## 4 WPAN MAC: Designing and Analysis
### Analytical Model  
As we have shown in Figure 2. WPAN MAC is a hybrid MAC protocol based on the CSMA/CA method and TDMA scheme. There are number of papers on analytical models for CSMA/CA and TDMA. However these models cannot be directly adopted for the performance analysis of WPAN MAC because these models do not address the overall performance of both CSMA/CA and TDMA schemes, Thus, to study the performance analysis of WPAN MAC, we present an analytical model of WPAN MAC in two parts: CSMA/CA method during CAP and TDMA scheme (in this paper, we use the words 'algorithm','scheme', and'method' interchangeably) during CTA period. We use the ground work of [6,12] and [13] to present our analytical model of CTA. Table 1 shows the notations used for our analytical model. Readers are advised to have a look at Table 1 when referring to equations for parameter notation. The theoretical throughput is given by:  
\[\textit{Th}=\frac{\textit{Transmitted Data}}{\textit{Transmission Cycle Duration}} \tag{1}\]  
From [12], a frame with a length \(L\) in bits, the probability that the frame is successfully transmitted can be calculated as:  
\[p_{s}=\left(\textit{l}-p_{e}\right)^{L} \tag{2}\]  
Similarly, we can define \(p_{s}\) for different ACK mechanisms as follows (it is noteworthy that header--all kind of headers, trailers, beacons, etc. or control signals - and data frames are transmitted \begin{table}
\begin{tabular}{c l} \hline \(\tau_{\textit{ars}}\) & Short Inter Frame Space (SIFS) time \\ \(\tau_{\textit{aws}}\) & Distributed Coordinate Function Inter Frame Space (DIFS) \\ \(\tau_{\textit{aws}}\) & Minimum Inter Frame Space (MIFS) time, Usually MIFS \(\leq\) \\ \(CW_{\textit{min}}\) & Minimum back-off window size \\ \(\tau_{\textit{rr}}\) & Transmission time of the physical preamble \\ \(\tau_{\textit{rr}}\) & Transmission time of the PHY header \\ \(L_{\textit{usc-it}}\) & MAC overhead in bytes \\ \(L_{\textit{ctx}}\) & ACK size in bytes \\ \(L_{\textit{max}}\) & Payload size in bytes \\ \(\tau_{\textit{usc-it}}\) & Transmission time of MAC overhead \\ \(\tau_{\textit{ctx}}\) & ACK transmission time \\ \(\tau_{\textit{hour}}\) & Transmission time for the payload \\ \(\tau_{\textit{r-ctr}}\) & The time for a transmission considered failed during CAP \\ \(\tau_{\textit{r-ctr}}\) & The time for a transmission considered successful during CAP \\ \(\tau_{\textit{r-ctr}}\) & The time for a transmission considered failed during CAT \\ \(\tau_{\textit{r-ctr}}\) & The time for a transmission considered successful during CAT \\ \(T_{\textit{acx-10}}\) & The time-out value waiting for an ACK \\ \hline \end{tabular}
\end{table} 
Table 1: Parameter Notations. with different rates; generally, data rate is quite high compared to control signal, so we also considered two different signal rates for data and control signals, respectively, as listed in Table 2):  
\[ \begin{split} P_{s-\text{{\scriptsize{Imm-ACK}}}}=&(1-p_{ \varepsilon})^{(L_{\text{{\scriptsize{Data}}}}+L_{\text{{\scriptsize{MAC-H}}} }+L_{\text{{\scriptsize{ACK-Lmm}}}})\,\text{{\tiny{$\
u$8}}}}\\ P_{s-\text{{\scriptsize{No-ACK}}}}=&(1-p_{ \varepsilon})^{(L_{\text{{\scriptsize{Data}}}}+L_{\text{{\scriptsize{MAC-H}}} }+L_{\text{{\scriptsize{ACK-Ly}}}})\,\text{{\tiny{$\
u$8}}}}\\ P_{s-\text{{\scriptsize{Dy-ACK}}}}=&(1-p_{ \varepsilon})^{(L_{\text{{\scriptsize{Data}}}}+L_{\text{{\scriptsize{MAC-H}}} }+L_{\text{{\scriptsize{ACK-Ly}}}})\,\text{{\tiny{$\
u$8}}}}\\ P_{s-\text{{\scriptsize{$\kappa-Dly-ACK-AGG}}}}=&( 1-p_{\varepsilon})^{(L_{\text{{\scriptsize{Data}}}}+L_{\text{{\scriptsize{MAC-H}}} }+L_{\text{{\scriptsize{MAC-H}}}}+L_{\text{{\scriptsize{ACK-Ly}}}})\,\text{{ \tiny{$\
u$8}}}}\\ \end{split}  \tag{3}\]  
Here, we use Imm-ACK, No-ACK, Dly-ACK, and K-Dly-AGG-ACK to denote the immediate acknowledgement, No acknowledgement, delay acknowledgement, and delay acknowledgement with aggregation, respectively. A successful transmission time during CTA is given by:  
\[