## 5 Analytical Models for the Proposed MAC
\[P\{S_{j+1,k}|S_{j,0}\}=\frac{p_{2}}{CW_{i}}\;\;for\;0\leq j\leq NB_{max}\;and \;\;0\leq k\leq CW_{i}-1. \tag{7}\]  
The tagged device starts data transmission after the CCA is successful. The transition probability for this case can be derived as follows. \[P\{T|S_{j,-1}\}=1-\alpha\;\;for\;0\leq j\leq NB_{max}. \tag{8}\]  
However, the tagged device will go for random backoff again, if the current CCA is failed. The corresponding transition probability can be derived as follows. \[P\{S_{j+1,k}|S_{j,-1}\}=\frac{\alpha}{CW_{i}}\;\;for\;0\leq j\leq NB_{max}\;and \;\;0\leq k\leq CW_{i}-1. \tag{9}\]  
Once the device reaches the maximum backoff attempts for transmission, the device returns back to the idle state no matter whether the transmission attempt is successful or failed. Hence, the corresponding transition probability can be deduced as follows. Figure 8:0Considering these transition probabilities, the probability of a device at state \(S_{j,k}\) can be derived as follows, where \(S_{j,k}\) is the stationary distribution of the Markov chain. After the channel access failure in the current backoff state, the tagged device enters into the next backoff state, and the probability for the corresponding state is:  
Figure 8:1  
Once the device finds the channel idle during the CCA without attaining the retry limits, it will enter into the initial backoff state, and the corresponding state probability can be deduced as:  
\[ \begin{split} S_{0,k}=\frac{CW_{i}-k}{CW_{i}}[& \{\sum_{j=0}^{NB_{max}}S_{j,0}\}(1-p_{2})(1-\alpha)\{(1-P_{5})+P_{5}p_{1}\}\\ +& S_{NB_{max},0}(p_{2}+(1-p_{2})\alpha p_{1}];\ \ 0\leq k\leq CW_{i}-1.\end{split}  \tag{12}\]  
Normally, a device defers to data transmission if the remaining time in the contention period is insufficient to complete the transmission, and we can derive the corresponding state probability as follows. \[S_{j,-1}=(1-p_{2})S_{j,0};\ \ \ for\ 0\leq j\leq NB_{max}. \tag{13}\]  
The tagged device starts transmission after the channel access is successful during the \(j\)-th backoff state. Hence, the probability for the corresponding state can be deduced as follows. \[T=(1-\alpha)S_{j,-1};\ \ \ for\ 0\leq j\leq NB_{max}. \tag{14}\]  
A device enters into the idle state after completion of the data transmission or has no new packet to transmit, and the corresponding state probability is deduced as:  
\[I=\frac{1}{p_{1}}(P_{5}T+(p_{2}+(1-p_{2})(1-\alpha))S_{NB_{max},0}). \tag{15}\]  
A device enters into the ready state whenever it has a new packet or needs to retransmit the failed packet. Hence, the corresponding state probability can be derived as follows. \[R=(1-P_{5})T+p_{1}I. \tag{16}\]  
The sum of the probabilities of all states should be equal to one based on the rule of the probability for which we can get the following equation. \[I+R+\sum_{j=0}^{NB_{max}}\sum_{i=0}^{CW_{i}-1}S_{j,i}+\sum_{j=1}^{NB_{max}}S_{ j,-1}+T=1 \tag{17}\]  
We can deduce the probabilities of all states by solving these equations. The probability for a device that attempts the CCA for the first time can be calculated as follows. \[\phi=\sum_{j=0}^{NB_{max}}S_{j,0}(1-p_{2}). \tag{18}\]