# Enhanced Beacons Dynamic Transmission over TSCH
## 4 Proposed Enhanced Beacons Dynamic Transmission Mechanism  
In [31], the contribution consists of a theoretical model to estimate the average time required for a node to associate with a synchronizer. This depends on the available channels and the average number of EBs transmitted until the association is achieved. The theoretical analysis is based on demonstrating that both the channel used by a synchronizer to send the _jth_ EB and the channel being scanned by the join-seeker when the _jth_ EB is sent are independent and identically distributed random variables. Results in [31] show that the average association time depends on \(m\) and the average time between EB transmissions. There is a trade-off between the average association time, the time between EB transmissions, and the number of channels used for communication. The use of fewer channels reduces the association time, but also the resilience of the network in the presence of interference. On the other hand, for a fixed number of channels, sending EBs more frequently reduces association time. However, when the association process ends, the frequent EB transmissions cause unnecessary energy consumption in the nodes. Additionally, a high EB transmission rate increases network traffic, which could limit slot availability in the schedule and communications between other network devices in networks with high node density. This enhanced beacons dynamic transmission over TSCH (EBDT-TSCH) algorithm aims to make dynamic _Teb_ in two phases: the intensive phase and the predetermined phase. The predetermined phase has a transmission period of EBs, upper-bounded by _Teb_ similarly to the 6 TiSCH. The intensive phase, on the other hand, presents a period of intensive transmission of EBs in which the maximum time between EBs represents a fraction of _Teb_, so the EBs transmission period in the intensive phase is \(Tebi=\alpha\)_Teb_, where \(0<\alpha<1\). This algorithm intends to use the intensive phase to decrease the network formation time, whereas the predetermined phase is used to avoid excess power consumption after network formation. A synchronizer running EBDT-TSCH starts in the intensive phase to associate, in a short time, as many nodes within its reach as possible. As part of the EBDT-TSCH implementation, a threshold parameter (\(u\)) delimits the end of the intensive phase and the beginning of the predetermined phase. The \(u\) parameter constitutes the number of EBs that will be sent during the intensive phase and is computed as \(u=\lfloor\beta m\rfloor\), whereas the previously discussed \(m\) is the number of available network channels and \(\beta\in\Re,\beta\geqslant 0\). After \(u\) EBs have been sent, the predetermined phase begins, in which the synchronizer reconfigures the maximum EB transmission period to _Teb_. Let us denote \(t_{i}\sim U(\rho\textit{Tebi},Tebi)\) and \(t_{p}\sim U(\rho\textit{Teb},Teb)\)\(0<\rho<1\), as the time between successive EB transmission in the intensive phase and the predetermined phase, respectively. Algorithm 1 shows the steps to be followed by a synchronizing node that implements EBDT-TSCH. Figure 3 illustrates an example of the timeline of EB transmissions performed by a synchronizing node following the EBDT-TSCH algorithm. In this example, \(m=4\), \(\alpha=0.5\), \(\rho=0.75\), and \(\beta=1\) are considered. In this way, four EBs are sent (\(u=4\)) in the intensive phase, and the time between them (\(t_{i}\)) follows a uniform distribution and maximum period equal to \(Tebi=0.5\textit{Teb}\). After that, the predetermined phase starts, and the time between successive EBs is also uniformly distributed, but with maximum value _Teb_. The proposed EBDT-TSCH does not imply a significant increase in implementation complexity and can be used in any network that uses the TSCH association mechanism. Achieving rapid association is particularly important in TSCH-based solutions that consider mobility in the network nodes.