## 3 Distributed Transmission Rate Adjustment  
Initially an arbitrary node \(i\), with no prior information, tentatively transmits data at the rate \(\lambda_{i}=t_{i}\). After a certain amount of time, node \(i\) can estimate the busy channel probability \(\beta_{i}\) by recording the CCA information. The estimation is performed by calculating the ratio of the number of busy CCAs to the number of all the CCA for a certain period and using this ratio as the busy channel probability \(\beta_{i}\). \(\beta_{i}\) is then used to determine the optimal \(\lambda_{i}\)[6, 8]. From Equation (3), once \(\beta_{i}\) is known, \(\tau_{i}\) can be determined as:  
\[\tau_{i}=1-\prod_{j\
eq i}\bigl{(}1-\alpha_{j}\bigr{)}=\frac{\beta_{i}}{(1- \beta_{i})(1+L)} \tag{4}\]  
The term \(\prod_{j\
eq i}\bigl{(}1-\alpha_{j}\bigr{)}\) contains transmission rate information of all other nodes \(j\
eq i\) and can be rewritten using Equation (2) as:  
\[\prod_{j\
eq i}\bigl{(}1-\alpha_{j}\bigr{)}=\prod_{j\
eq i}(1-\frac{1-\beta_{j }^{m+1}}{(1-\beta_{j})\lambda_{j}}) \tag{5}\]  
One important observation is that for any 2 arbitrary node \(i\) and \(r\), the backoff failure probability \(\beta_{i}\) and \(\beta_{r}\) are almost equal as long as the network size is not too small, or when the nodes are of homogeneous nature. This is because \(\beta_{i}\) and \(\beta_{r}\) are determined by \(\prod_{j\
eq i}\bigl{(}1-\alpha_{j}\bigr{)}\) and \(\prod_{j\
eq r}\bigl{(}1-\alpha_{j}\bigr{)}\) respectively and the ratio \(\frac{\prod_{j\
eq i}(1-\alpha_{j})}{\prod_{j\
eq r}(1-\alpha_{j})}=\frac{(1- \alpha_{r})}{(1-\alpha_{i})}\) is very close to 1 as \(\alpha_{r}\) and \(\alpha_{i}\) are typically smaller than 0.01 [29, 33]. Thus the subscript \(i\) for every \(\beta_{i}\) can be dropped without affecting the analysis. Combining Equations (4) and (5), one can get:  
\[\beta=\frac{(1-\prod_{j\
eq i}(1-\frac{1-\beta^{m+1}}{1-\beta}\frac{1}{ \lambda_{j}}))(1+L)}{1+(1-\prod_{j\
eq i}(1-\frac{1-\beta^{m+1}}{1-\beta}\frac{ 1}{\lambda_{j}}))(1+L)} \tag{6}\]  
Our interest lies in finding a detailed characterization of the aggregate transmission rate \(\sum_{j\
eq i}(1/t_{j})=\sum_{j\
eq i}(1/\lambda_{j})\) for \(\forall j\
eq i\). However, it cannot be directly derived from Equation (6). Notice that in Equation (6), every \(\frac{1-\beta^{m+1}}{1-\beta}\frac{1}{\lambda_{j}}\) is sufficiently small (typically far smaller than 0.01 for non-saturated applications [29, 33]) and when expanding the polynomial \((1-\prod_{j\
eq i}(1-\frac{1-\beta^{m+1}}{1-\beta}\frac{1}{\lambda_{j}}))\), the second and higher order terms can be neglected without compromising the accuracy (e.g., the error when \(\frac{1-\beta^{m+1}}{1-\beta}\frac{1}{\lambda_{j}}=0.01\) and \(N=10\) is less than 0.005). As such, Equation (6) can be rewritten in order to \"isolate\" \(\lambda_{j}\) from other variables as follows:  
\[\beta\approx\frac{\frac{1-\beta^{m+1}}{1-\beta}\,\sum_{j\
eq i}(1/\lambda_{j}) \,(1+L)}{1+\frac{1-\beta^{m+1}}{1-\beta}\,\sum_{j\
eq i}(1/\lambda_{j})\,(1+ L)} \tag{7}\]  
From Equation (7), \(\sum_{j\
eq i}(1/t_{j})=\sum_{j\
eq i}(1/\lambda_{j})\) can be given as:  
\[\sum_{j\
eq i}(1/\lambda_{j})=g(\beta)=\frac{\beta}{(1-\beta^{m+1})(1+L)} \tag{8}\]From Equation (8), the aggregate transmission rate is explicitly related to the busy channel probability. Once \(\sum_{j\
eq i}(1/t_{j})\) is determined, estimation of the busy channel probability at the optimal operating point can be performed. In the following analysis, the superscript \(*\) is used to denote variables at the optimal operating point. The optimal transmission rate for node \(i\) can be expressed as \(1/{\lambda_{i}}^{*}=\Delta^{*}/t_{i}\), where \(\Delta^{*}\) is ratio of the optimal transmission rate \(1/{\lambda_{i}}^{*}\) to \(1/t_{i}\).