# 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. Based on the constraints outlined in R1-R4, a relationship between the number of nodes (N) and number of rings (\(Nr\)) can be established, as shown in Table 2. This table outlines the minimum and maximum number of nodes required to vary the ring numbers, thereby facilitating the derivation of Equation (1). Conversely, if the number of network nodes (N) can be obtained, Equation (1) can be used to calculate the potential number of rings (\(Nr\)) formed. \[9\times N_{r}-3\leq N\leq 18\times N_{r}+3 \tag{1}\]  
The derivation process of Equation (1) can be categorized into scenarios with minimum and maximum numbers of nodes, as explained below. Each basic ring member has six nodes comprising three masters and three intra-ring bridges. To achieve the minimum number of basic members, three B_inter bridges must be added between the rings. In addition, each piconet can accommodate additional slave members. For the innermost and outermost rings, where the masters are connected to two B_intra and one B_inter, respectively, considering that each piconet can connect up to seven  
Figure 1: An example of four interconnected triangular ring networks. Figure 2: The packet format for piconet and scatternet addressing. nodes, these rings can accommodate four slave members each, totaling eight slaves. For the remaining rings, excluding the innermost and outermost rings, the masters were connected to two B_intra and two B_inter bridges, allowing each piconet to include up to three slave members. The total number of slaves for these rings was \((N_{r-2})\times 3\times 3\). Adding these numbers to the minimum number of basic members yields the maximum node capacity, which is the sum of these numbers: \(18\times N_{r}+3\). Based on Table 2 and Formula (1), the networks with the same number of nodes N can exhibit varying numbers of rings \(Nr\), which could be m \(-1\), m, or m + 1. For example, if N = 33, the network can potentially produce two, three, or four rings. In such a scenario, selecting a network configuration with the minimum number of rings is preferable. This decision was guided by the principle that a higher number of rings implies more piconets, leading to increased operational costs. Therefore, the least number of rings were selected as the most efficient configuration for MTRSRP. Case 2: One B_INTRA BRIDGE CONNECTS TWO Piconets WITHIN A RING, Whereas Two B_INTER BRIDGE CONNECT PICONETS ACROSS THE RINGS. Following constraints R1-R4, Table 3 outlines the relationship between the number of nodes and number of rings for Case 2. Additionally, using Table 3, Equation (2) can be deduced. The derivation of Equation (2) is as follows. The initial ring contains six basic members, including three masters and three intra-ring bridges. With each additional ring, the number of basic members increases by 12, comprising three masters, three B_intra bridges, and six B_inter bridges between rings. Thus, the minimum lower limit of the basic members was \(12\times N_{r}-6\). In addition, each piconet can include additional slave members. For the masters on the innermost and outermost rings already connected to two B_intra and two B_inter bridges, each piconet can add up to three slave members, totaling \(3\times 3\times 3\) slave members. Except for the innermost and outermost rings, the piconets on the other rings are connected to two B_intra and four B_inter bridges, allowing each piconet to add at most one slave member, totaling \((N_{r}-2)\times 1\times 3\) slaves. Adding these to the minimum basic \begin{table}
\begin{tabular}{c c} \hline \hline
**Number of Nodes** & **Number of Rings** \\ \hline
6–21 & 1 \\ \hline
15–39 & 2 \\ \hline
24–57 & 3 \\ \hline
33–75 & 4 \\ \hline
3. \(\ldots\ldots\) & \(\ldots\ldots\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relationship between the number of nodes N and number of rings \(Nr\) for Case 1.