## 4 Proposed Adaptive Scheduling Technique
\[B_{Tkl}=\lfloor\ log_{2}D_{Tk}\ \rfloor+1 \tag{4}\]  
Node \(k\) allocates Tx cells using its own MAC address along with address of the receiver node \(l\), and allocates Rx cells based on its own MAC address and those of its child nodes. The sender node \(k\) communicates the number of scheduled Tx cells to the receiver node \(l\) via a reserved field in the packet header. To minimize end-to-end latency, each node allocates Rx cells within one segment and Tx cells in the immediately following segment. If the hop count of the sender node \(k\) is \(h\), the Tx cells are allocated in the \((H-h)^{th}\) segment of the slotframe. Figure 4: Proposed adaptive and autonomous scheduling slotframe structure using segmentation scheme for 6T3 CH networks. Figure 3: 6 TiSCH data collection mesh network represented as an RPL tree with varying hop distances and traffic loads. Let \(n_{h}\) denote the node density at hop distance, \(h\). The total number of cells, \(B_{h}\), allocated for all links between the sender nodes, \(n_{h}\), and their respective parent nodes, \(l\), within the corresponding segment is given by  
\[B_{h}=\sum_{\forall h\in m_{h}}B_{Tkl} \tag{5}\]  
This approach adapts the number of scheduled cells based on the pending packets in a node's queue and the corresponding link-layer PDR. However, the number of allocated cells increases logarithmically rather than linearly, thereby preventing overprovisioning and reducing contention. Furthermore, to minimize end-to-end latency and improve slotframe utilization, we propose the use of variable-length slotframe segments for links at different hop levels. By utilizing available channel offsets, \(C_{nn}\), the minimum size of a slotframe segment is determined by ensuring sufficient allocation of Tx/Rx cells for nodes at hop distance, \(h\), as given by  
\[L_{hs_{min}}\geq\left\lceil\frac{n_{h}}{C_{nn}}+log_{2}(H-h+1)\right\rceil \tag{6}\]  
A larger slotframe segment is assigned to RPL levels with higher node density. Moreover, nodes located closer to the root typically handle greater traffic than other nodes, often resulting in buffer overflows. To mitigate this, segment sizes are increased for nodes nearer to the root. Taking into account the higher traffic at lower hop distances, the segment size is computed as the ceiling value derived from node density, the number of available channel offsets, and the logarithm of the hop distance. On the contrary, links at higher hop distances are assigned smaller segment lengths. The minimum slotframe length, \(SF_{min}\), is determined as the sum of the lengths of all segments. \[SF_{min}=\sum_{h=1}^{H}L_{hs_{min}} \tag{7}\]  
The time offset for each cell is determined using a hash function applied to the addresses of the sender and receiver. According to (4), \(B_{Tkl}\) cells are scheduled for the link \((k\to l)\), where the time offset of the \(i^{th}\) cell within a segment of size \(L_{hs}\) given by  
\[to_{kl}(i)= \begin{cases}to+\sum_{d=(h+1)}^{H}L_{ds},&h=1\;to\;H-1\\ \;to\;,&h=H\end{cases}  \tag{8}\]  
where \(to=hash[(k_{MAC}+i)+l_{MAC}]\%L_{hs}\) for \(i=0,\;1,\ldots\ldots,(B_{Tkl}-1)\), \(k_{MAC}\) and \(l_{MAC}\) are MAC addresses of node \(k\) and \(l\), respectively. The hash function used to determine the time offset ensures that cell allocation is uniformly distributed across the segment by combining sender and receiver MAC addresses along with segment length. This uniform distribution minimizes the collision among multiple nodes within the same segment. Additionally, since the hash function depends on unique MAC pairs, it avoids overlap across dissimilar links. The use of a modulo operator with the segment length ensures that cell allocation remains within the same segment. As derived in (5), the proposed AAS strategy schedules \(B_{h}\) unique cells within the \((H-h)^{th}\) segment for all links connecting sender nodes at hop distance \(h\) to their respective parent nodes.