# A Delay-Aware and Reliable Data Aggregation for Cyber-Physical Sensing
## 4 Design of the TSR-DARDA Scheme
### Tsr-Darda Scheme
#### 4.2.2 Distributed In-Network Aggregation  
When the circular tier and cell partitioning is finished, collecting nodes collect the data sensed by non-collecting nodes in each cell. Besides, these collecting nodes serve the function of data aggregation and information forwarding to the sink. Figure 5: Illustration of parameters relationship for tiers and cells partition. Assume \(H(k,m)\) denotes the subset of nodes in tier \(T_{k}\) that could transmit data simultaneously, and \(|\cdot|\) represents the cardinality of the set. Let one node from every three cells to be included in \(H(k,m)\), i.e., \(H(k,m)\) has a node from cells \(C_{k}^{m},C_{k}^{m+3},\ldots,\) so on. Because any two nodes in \(H(k,m)\) are with distance more than \(2r\), their transmissions are without interference with each other. Furthermore, according to the cells partition in tier \(T_{k}\) mentioned in the above section, all cells have the same number of nodes (possibly except one cell, which may have a smaller number of nodes), so each \(H(k,m)\) has an identical number of nodes and the number of nodes would be about a third of the number of cells, namely, it would be about a third of the number of \(C_{k}^{m}\):  
\[|H(k,m)|=\left\lfloor\frac{1}{3}\left\lceil\frac{2\pi\gamma_{k,m}}{r}\right\rceil \right\rfloor\quad m=1,2,...,\hbar_{k} \tag{17}\]  
where, \(\hbar_{k}=\left\lceil\xi_{k}\right\rceil\). Since the wireless channels between two adjacent tiers are independent across links as described in Section 3.1, the subset of nodes \(H(k,m)\) in \(T_{k}\) could transmit data simultaneously with subsets in other tiers. Thus, we could ensure simultaneous node transmission by as many as possible. When the collecting nodes in the outer layer finish transmission to the parent collecting nodes in the inner layer and the collecting nodes in the inner layer just finish data aggregation, the collecting nodes in the inner layer go on forwarding the aggregated information together to their parent collecting nodes in a more inner layer until destination sink is reached. The TSR-DARDA scheme is implemented as shown in detail in Figure 6. As shown in Figure 6, the implementation of TSR-DARDA scheme consists of two stages of circular tier, cell partitioning and distributed in-network aggregation. ```
Input: An envisioned half sphere network with the radius of \(R\), the node density of \(\rho\), transmission range of \(r\) and nonzero packet loss reliability \(p\), retransmission times \(\mu\). Output: Data aggregation route Stage 1: Circular tiers and cells partition of the surface of the half sphere 1. According to Formula (16) the network is partitioned into tiers \(\left\{T_{k}^{*}\right\}\)(\(d\) = 1, 2, 3,..., \(k\)) with different width; 2. for each tier \(T_{k}\)
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Tier \(T_{k}\) is partitioned into cells \(\left\{C_{k}^{*}\right\}\) according to width \(r\) at the boundary of the inner tier;  Select a node from cells \(C_{k}^{*},C_{k}^{*-1},...,\) so on to construct \(H(d,m)\), that is \(\cup_{m=1}^{\hbar_{k}}H(d,m)\) = \(T_{k}\);  end for Stage 2: Distributed in-network aggregation (3) for\(d\) = \(k\) to 1 for\(m\) = 1 to \(\hbar_{k}\)  each non-collecting node \(i\) in \(H(d,m)\) unicasts its original or aggregated information (1 + \(\mu\)) times to its collecting node \(\lambda\) in the same cell. if collecting node \(\lambda_{i}\) receives all the packet from non-collecting nodes in the same cell, then node \(\lambda_{i}\) does aggregation and broadcasts the information (1 + \(\mu\)) times to its parent collecting nodes in inner tier. end if endfor  endfor (4) Output results;
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Figure 6: Implementation of the proposed TSR-DARDA scheme. **Remark. **_The tiered structure scheme is studied in [3] combining broadcast and unicast. However, it does not consider the careful design of tier width and parallel transmission, which play an important role in delay-aware data collection. Our proposed scheme takes these issues into consideration. The VWTSR scheme proposed in [27] was aimed at flat networks.