## 4 Discrete Time Markov Chain Models and Analysis
### Multi-Link Access with the Longest Backoff (MLA-L)  
We design a 3D Markov Chain model of MLA-L in Figure 7 for an MLD with two stations attached. Same with the MLA-S case as mentioned in Section 4.2, let \(\{s,b_{1},b_{2}\}\) represent the current backoff stage, the remaining backoff counter of the first station, and the remaining backoff counter of the second station, respectively. Unlike MLA-S, two backoff counters, \(b_{1}\) and \(b_{2}\) decrease to zero at the beginning of each time slot until all backoff counters reaches to zero. Hence this case can be expressed as follows:  
\[P\{i,k_{1}-1,k_{2}-1|i,k_{1},k_{2}\}=1,\quad\text{where}\quad i\in[0,m],\quad k _{1},k_{2}\in[0,W_{i}-1],\quad k_{1}+k_{2}\
eq 0. \tag{24}\]  
The states where the above-mentioned transmission occurs are represented as blue areas in the figure. After the successful transmission on the primary link, with probability \(1-p\), the state will be moved to any state in the initial stage. Since there are \({W_{0}}^{2}\) states on the stage 0, the transition probability for transmission success case is expressed as follows:  
\[P\{0,l_{1},l_{2}|i,0,0\}=\frac{1-p}{{W_{0}}^{2}},\quad\text{where}\quad i\in[ 0,m],\quad l_{1},l_{2}\in[0,W_{0}-1]. \tag{25}\]  
In cases of transmission failures, the transmissions on the auxiliary link are also considered as failure so that the CW should be doubled. When the stage was from zero to \(m-1\), the one-step transition probability for the transmission failure case is as follows:  
\[P\{i+1,l_{1},l_{2}|i,0,0\}=\frac{p}{{W_{i}}^{2}},\quad\text{where}\quad i\in[ 1,m],\quad l_{1},l_{2}\in[0,W_{i}-1]. \tag{26}\]Similar to Equation (26), we can derive one-step transition probability for transmission failure case when the backoff stage is \(m\) as follows:  
\[P\{m,l_{1},l_{2}|m,0,0\}=\frac{p}{W_{m}},\quad\text{where}\quad l_{1},l_{2}\in[0,W_{m}-1]. \tag{27}\]  
In the same manner as MLA-S, we can reduce this 3D dimensional Markov Chain model to 2D Markov Chain as shown in Figure 8. In this model, the state \(\{s,b\}\) represents the current backoff stage and the longest remaining backoff counter of an MLD. Figure 7: A 3D Markov Chain model of MLA-L when the two stations are affiliated with an MLD. The difference from MLA-S is that, \(b\) for MLA-L is the longest remaining backoff counter among the MLD since the MLD have to wait during the longest remaining backoff counter. The one-step transmission probabilities for the Markov Chain of MLA-L with two stations affiliated with an MLD are the following:  
\[\left\{ \begin{array}{l}P\{i,k|i,k+1\}\\ =1,\\ P\{0,k|i,0\}\\ =\frac{2k+1}{{W_{0}}^{2}}\cdot(1-p)\\ P\{i,k|i-1,0\}\\ =\frac{2k+1}{{W_{1}}^{2}}\cdot p,\\ P\{m,k|m,0\}\\ =\frac{2k+1}{{W_{m}}^{2}}\cdot p,\\ =\frac{2k+1}{{W_{m}}^{2}}\cdot p,\\ \end{array} \right.\quad i\in[0,m],\quad k\in[0,W_{i}-2] \tag{28}\]  
The first equation in Equation (28) accounts for decreasing remaining backoff counter for every time slots. The second, the third, and the fourth equations account for one-step transition probability after every transmission. Similar to the MLA-S case, we denote a discrete random variable as follows:  
\[B_{N,j}^{MLA-L}=\max(X_{1,j},X_{2,i},\ldots,X_{N,j}). \tag{29}\]  
Figure 8: A simplified Markov Chain model for the backoff window size in MLA-L when the two stations are affiliated with an MLD. with Equation (29), we can derive CDF of \(B_{N,i}^{MLA-L}\) as follows:  
\[ \begin{split} P\Big{(}B_{n,i}^{MLA-L}\leq k\Big{)}&=P( \max(X_{1,i},X_{2,i},...,X_{N,i})\leq k)\\ &=P(X_{1,i}\leq k,X_{2,i}\leq k,...,X_{N,i}\leq k)\\ &=\prod\limits_{j=1}^{N}P\big{(}X_{j,i}\leq k\big{)}\\ &=\prod\limits_{j=1}^{N}\frac{k+1}{W_{i}}\\ &=\bigg{(}\frac{k+1}{W_{i}}\bigg{)}^{N}.\end{split}  \tag{30}\]  
Notice that discrete random variable \(B_{N,i}^{MLA-L}\) is the maximum value of \(N\) backoff counters, contrary to \(B_{N,i}^{MLA-S}\). With Equation (29), we can derive CDF of \(B_{N,i}^{MLA-L}\) as follows:  
\[