## 4 Numerical Analysis with Results
### Delay Analysis  
Transmission delay of a node is the duration, when a node has data to transmit till its successful transmission to the PAN coordinator. Suppose, a node \(j\) needs to send \(L\) amount of data just at the beginning of the beacon frame, its delay in \(ESS_{ADC}\) (\(D_{ESS_{ADC}}\)) is calculated as:  
\[D_{ESS_{ADC}}^{j}=BI_{ESS}+(\sum_{b=1}^{b=j}K_{b}\times t_{ESS_{ADC}}) \tag{25}\]  
Here,  
\[t_{ESS_{ADC}}(sec)=4.8\times 10^{-4}\times 2^{SO}\]  
and \(K_{b-1}\) describes the total slots assigned to node \(j\) and previously allocated slots to the preceding nodes. ESS follows the same formula in finding its delay, however its SO is fixed whereas, \(ESS_{ADC}\) adapts its SO and BO according to the data requirement. Delay of ESS and \(ESS_{ADC}\) will only be same,  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Number of Nodes & 21 \\ Network Size & 100 m \(\times\) 100 m \\ Data Rate & 250 Kbps \\ Offered Load/node (Bytes) & 20:20:300 \\ Superframe Order in ESS and 802.15.4 & 2:1:4 \\ Superframe Order (initial) in \(ES_{ADC}\) & 0 \\ Beacon Order in ESS and 802.15.4 & 3:1:5 \\ Beacon Order (initial) in \(ES_{ADC}\) & 0 \\ Duty Cycle in 802.15.4 and ESS & 50\% \\ Duty Cycle in \(ESS_{ADC}\) & 50\% \\ GTS Duration in 802.15.4 (sec) & \(9.6\times 10^{-4}\times 2^{SO}\) \\ GTS Duration in \(ESS_{ADC}\) and ESS (sec) & \(4.8\times 10^{-4}\times 2^{SO}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation Parameters.  
when both have same values of SO and BO. If \(Y\) nodes are successfully scrutinized for GTS allocation, then accumulated delay of the PAN in \(ESS_{ADC}\) (\(D_{ESS_{ADC}}^{max}\)) is computed as:  
\[D_{ESS_{ADC}}^{max}=\sum_{i=1}^{i=Y}[BI_{ESS_{i}}+(\sum_{b=1}^{b=i}K_{b}\times t _{ESS_{ADC}})] \tag{26}\]  
However, in IEEE 802.15.4 standard, delay of the same node \(j\) in transmitting its data (\(D_{15.4}\)) is calculated as:  
\[D_{15.4}=BI+SD-(\sum_{b=1}^{b=j}K_{b-1}\times t_{15.4}) \tag{27}\]  
Here,  
\[t_{15.4}=9.6\times 10^{-4}\times 2^{SO} \tag{28}\]  
If \(X\) nodes are successfully scrutinized for GTS allocation, then accumulated network delay of the PAN in IEEE802.15.4 standard (\(D_{15.4}^{max}\)) is determined as:  
\[D_{15.4}^{max}=\sum_{i=1}^{i=X}[(BI+SD)_{i}-\sum_{x=1}^{x=i}K_{x-1}\times t_{ 15.4}] \tag{29}\]  
Figure 6 comprises of 4 sub-figures that shows delay comparison of \(ESS_{ADC}\) with \(ESS\) and IEEE 802.15.4 standard for four different values of \(BO\) such as 4, 5, 6, and 7. In each subplot, accumulated delay of nodes between all three schemes is evaluated for all possible values of \(SO\) against each \(BO\), when each node has 20 bytes of data. The results verified that accumulated delay in \(ESS_{ADC}\) is less than \(ESS\) and IEEE802.15.4 standard. This significant different is due to fixed \(SO\) and \(BO\) values in \(ESS\) and the standard. However, \(ESS_{ADC}\) allows PAN coordinator to adapt its duty cycle to accommodate GTS requests with minimum \(SO\) and \(BO\) values. Consequently, resulting in a significant decrease in the accumulated delay as compared to other two.  
The results shown in Figure 7 further verifies that accumulated delay of all nodes within the data range of 25 to 50 bytes are significantly less in the proposed \(ESS_{ADC}\) as compared to the other two schemes when they both have SO = 3 and BO = 4.  
Figure 6: Accumulated delay comparison against all ranges of 4 different BO values.