## 4 Discrete Time Markov Chain Models and Analysis  
Since a derivation procedure is applied to derive analytical models for multi-link DCFs in common, we first introduce the throughput derivation methodology here. In the header of this section, we first introduce a superb derivation with elementary conditional probability arguments [15], which is a revised model from a two-dimensional Markov chain analysis [14] to assist in understanding. In order to calculate throughput, we first should solve two non-linear equations. The first is the probability \(\tau\) that a station transmits in a randomly chosen slot time, and the second is the value of \(p\) that is the conditional collision probability, given that the frame has been transmitted. We hereafter introduce an elementary conditional probability approach proposed in the literature. First, to derive \(\tau\), the event that an MLD is transmitting a frame in a randomly selected time slot is denoted as \((TX)\). Given that the maximum retry number is \(R\), the event that the MLD is found in backoff stage is denoted as \((s=i)\), where \(i\in(0,1,...,R)\). To find the value of \(\tau\), we can derive the below equation with [PERSON] theorem as follows:  
\[P(TX)\cdot\frac{P(s=i|TX)}{P(TX|s=i)}=P(s=i). \tag{1}\]  
Since Equation (1) also holds for each _is_, the equation can be written as follows:  
\[P(TX)\cdot\sum_{i=0}^{R}\frac{P(s=i|TX)}{P(TX|s=i)}=\sum_{i=0}^{R}P(s=i). \tag{2}\]  
Thus, \(\tau\) can be represented as follows:  
\[\tau=P(TX)=\frac{1}{\sum\limits_{i=0}^{R}\frac{P(s=i|TX)}{P(TX|s=i)}}. \tag{3}\]  
Meanwhile, the conditional collision probability \(p\) means that when a station is trying to access the channel with zero of the backoff counter, at least one remaining station tries to capture the channel with a zeroed backoff counter in the same time slot. Hence, \(p\) can be expressed as follows:  
\[p=1-(1-\tau)^{n-1}. \tag{4}\]  
where \(n\) is the number of contenting stations. \(P(s=i|TX)\) is a probability that a certain station is on the \(i\)th stage given that the station has attempted to transmit due to its zeroed backoff counter. If a recipient station has successfully received the transmission, the number of stage \(i\) will be returned to zero. Otherwise, the number of stage \(i\) increased by one only if a pre-defined retry number, say \(R\), has not been exceeded. Furthermore, since \(P(s=i|TX)\) follows geometric distribution, \(P(s=i|TX)\) can be expressed as follows:  
\[P(s=i|TX)=\frac{(1-p)p^{i}}{1-p^{R+1}},\quad i\in(0,1,...,R) \tag{5}\]  
where \(p\) is the conditional collision probability defined in Equation (4). Meanwhile, \(P(TX|s=i)\) is the probability that the station is subject to attempt to access the channel in the \(i\)th stage. Let us explain with a detailed example. The backoff counter will be selected based on uniform distribution. Assume that CW on the initial stage is 7, the backoff counter will be in the interval of \([0,7]\). As a result, an expected time slot duration to await becomes \(1+(0+7)/2=4.5\). Hence, \(P(TX|s=0)=\frac{1}{4.5}\). In this way, \(P(TX|s=i)\) can be calculated as follows:  
\[P(TX|s=i)=\frac{1}{1+E[b_{i}]} \tag{6}\]  
where \(E[b_{i}]\) is the average value of backoff counter for stage \(i\). In the baseline of 802.11, the initial CW, CWmin and the maximum CW, CWmax values are defined as 15 and 1023, respectively, and the retry number is 6. Since the selected backoff counter follows a uniform distribution, then \(E[b_{i}]\) values for all \(i\) can be represented as [7.5, 15.5, 31.5, 63.5, 127.5, 255.5, 511.5], respectively. We can solve two non-linear equations using Equations (3)-(6) with numerical techniques.