## 3 O-MAC Protocol Design Goals and Architecture
### Interior Communication
#### 3.1.2 Preamble Tradeoff Validation Model
\begin{table}
\begin{tabular}{c c} \hline
**Notation** & **Description/Definition** \\ \hline \((2^{\gamma}-1)\) & Total number of slot waits \\ \(\beta\) & Required time for sending each symbol \\ \(c\) & Congestion \\ \(C_{at}\) & Channel access time \\ \(D_{p}\) & Data payload \\ \(F_{dil}\) & Frame delimiter used for giving the signal of the preamble \\ \(F_{l}\) & Frame length \\ \(M_{pdu}\) & Medium access control protocol’s data unit length \\ \(P_{l}\) & Packet length \\ \(P_{max}\) & Maximum number of packets \\ \(T_{retry}\) & Retry transmission \\ \(T_{f}\) & Data frame transfer time \\ \(T_{ti}\) & Total time in the ideal scenario \\ \(T_{time}\) & Total time spent in the worst-case scenario \\ \(T_{s\Delta}\) & Total symbol wait time for the acknowledgement \\ \(T_{tw}\) & Total time in the worst-case scenario \\ \(T_{turn}\) & Processing time \\ \(\omega\) & Number of symbols \\ \(\Delta R\) & Standard transfer rate \\ \(\Delta\varphi\) & Initial back-off time used to give space to other nodes to access the channel \\ \hline \end{tabular}
\end{table} 
Table 2: Description of the notations used in the ODFF. We analytically proved in the previous section that use of a short preamble consumes less time for several parameters; here, we validate the energy consumed by a short preamble and by the sending and receiving of data over the channel. The energy consumed for the short preamble can be obtained as:  
\[E_{sp}=\Delta\mathrm{p}+\ E_{syn}^{2}\times\ C_{drift} \tag{10}\]  
The node that transmits its clock to the one-hop neighbor during interior-region-communication is called the source node, and the node that receives the clock in the one-hop neighborhood is called the particular node (either the principal or backup node). The synchronized nodes send a short preamble prior to sending data without using the target address because a short preamble is sent to particular nodes in the one-hop neighborhood, which reduces the energy consumption. Let us assume that the energies consumed in one work cycle by the source and the particular nodes are \"\(\beta\)\" and \"\(\delta\)\", respectively. The average short preamble reception time could be reduced because the particular node wakes up based on the stored schedule. Thus, the source node and particular node consume an amount of energy that can be calculated as follows:  
Figure 4: Timeline comparison of low-duty-cycle hybrid MAC protocols. \[\beta=\sum_{i=0}^{n}S_{j}\,\frac{(\Delta\varphi.\,\mu\times\Delta v^{2}\,)\times \big{(}\,{E_{syn}}^{2}\times\,{C_{drift}}\big{)}+(E_{ct\ \ +\Delta\text{p}})}{\Delta_{t}} \tag{11}\]  
Equation (11) yields the energy consumed by the source node. \[\delta=\sum_{i=0}^{n}S_{j}\,\frac{(\Delta\varphi.\,\mu\times\Delta v^{2}\,) \times\big{(}\,{E_{syn}}^{2}\times\,{C_{drift}}\big{)}+(E_{ct\ \ +\Delta\text{p}})}{\Delta_{t}} \tag{12}\]  
Equation (12) yields the energy consumed by the particular node that is available at the one-hop destination. Table 3 provides the factors used in the preamble tradeoff validation model. To validate the analytical model, we have substituted the equations and compared the results for X-MAC, IEEE 802.15.4, B-MAC and WiseMAC. We are interested in determining the energy consumed for the metrics, such as the channel access, the short preamble, and data forwarding, which are calculated and given in Table 4.