## 4 Performance Evaluation  
### Analysis of End-to-End Delay  
We calculated the worst-case end-to-end delay (\(E2 ED\)) that corresponded to BP for the four broadcast protocols: Glossy [4], DPFNI [6], RSBP [14], and SSMAb. In Glossy [4], the time to send a packet, \(T_{tx}\), is given as follows:  
\[T_{tx}=T_{sw}+T_{cal}+T_{pr}+T_{f}+T_{l}+T_{m}=375.5(\mu s)+T_{m} \tag{15}\]  
where, \(T_{sw},T_{cal}\), \(T_{pr}\), \(T_{f}\), \(T_{l}\), \(T_{m}\) are software delay, calibration time, preamble time, SFD (start of frame delimiter) time, length in time, and MAC protocol data unit (MPDU) time, respectively. Note that in Equation (15), \(T_{m}\) is the time to send payload of size \(p\). Therefore, in the 802.15.4 physical layer \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Features** & **Flooding** & **DPFNI** & **RSBP** & **SSMAb** \\ \hline \multirow{3}{*}{\begin{tabular}{c} Reliability of \\ message delivery \\ \end{tabular} } & Multiple chances of & Multiple chances of & A single chance of & Multiple chances of \\  & receiving a & receiving with & receiving with no & receiving with \\  & message & probability & collision & reduced collision \\  & Very high & High & Mid & High \\ \hline \multirow{3}{*}{\begin{tabular}{c} Delay in message \\ delivery \\ \end{tabular} } & \multirow{3}{*}{Random delay} & \multirow{3}{*}{Random delay} & Time bound by slot & Time bound by \\  & & & schedule & schedule \\  & Low & Low & High & High \\ \hline \multirow{3}{*}{\begin{tabular}{c} Concurrency of \\ broadcast \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} Free broadcast after \\ random delay \\ \end{tabular} } & Free broadcast after & Scheduled & \\  & & random delay & broadcast & same tree level \\  & High & High & Zero & Mid \\ \hline \multirow{3}{*}{\begin{tabular}{c} Energy \\ consumption \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} Rebroadcasting by \\ all nodes and long \\ active time \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} Rebroadcasting by \\ with probability \\ and long active \\ \end{tabular} } & Rebroadcasting by & Rebroadcasting by \\  & & with probability & internal nodes only \\  & all nodes and long active & and managed & and managed active \\  & & time & active time & time \\  & High & High & Optimal & Low \\ \hline \multirow{3}{*}{\
\begin{tabular}{c} Responsiveness to \\ dynamic topology \\ \end{tabular} } & \multirow{3}{*}{No topology} & Topology-dependent & Tree topology and & Tree topology and \\  & & probability & topology-dependent & topology-independent \\  & High & Mid–High & Low & Mid–High \\ \hline \multirow{3}{*}{
\begin{tabular}{c} Scheduling \\ overhead \\ \end{tabular} } & \multirow{3}{*}{No scheduling} & No scheduling & Centralized & Distributed \\  & & & scheduling \\ \cline{1-1}  & No & No & High & Almost negligible \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the characteristics of SSMAb and other approaches. standard, the bandwidth was 250 kbps and each packet took an additional 6 bytes for synchronization header, PHY header, and MAC header and footer. Thus, the \(E2 ED\) of Glossy is given as follows:  
\[E2 ED(Glossy)=H\times T_{tx}=H\times(0.3755+(6+p)\times 0.032)(ms) \tag{16}\]  
In DPFNI [6], it uses the back-off time before retransmission, and its \(E2 ED\) is given as follows:  
\[ \begin{array}{ll}E2 ED(DPFNI)&=H\times T_{B}\\ &=H\times\left(T_{Backoff}+T_{ppd}+T_{tx}\right)=H\times(3.192+(6+p)\times 0.0 32\right)(ms)\end{array}  \tag{17}\]  
where \(T_{B}\) is a broadcast time as the sum of the back-off time (\(T_{Backoff}\)), the physical layer processing delay (\(T_{ppd}\)), and transmission time (\(T_{tx}\)), and \(T_{Backoff}\) in the paper is not bigger than 3 ms. In RSBP [14], only non-leaf nodes rebroadcast a message and one distinct \(BS\) is allocated to each of the non-leaf nodes. Let \(nBNodes\) denote the number of broadcast nodes.