# TSCH and RPL Joining Time Model for Industrial Wireless Sensor Networks
## 4 TSCH Analytical Model  
The objective of the analytical model is to be able to correctly estimate the average time that the synchronization process can take for a new node to join the network, and in the same way to estimate the maximum time that the formation of a complete network will take. Furthermore, knowing the schedule, it is possible to estimate the consumption of the nodes, since each platform has an estimated consumption for the transmission/reception processes. The synchronization process will be affected by the following parameters:  
* EB transmission period (\(T_{EB}\)). This period marks the time when a TSCH beacon is generated. Its value does not have to coincide with the size of the slotframe, since this message is sent in the next active slot once the beacon has been generated. The beacon transmission period can improve the processes of synchronization and deployment of the network by following a dynamic adjustment of the period, as demonstrated by the bell-shaped pattern in article [25]. * Scanning period (\(T_{SCAN}\)). This period marks the time that each channel is scanned individually. Nodes that want to join the network listen to the same channel, configuring channels at random and repeating this process until an EB message is received by the node. * Number of neighbors (\(N\)). The number of nodes that are within the coverage area of the node that wants to join. Being within the coverage area, the node that joins the network will be able to receive EB messages from the N nodes. * Number of channels (\(C\)). This represents the subset of channels used, both in the normal operation of the network and in the scanning phase. There are a total of 16 channels in the 2.4 GHz band, but a subset can be selected to limit the channels used during normal network operation. This number of channels will affect the synchronization time as the new node must search a larger number of channels to find a beacon to synchronize with. * Packet Delivery Rate (\(PDR\)). This parameter represents the probability of success of receiving a packet correctly. Signaling messages may be lost due to a collision with another message or due to interference in the transmission channel. Table 1 summarizes these main variables along with other parameters that influence the synchronization process. Starting from a schedule where a slotframe of size \(SF_{SIZE}\) is configured, each node that is part of the network is assigned a guaranteed slot for beacon transmission thanks to the help of the autonomous scheduler Orchestra. This slot will be assigned from the node identifier, so that a maximum of \(SF_{SIZE}\) nodes can join the network using a guaranteed slot for the same channel offset. In this way, it is ensured that there are no collisions between beacon messages. Assuming that the scanning period \(T_{SCAN}\) is high enough (\(T_{SCAN}>>T_{EB}\)) not to change channels until a beacon is correctly received, we can calculate the average number of slotframes needed to correctly receive a beacon. If a node is listening on a fixed channel 100% of the time, at most it will be necessary to send \(C\) beacons if a different channel is