## 4 WPAN MAC: Designing and Analysis
### Analytical Model
\begin{split} T_{\text{{\scriptsize{CS-CH}}}}=& \begin{pmatrix}T_{\text{{\scriptsize{MS-CH}}}}+T_{\text{{\scriptsize{MA-CH}}} }+T_{\text{{\scriptsize{pre}}}}+T_{\text{{\scriptsize{PA}}}}\\ \end{pmatrix}&\text{{\scriptsize{for}}}\,\text{{\scriptsize{D-ACK}}}\\ \begin{pmatrix}2^{\text{{\tiny{PA}}}}T_{\text{{\scriptsize{MS-CH}}}}+T_{\text{{ \scriptsize{MA-CH}}}}+2^{\text{{\tiny{PA}}}}T_{\text{{\scriptsize{PA}}}}+T_{ \text{{\scriptsize{MP}}}}+T_{\text{{\scriptsize{MP}}}}+T_{\text{{\scriptsize{ MC-ACK}}}}+T_{\text{{\scriptsize{TH-ACK}}}}+T_{\text{{\scriptsize{TH-ACK}}}}+T_{\text{{ \scriptsize{TH-ACK}}}}\\ \begin{pmatrix}K\text{{\scriptsize{TH-ACK}}}+T_{\text{{\scriptsize{MA-CH}}}}+T_{ \text{{\scriptsize{MC-H}}}}+T_{\text{{\scriptsize{PR}}}}+T_{\text{{\scriptsize{MP }}}}+T_{\text{{\scriptsize{MP}}}}+T_{\text{{\scriptsize{MP-ACK}}}}+T_{\text{{ \scriptsize{TH-ACK}}}}+T_{\text{{\scriptsize{TH-ACK}}}}+T_{\text{{\scriptsize{ DP}}}}\\ \end{pmatrix}&\text{{\scriptsize{for}}}\,\text{{\scriptsize{D-ACK}}}\\ \begin{pmatrix}K\text{{\scriptsize{TA-CH}}}+T_{\text{{\scriptsize{MA-CH}}}}+T_{ \text{{\scriptsize{MA-CH}}}}+T_{\text{{\scriptsize{MC-H}}}}+T_{\text{{ \scriptsize{MA-CH}}}}+T_{\text{{\scriptsize{MP}}}}+T_{\text{{\scriptsize{MP}}}}+T_{ \text{{\scriptsize{MP}}}}+T_{\text{{\scriptsize{MP-ACK}}}}+T_{\text{{\scriptsize{ $\kappa-Dly-ACK-AG$}}}\\ \end{pmatrix}&\text{{\scriptsize{for}}}\,\text{{\scriptsize{K-Dly-ACK-AG$}}} \end{pmatrix}\end{split}  \tag{4}\]  
In Equation (4) \(T_{\text{{\scriptsize{MAC-H}}}}\) and \(T_{\text{{\scriptsize{Data}}}}\) are calculated as \(T_{\text{{\scriptsize{MAC-H}}}}=L_{\text{{\scriptsize{MAC-CH}}}}\)/\(C_{\text{{\scriptsize{rate}}}}\), and \(T_{\text{{\scriptsize{Data}}}}=L_{\text{{\scriptsize{Data}}}}\)/\(D_{\text{{\scriptsize{rate}}}}\), respectively. \(C_{\text{{\scriptsize{rate}}}}\) and \(D_{\text{{\scriptsize{rate}}}}\  
at the start of every idle backoff slot. It is important to note that the time scale for _b(t)_ doesn't represent real time but it observes only backoff slots and its suspended for the duration of all transmissions and interframe spaces (_i.e._, SIFS). Whenever _b(t)_ reaches zero the station transmits and regardless of the outcome of the transmission, uniformly chooses a new value for _b(t)_ from (0,1,...,_CW\({}_{j}\) - 1)(_i.e._, a new backoff counter value). Here, we define _p\({}_{c}\)_ as the conditional collision probability and we also assume that it is independent and constant, regardless the number of retransmissions attempted. _P\({}_{\rm c}\)_ also represents the probability of detecting the channel busy. Thus, the two dimension process, {_s(t)_,_b(t)_}, is a discrete-time Markov Chain. Therefore, the state of each node is described by {_j_,_k_}, where \(j\) stands for the backoff stage, and \(k\) stands for the backoff timer value. The state transition diagram of the Markov chain model shown in Figure 6 has the following transition probabilities:  
\[\begin{array}{l}\left[ \begin{array}{ll}P\left\{j,k\mid j,k+1\right\}=(1-p_{ c})&k\in(0,CW_{j}-2)&j\in(0,3)\\ P\left\{j,k\mid j,k\right\}=p_{c}&k\in(1,CW_{j}-1)&j\in(0,3)\\ P\left\{j,k\mid j-1,0\right\}=p_{c}/CW_{j}&k\in(1,CW_{j}-1)&j\in(1,3)\\ P\left\{m,k\mid m,0\right\}=p_{c}/CW_{j}&k\in(1,CW_{j}-1)\end{array} \right. \tag{8}\]  
The first equation in (8) indicates that at the beginning of each slot time, the backoff counter is decreased if the channel is sensed as idle. The second equation shows that the backoff counter is frozen if channel is sensed as busy. The third and fourth equations, respectively, indicate that following an unsuccessful transmission, the node back off stage (_j_ - 1) selects a backoff interval uniformly in the range of (0,_CW\({}_{j}\)_ - 1) and when the backoff stage reaches m, _CW\({}_{m}\)_ stays constant. From the given backoff algorithm framework we can calculate the failure probability, the success probability, and the busy probability of a transmission during CAP. Here we derive throughput, efficient bandwidth utilization, and delay with these three basic probabilities.