# A Delay-Aware and Reliable Data Aggregation for Cyber-Physical Sensing
## 4 Design of the TSR-DARDA Scheme
### Tsr-Darda Scheme
#### 4.2.1 Circular Tier and Cell Partitioning of the Surface of the Half Sphere  
For the envisioned half sphere as described in Section 3.1, the sink is located at the spherical vertex and assume nodes' interference radius is \(2r\). To ensure a clear statement, the three-dimensional circular tier and cell partitions are projected onto a two-dimensional plane from top to bottom as shown in Figure 4. The width of each tier is firstly calculated according to the constraint that the collecting nodes in the inner layer just finish data aggregation when the collecting nodes in the outer layer finish transmission to the parent collecting nodes in the inner layer. That means \(\tau_{k-1,n}=\tau_{k}\) and \(\tau_{k}=\tau_{k,n}+\tau_{k,g}\), where \(\tau_{k}\) represents the data aggregation time needed for tier \(k\) and it equals the sum of time required by non-collecting nodes aggregates data to collecting nodes, which denoted by \(\tau_{k,n}\), and time of collecting nodes aggregates data to parent collecting nodes in inner tier, which denoted by \(\tau_{k,g}\). \(\tau_{k-1,n}\) represents the time needed for tier \(k-1\) to finish non-collecting nodes data aggregation to collecting nodes in the same tier. Accordingly, the surface could be divided into circular tiers \(\{T_{k}\}\). Assume the width of the outmost tier \(T_{k}\) is \(\delta r\). That is \(d_{k}=\delta r\) (the value of \(\delta\) will be discussed in following Discussion section). It is partitioned into cells according to the width of \(r\) at the boundary of the inner tier as shown in Figure 4. The \(m\)-th cell in \(k\) tier is denoted by \(C_{k}^{m}\). According to the cell partition method described above, nodes from every three cells do not interfere with each other. Figure 3: The overall approach. We use \(d_{i}\) to denote the surface width of tier \(i\) (\(i=1,2,...,k\)), \(h_{i,w}\) and \(h_{i,n}\) respectively denote the vertebral distance from sink to the bottom transverse plane with radius of \(\gamma_{i,w}\) and to up the transverse plane with radius of \(\gamma_{i,n}\). Figure 5 illustrates the relationship among these parameters of tier \(T_{k}\). As shown in Figure 5:  
\[\gamma_{k\_n}=R\cos\frac{d_{k}}{R}=R\cos\frac{\delta r}{R}, \tag{4}\]  
and:  
\[\gamma_{k\_w}=R \tag{5}\]  
so, the number of cells in tier \(T_{k}\) is:  
\[num(T_{k})=num_{k}=\frac{2\pi\gamma_{T_{k},n}}{r}=\frac{2\pi R\cos\frac{\delta r }{R}}{r} \tag{6}\]  
Actually, it is an integer no smaller than \(num_{k}\). Because:  
\[h_{k\_n}=R-R\sin\frac{\delta r}{R} \tag{7}\]  
and:  
\[h_{k\_w}=R \tag{8}\]  
then, we can get the area of each cell is:  
\[s_{cell\_k}=\frac{r}{2\pi\gamma_{T\_n}}(2\pi\gamma_{T\_n}\cdot h_{k\_w}-2\pi \gamma_{T\_n}\cdot h_{k\_n}) \tag{9}\]  
Furthermore, a simplified formula is obtained as follows:  
Figure 4.0  
Finally, we can get:  
Figure 4.1  
Therefore, the number of nodes in each cell is \(\xi_{k}=\rho\cdot s_{cell\_k}\), where \(\rho\) denotes the node density. In addition, \(\mu\) denotes the maximum retransmission times as described in the system model. Accordingly, we can get the time slots required for collecting nodes in the outmost tier to finish data aggregation of the tier under the assumption that there is only one collecting node within each cell. The time  
Figure 4.2 Projection from the three-dimensional space to two-dimensional plane of tiers and cells partition. slot is \(\tau_{k\_n}=3\xi_{k}(1+\mu)\) considering the above assumption that the nodes' interference radius is \(2r\) so nodes from every three cells do not interfere with each other. The collecting nodes need to deliver the aggregated information to the collecting nodes in inner layer by broadcast and retransmission. Thus, the time for forwarding is \(\tau_{k\_g}=3(1+\mu)\), so the total time required for all nodes in the outmost tier to finish transmission to the inner layer is computed by the following formula:  
Figure 4.3  
For the inner tier \(T_{k-1}\), similar calculations can be done, so he following results can be obtained.