## 4 Discrete Time Markov Chain Models and Analysis
### Multi-Link Access with the Shortest Backoff (MLA-S)  
For multi-link access, channel acquisition procedures occur on multiple links. Hence, we need to construct the multi-dimensional Markov Chain model, rather than the conventional 2D Markov Chain. We design a 3D Markov Chain model of MLA-S in Figure 5 for an MLD with two stations attached. 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. Two backoff counters, \(b_{1}\) and \(b_{2}\) are decremented at the beginning of each time slot until either backoff counter reaches to zero. Hence this case can be expressed as follows  
\[P\{i,k_{1}-1,k_{2}-1|i,k_{1},k_{2}\}=1,\quad\text{where}\quad i\in[0,m],\quad k _{1},k_{2}\in[1,W_{i}-1]. \tag{11}\]  
Recall that in case of MLA-S, when just one of backoff counters reaches zero, transmissions are attempted on both links. The states where the 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 randomly selected state in stage 0, which is the initial stage. Since there are \({W_{0}}^{2}\) states in stage 0, the transition probability for transmission success case is expressed as follows:  
\[P\{0,l_{1},l_{2}|i,k_{1},k_{2}\}=\frac{1-p}{{W_{0}}^{2}},\quad\text{where} \quad i\in[0,m],\quad k_{1}\cdot k_{2}=0,\quad l_{1},l_{2}\in[0,W_{0}-1], \tag{12}\]  
There may be cases of transmission failure. If a transmission on the primary link is failed, the transmissions on the other link is 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 case of transmission failure is as follows:  
\[P\{i+1,l_{1},l_{2}|i,k_{1},k_{2}\}=\frac{p}{{W_{i+1}}^{2}},\quad\text{where} \quad i\in[0,m-1],\quad k_{1}\cdot k_{2}=0,\quad l_{1},l_{2}\in[0,W_{i+1}-1]. \tag{13}\]Similar to Equation (13), we can derive one-step transition probability for the transmission failure case when the backoff stage is \(m\). \[P\{m,l_{1},l_{2}|m,k_{1},k_{2}\}=\frac{p}{W_{m}}^{2},\ \ \ \ \text{where}\ \ \ k_{1}\cdot k_{2}=0,\ \ \ l_{1},l_{2}\in[0,W_{m}-1]. \tag{14}\]  
Figure 5: A 3D Markov Chain model of MLA-S when the two stations are affiliated with an MLD. To make numerical analysis of this model simpler, we simplify the Markov Chain model as follows. Figure 6 shows simplified 2D Markov Chain model for MLA-S. In this model, the state \(\{s,b\}\) represents the current backoff stage and the shortest remaining backoff counter of an MLD, that is, \(b=\min(b_{1},b_{2})\). With this simplified 2D Markov Chain, the one-step transition probabilities with two stations affiliated with an MLD are the following:  
\[\left\{ \begin{array}{ll}P\{i,k|i,k+1\}&i\in[0,m],\ \ k\in[0,W_{i}-2]\\ =1,&i\in[0,m],\ \ k\in[0,W_{i}-2]\\ P\{0,k|i,0\}&\\ =\frac{2(W_{0}-k)-1}{{W_{0}}^{2}}\cdot(1-p),&i\in[0,m],\ \ k\in[0,W_{0}-1]\\ P\{i,k|i-1,0\}&\\ =\frac{2(W_{i}-k)-1}{{W_{i}}^{2}}\cdot p,&i\in[1,m],\ \ k\in[0,W_{i}-1]\\ P\{m,k|m,0\}&\\ =\frac{2(W_{m}-k)-1}{{W_{m}}^{2}}\cdot p,&k\in[0,W_{m}-1].\end{array} \right. \tag{15}\]  
The first equation in Equation (15) explains that when the backoff counter remains, it decreases for every time slots. The second equation account for the fact that the minimum backoff counter will be distributed after successful transmission with respect to the formula. The third and the fourth equations stand for the transition probabilities when the previous transmissions are failed. Let us elaborate the formula derivations for the second, the third, and the fourth equations as follows. Let \(X_{n,i}\) be a discrete random variable of initially selected backoff counter right after transmission or collision for the particular link \(n\) when the stage of the backoff counter is bound for the \(i\)th stage.