## 4 Control Packet Collision
### Collision Avoidance with Proactive Interval Setting  
Analytical derivation of the probability of control packet collision  
In analyzing the probability of control packet collision, we introduce the following assumptions. * All nodes possess complete information about the network topology and contain a static routing table based on this information. Here, we use the topology shown in Figure 8, where node \(R\) is a sink node. Thus, the forward node of node \(A\) is node \(R\), and its sideward nodes are node \(B\) and node \(C\). * Each sensor node generates a data packet in accordance to a Poisson process with intensity \(\lambda\), and subsequently sends the data to the sink node. In addition, when nodes forward data, they always select forward nodes, and any forward node is equally likely to be chosen as the receiver. * When packet collisions occur, the receiver of the packets always discards all packets involved in the collision. * Each node sends ID packets at a regular intermittent interval denoted as \(T\). Moreover, all nodes perform the \"clear channel assessment\" (CCA) procedure when sending any type of packet. Neither ID packets nor SREQ packets are transmitted if the CCA has indicated that the wireless channel is busy. If the wireless channel is idle, nodes transmit an ID or an SREQ packet after a random backoff period of time. After it is ensured that receivers can obtain SREQ packets correctly, collisions between data, RACK and DACK packets and other packets occur less frequently. However, if a collision occurs, a receiver must wait for the following ID packet, which increases the total amount of time spent by the affected sender node in waiting for an ID packet. Therefore, data packets, RACK packets and DACK packets are transmitted by using binary exponential backoff in order to prevent collisions with other packets (especially ID packets). From the above assumptions, we can calculate \(G(R)\), which is the approximate average number of data packets received by node \(R\) in one second. \(G(R)\) depends on the number of backward nodes for node \(R\) and its traffic load. Here, we define \(N_{b}(R)\) as the set of backward nodes of node \(R\) and \(|N_{f}(n)|\) as the number of forward nodes of node \(n\). The probability with which a node (denoted as \(n\)) selects node \(R\) as its receiver is \(\frac{1}{|N_{f}(n)|}\), and therefore \(G(R)\) is expressed as follows:  
\[G(R)=\sum_{n\in N_{b}(R)}\frac{1}{|N_{f}(n)|}\{G(n)+\lambda\}. \tag{1}\]  
SREQ collisions occur when two or more neighboring nodes send SREQ packets simultaneously. We assume that all nodes use the CSMA/CA mechanism, which can reduce the number of SREQ collisions. However, SREQ collisions can still occur, unless there are no hidden nodes, since SREQ packets can be returned at once. In the CSMA/CA mechanism with exponential backoff, the number of time slots chosen at random by each node is \(2^{BE}\), where \(BE\) is a moderate integer value. If the wireless channel is idle, the sender node transmits an SREQ packet (or an ID packet) after a CCA and a random backoff period, as described in Section 3.1. In this regard, a time slot with a range of \(2^{BE}\) is utilized for the random backoff period. Here, we assume that node \(R\) receives the same number of data packets from each of its backward nodes. Therefore, the probability with which a node returns an SREQ packet upon receiving an appropriate ID can be expressed as \(1-e^{-G_{b}(R)T}\), where \(G_{b}(R)\) is \(\frac{G(R)}{|N_{b}(R)|}\). Furthermore, the probability with which the node does not return an SREQ packet can also be expressed as \(e^{-G_{b}(R)T}\). \(P_{SREQ}\), which is the probability with which SREQ collisions occur, is also the probability with which at least two neighboring nodes of node \(R\) receive a data packet. However, the CSMA/CA mechanism cannot avoid SREQ collisions when node \(R\) sends an ID packet.