8), it is possible to express the frame-existence probability \(q_{i}\) as  
\[q_{i}=\left\{ \begin{array}{ll}q_{i}^{\prime},&non-saturation\\ 1,&saturation\end{array} \right.