# Multihop Latency Model for Industrial Wireless Sensor Networks Based on Interfering Nodes
## 3 DAO Multi-Hop Analytical Model  
The time it takes for the new node to form the downstream routes was then modelled by sending the DAO message. This message originates from the node that has just joined the RPL topology after receiving a DIO message and must be transmitted up to the RPL  
Figure 3: Comparison between single-hop and multi-hop exchange of control messages. Figure 2: Example of transmission of DIO messages based on the Trickle Timer. topology coordinator via the default route. In this way, all nodes in the path add that route to the new node. Table 1 shows the different parameters that influence the process of sending the DAO message. These parameters have been taken into account when defining the analytical model, which allows the time it takes for each node to send this message to the root to be characterized. Taking into account the probability of successful transmission using the Packet Delivery Rate (\(PDR\)) parameter, it is possible to determine the time it takes to send a DAO message in ideal situations on a single-hop link. In the best case, this message will be transmitted in the next available slotframe, whose time is determined by the variable \(SF_{RPL}\), which represents the number of timeslots dedicated to the RPL slotframe. Since the time at which the DAO message is generated can occur at any point in the slotframe, this time is randomly distributed between 0 and \(SF_{RPL}\), so that on average the time it takes to transmit this message on a single-hop link will be \(SF_{RPL}/2\), with a probability determined by the parameter \(PDR\). In the case that the message is lost, it will be sent in the next slotframe, with a probability (\(1-PDR\)). It is possible to resend the message up to four times. The time taken to send the DAO message, \(t_{pdr}^{DAO}(k)\), will have a different behaviour in the first hop with respect to the rest, since the DAO can be generated at any instant within the slotframe, while the retransmissions will always be carried out in the next slotframe. This behaviour is characterized by the parameter \(k\), which will have a value of 1 in the first hop and 0 for the rest of the cases. \[t_{pdr}^{DAO}(k)=\sum_{i=0}^{3}\left(SF_{RPL}\cdot i+\frac{SF_{RPL}}{2^{k}}.PDR \right)\cdot\left(1-PDR\right)^{i} \tag{2}\]  
Using as an example the case in which \(k=1\), Equation (2) would be represented as shown below,  
\[t_{pdr}^{DAO}(1)=\underbrace{\frac{SF_{RPL}}{2}.PDR}_{\text{First attempt ($i=0$)}}+\underbrace{\left(SF_{RPL}+\frac{SF_{RPL}}{2}.PDR\right)\cdot\left(1- PDR\right)}_{\text{Second attempt ($i=1$)}}+\cdots \tag{3}\]  
where the first operand represents the first attempt to send the DAO message and the following are the forwarding messages if a collision occurs. This parameter only indicates the time to send it over a single hop link, so to reach the RPL root it will be necessary to multiply it by the number of hops to the root node. However, due to the configuration of the Orchestra slotframe [13] dedicated to RPL control traffic, DAO messages are sent through a single shared slot, so they may collide with other types of messages such as DIO, which will be sent by all nodes within the coverage area following the pattern set by the Trickle Timer. In this way, in each hop the node that has to resend the message will have a series of interfering nodes within its coverage area, which do not have to coincide with the interferers of another hop. To model this behaviour, a vector of inter-hop nodes has been defined, where the size of the vector indicates the