## 4 Control Packet Collision
### Collision Avoidance with Proactive Interval Setting
Thus, \(P_{SREQ}\) can be calculated as follows:  
\[P_{SREQ}=1-\sum_{k=0}^{|N_{b}(R)|}C(R,k)e^{-(|N_{b}(R)|-k)G_{b}(R)T}(1-e^{-G_{b }(R)T})^{k}, \tag{2}\]  
where \(C(R,k)\) indicates the number of combinations of \(k\) different nodes out of \(N_{b}(R)\), which addresses the hidden node problem under CSMA/CA. Here, we consider only the case where \(k\) is less than three because the term \(e^{-(|N_{b}(R)|-k)G_{b}(R)T}(1-e^{-G_{b}(R)T})^{k}\) is exceedingly small and can be ignored for large \(k\). \(C(R,k)\) is defined as follows:  
\[C(R,k)=\left\{ \begin{array}{ll}1&(k=0)\\ |N_{b}(R)|&(k=1)\\ \frac{2^{BE}-1}{2^{BE}}h(R)&(k=2),\end{array} \right. \tag{3}\]  
where \(h(R)\) is the number of couples of nodes out of \(N_{b}(R)\) in relation to the number of hidden nodes. Next, we target collisions of ID packets at node \(R\). A collision of ID packets occurs when ID packets are sent by the neighbors of node \(R\) while node \(R\) is receiving an SREQ or a data packet. Note that it is not necessary to consider the backoff time slot of CSMA/CA as discussed in \(P_{SREQ}\) since ID packets are rarely transmitted simultaneously by multiple nodes. Here, we define \(H(R)\) as the average number  
Figure 8: Simple network model. of hidden nodes for node \(R\) for the time when node \(R\) is receiving SREQ or data packets. \(H(R)\) is represented as follows:  
\[H(R)=\frac{1}{|N_{a}(R)|}\ \sum_{n\in N_{a}(R)}h(R,n), \tag{4}\]  
where \(N_{a}(R)\) is the set of adjacent nodes for node \(R\), \(|N_{a}(R)|\) is the number of elements of \(N_{a}(R)\) and \(h(R,n)\) is the number of hidden nodes for node \(n\) included in \(N_{a}(R)\). The average interval for receiving ID packets while node \(R\) is receiving SREQ or data packets can be computed as \(\frac{T}{H(R)}\) because \(H(R)\) nodes can send ID packets even while node \(R\) is receiving other packets. Here, we define \(T_{r}\) as the reception time for SREQ and data packets, in which case the probability of ID collisions, denoted as \(P_{ID}\), is expressed as follows:  
\[P_{ID}=\frac{T_{r}H(R)}{T}. \tag{5}\]  
Procedure for determining the proper transmission interval  
In order to determine the proper transmission interval, we modify Equation (2). Equation (2) shows the probability with which an SREQ collision occurs when an ID packet is sent by node \(R\), and Equation (5) shows the probability with which an ID collision occurs when node \(R\) receives an SREQ or a data packet. Therefore, we introduce \(P^{\prime}_{\textit{SREQ}}\) (the product of \(P_{\textit{SREQ}}\) and \((G(R)T)^{-1}\)), which corresponds to the probability with which an SREQ collision occurs when receiving an SREQ or a data packet (Equation (6)). \[P^{\prime}_{\textit{SREQ}}=\frac{1-\sum_{k=0}^{2}C(R,k)e^{-(2-k)G_{b}(R)T}(1-e^ {-G_{b}(R)T})^{k}}{G(R)T}. \tag{6}\]  
Then, we can obtain \(T^{*}\) by minimizing \(P_{\textit{CTRL}}\), which is the probability of control packet collisions, as follows:  
\[P_{\textit{CTRL}}=P^{\prime}_{\textit{SREQ}}+P_{ID}. \tag{7}\]  
Unfortunately, an explicit expression of \(T^{*}\) which minimizes Equation (7) cannot be given; instead, we can compute the approximate value of \(T^{*}\) by calculating the minimum value of the sum and subsequently computing \(T^{*}\) every 10 ms in the semi-open interval (0.0 s, 2.0 s]. Figure 9 shows the results of the analysis and simulation of control packet collisions for the network topology shown in Figure 8, where \(\lambda=0.024\), \(BE=3\) and the error bar corresponds to the 95% confidence interval. From the results shown in Figure 9, it can be concluded that the analysis and the simulation of both \(P_{ID}\) and \(P_{\textit{SREQ}}\) correspond rather well, which indicates that our analysis is correct. However, for \(P_{\textit{SREQ}}\), as the intermittent interval becomes longer, the simulation results indicated superior performance than the analytical results due to the assumption that CSMA/CA can always prevent packet collisions, except in the presence of hidden nodes.