# Bell-X, An Opportunistic Time Synchronization Mechanism for Scheduled Wireless Sensor Networks
## 4 Problem Statement and Proposed Advertising Method
### Dynamic and Configurable Transmission Period Mechanism  
The proposed mechanism for the transmission of EB messages consists of a dynamic and configurable timer, whose temporal response resembles a stepped bell that is periodically repeated in time. Derived from the shape of the temporal representation of this beaconing timer, the mechanism is called Bell-X (with the X representing the tuneable parameter that defines the timer behaviour). In this mechanism, peak zones can be appreciated, in which the beacon transmission period is set to its maximum value, so that the consumption due to the transmission of advertising messages will be minimized. On the other hand, the timer will periodically go into valley areas to send more beacons during a certain time that can be preconfigured. Depending on the requirements of the application domain, there will be a compromise relationship between connection time and energy consumed. During this time in which the timer is reduced, a window is created in which the connection success probability of new nodes is favoured, while the peak areas allow for compensating the energy consumption that the sending of more messages requires. The behaviour of EB messages sent during the valley period could resemble the sending of message bursts to expedite the synchronization process. Between the zone of peak and valley, the transition is carried out in a staggered way, so that the beacon transmission period is doubling each time the timer set in each step expires, in the same way that the RPL Trickle Timer behaves. Figure 5 shows the temporal evolution of one possible configuration of the proposed mechanism, indicating the periods of valley, peak, and intermediate steps. As already mentioned above, this mechanism allows for configuring certain aspects of the bell's shape, so that desired behaviors can be configured depending on each type of application. Table 2 shows a list of the configuration parameters. The description of the operation of the proposed mechanism is as follows:  
1. Once the nodes are synchronized and connected to the topology, the dynamic mechanism starts the beacon transmission period by setting it to the value indicated by the Interval min parameter. 2. In addition to setting the beacon period, a timer is started with a duration equal to the value of the Interval min parameter multiplied by the value of the Valley Factor parameter. At this point, the beacon transmission period is set to its minimum value, allowing as many beacons as indicated in the Valley Factor parameter during the valley zone to be sent. 3. Once the configured timer has expired, the dynamic mechanism modifies the transmission beacon period by doubling the previous value of the beacon period. Again, a new timer is configured  
Figure 5: Example of a possible Bell-X configuration (Interval min: 4 —Doublings: 4 —Valley factor: 4 —Step factor: 4 —Peak factor: 8). while using the current value of the beacon period and the Step Factor parameter. This same process will be repeated as many times as the value of the beacon period is doubled. 4. When the beacon period is set to its maximum value, the timer used will be the current value of the beacon period multiplied by the Peak Factor parameter. The maximum value of the beacon period must be enough to maintain synchronization with the nodes that already belong to the network. This work [12] analyses the drift clock estimation that was used in TSCH networks, proposing an adaptive synchronization that obtains drifts estimation lower than 10 ppm in multi-hop networks. This allows configuration of transmission periods with values between 50 and 130 s, depending on the amplitude of this drift. 5. Once the time in which the beacon period is at its maximum has passed, the process will be reversed, which reduces the value of the beacon period until it reaches the valley area.