## 4 Calculation of the Average Idle Time  
Based on the above Equation (15), we will calculate the average idle time from the perspective of observer node B, as shown in Figure 3. Note that the value \(k\) of the backoff counter decreases in units of 1 when node B detects that the channel is idle, with a probability of \(1-p_{b}\). The counter is frozen when node B detects that the channel is busy with the probability, \(p_{b}\), and only when the counter's value is decremented from \(W_{i}-1\) to 0 will the node successfully transmit with the probability, \(p_{s}\). When the backoff stage reaches the maximum (or retransmission limit), the data frame still fails to transmit, i.e., \(i>m\), and the frame will be discarded immediately. Therefore, the following correlation probabilities were found:  
* The collision probability, \(p\), i.e., the probability that at least one of the remaining \(n-1\) competing nodes at a time slot sends a data frame, can be found as \(p=1-(1-\tau)^{n-1}\);
* The probability \(p_{b}\) that the channel is busy, i.e., the probability that at least one node transmits at a time slot, can be found as \(p_{b}=1-(1-\tau)^{n}\);
* Since the channel has no defects, the total collision probability, \(p_{c}\), is only related to the busy channel caused by the data frame transmission and data frames' collision, and it can be expressed as \(p_{c}=1-(1-p_{b})(1-p)\);
* The successful transmission probability, \(p_{s}\), i.e., under the condition that at least one node of the \(n\) nodes transmits at a time slot, the probability that exactly one node transmits successfully can be expressed as \(p_{s}=\frac{n\pi(1-\tau)^{n-1}}{1-(1-\tau)^{n}}\).  
Based on the above analysis, we note that the time interval between the two consecutive successful transmissions of node B (\(k\)-th and \((k+1)\)-th in Figure 3) may include events, such as failure of sending data frames by node B, the other \(n-1\) nodes successfully send data frames or fail to send data frames when B carries out the backoff procedure, and the channel is idle. Thus, the average idle time, \(\overline{BO}\), experienced by node B between two consecutive successful transmissions is defined as the average time that the ICS procedure was performed and the average time that the backoff counter was active during the backoff procedure. From this, it can be determined that the expression of \(\overline{BO}\) is:  
\[\begin{array}{l}\overline{BO}=\frac{1-p_{b}}{1-p_{c}^{2}}D+p_{b}\frac{(1-p_{ b})^{\frac{D}{2}}}{1-p_{c}^{2}}\frac{D}{2}x_{0}+\frac{1-p_{c}}{1-p_{c}^{2}}(x_{0}+x_{ 1})+p_{c}\frac{1-p_{c}}{1-p_{c}^{2}}(x_{1}+x_{2})\\ +p_{c}^{2}\frac{1-p_{c}}{1-p_{c}^{2}}(x_{1}+x_{2}+x_{3})+\ldots+p_{c}^{m-1} \frac{1-p_{c}}{1-p_{c}^{2}}(x_{1}+x_{2}+\ldots+x_{m}).\end{array} \tag{16}\]  
The first term of Equation (16) is the average idle time to complete the ICS procedure, denoted as \(T_{-1}\). The second item is the average idle time that the node may traverse, which performs the backoff procedure after the channel is sensed to be busy during the ICS procedure. Note that \(\sigma\in[0,D]\) follows a uniform distribution, with a mean of \(D/2\). Therefore, starting from the second term, it describes the average idle time of the backoff procedure, denoted as \(T_{bt}\), where \(x\) is a random variable subject to a uniform distribution and is the original length of the backoff counter at each retransmission attempt, and \(x_{i}\) and \(x_{j}\) are independent of each other.  
Let the probability generation function (PGF) of \(\overline{BO}\) be \(G_{\overline{BO}}\), and use the properties of PGF to obtain:  
\[G_{\overline{BO}}(z)=G_{T_{-1}}(z)+G_{T_{bt}}(z)=G_{T_{-1}}(z)+\frac{1-z^{W_{ 0}}}{(1-z)W_{0}}+\prod_{i=0}^{m-1}\frac{1-z^{2i}a_{i}W_{0}}{2^{i}W_{0}(1-z^{a_ {i}})}. \tag{17}\]  
where \(a_{i}=p_{c}^{i}\frac{1-p_{c}^{m-i}}{1-p_{c}^{2}}\). The above equation will be used to derive our modeling.  
Figure 3: Possible events that occurred between two consecutive successful transmissions for node B.