# MTRSRP: Joint Design of Multi-Triangular Ring and Self-Routing Protocol for BLE Networks
## 4 The Proposed MTRSRP
### Multi-Triangular Ring Formation
#### Case 1: Only One B_inter and One B_intra Bridge exists between two piconets.
\begin{table}
\begin{tabular}{c c} \hline \hline
**Number of Nodes** & **Number of Rings** \\ \hline
6–21 & 1 \\ \hline
18–36 & 2 \\ \hline
30–51 & 3 \\ \hline
42–66 & 4 \\ \hline
54–81 & 5 \\ \hline \(\ldots\ldots\) & \(\ldots\ldots\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Relationship between the number of nodes N and number of rings \(Nr\) for case 2.  
members yielded the maximum limit of the nodes that could be accommodated, which was \(16\times N_{r}+6\) nodes. Figure 3 illustrates an example of the two interconnected triangular ring networks with dual inter-bridges in Case 2. \[12\times N_{r}-6\leq N\leq 15\times N_{r}+6 \tag{2}\]  
_Case 3: Two B_intra Bridges interconnect two Piconets within a ring, and only one B_inter Bridge connects Piconets across the rings._  
Based on the above constraints, the data in Table 3 can be derived, which outline the relationship between the number of nodes and number of rings. In addition, Equation (3) can be derived from the data presented in Table 4. The relationship between the number of nodes (\(N\)) and number of rings (\(N_{r}\)) for Case 3 is:  
\[12\times N_{r}-3\leq N\leq 15\times N_{r}+3 \tag{3}\]  
The derivation process of Equation (3) can be derived as follows. Each basic member in a ring comprises nine nodes with three masters, six B_intra bridges, and three B_inter bridges between the rings. This provides the minimum basic membership of \(12\times N_{r}-3\). Each piconet can accommodate additional slave members. The masters in the innermost \begin{table}
\begin{tabular}{c c} \hline \hline
**Number of Nodes** & **Number of Rings** \\ \hline
9–18 & 1 \\ \hline
21–33 & 2 \\ \hline
33–48 & 3 \\ \hline
45–63 & 4 \\ \hline
57–78 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Relationship between the number of nodes (\(N\)) and number of rings (\(N_{r}\)) for Case 3. Figure 3: An example of two interconnected triangular rings with dual inter-bridges for case 2 networks. and outermost rings are already connected to four B_intra and one B_inter bridge, allowing each piconet to add two slave members, totaling \(2\times 3\times 3\). For rings other than the innermost and outermost ones, the piconets are connected to four B_intra and two B_inter bridges, allowing each piconet to add at most one more slave member, totaling \((N_{r}-2)\times 1\times 3\) slaves. Adding these to the minimum basic members yields the maximum limit of the nodes that can be accommodated, which is \(5\times N_{r}+3\) nodes. An example of two interconnected triangular ring networks with dual intra-bridges in Case 3 is shown in Figure 4.  
The operational scenario discussed in this study assumes that all nodes are within the transmission range. In such an environment, node mobility has a relatively minor impact on the network architecture because significant issues are avoided if the nodes remain within the communication range. To simplify this problem, the next step involves determining a method for establishing the desired network architecture. The process of creating a multi-ring interconnected network is divided into two main phases: (1) leader selection, which focuses on choosing a coordinator and providing them with information such as timing, addresses, and other relevant details about all nodes within the network, and (2) a scatternet formation phase, which utilizes the information provided by the coordinator, along with the three identified cases, to establish connections between nodes and facilitate packet transmission. (1) The leader election phase:  
In the initial stage of forming a Bluetooth network, establishing point-to-point connections requires neighbor node discovery and information exchange. BLE devices accomplish this by exchanging broadcast messages to collect the IDs and numbers of neighboring nodes. When nodes receive broadcast messages from their neighbors, they update their neighbor tables according to the number of neighbors. This study employs this process to select a coordinator using a competitive method and achieve information aggregation.