## 4 System Model
### Delay Analysis from Simplified Model  
Assume that each state in the simplified Markov chain can be expressed as a random variable of an initial UORA back-off value. This means that if an STA gets a random UORA back-off value, the value can be mapped to an appropriate value of the stage. Simply, let us set a random variable \(N\) as the number of UORA trigger intervals for transmission given \(R\)th retransmission. Now, the average overall delay can be described as  
\[E[T_{overall}]=E[T_{Initial}]+E[T_{contention}] \tag{23}\]  
\(T_{overall}\) is the total delay from generating a frame to delivering it or to discarding it after \(M\) retransmissions. \(T_{initial}\) is the waiting time for receiving the first UORA trigger frame after generating a frame. \(T_{contention}\) is the time delay from receiving the first UORA trigger frame to successive delivery or to \(M\)th retransmission failure. Because the time includes UORA back-off delays and retransmission delays, it is called contention delay in this paper.  
From the simplified model, we can obtain the probability distribution of the random variable \(N\) easily. For a given number of retransmission \(j\),  
\[P[N_{j}]=\left\{\begin{array}{ll}\frac{r+1}{W_{j}}&N_{j}=0\\ \frac{r}{W_{j}}&0<N_{j}<\left\lfloor\frac{W_{j}-2}{r}\right\rfloor\\ \frac{W_{j}-r\left\lfloor\frac{W_{j}-2}{r}\right\rfloor-1}{W_{j}}&N_{j}=\left \lfloor\frac{W_{j}-2}{r}\right\rfloor\end{array}\right. \tag{24}\]  
where \(j\in[0,M]\), \(r\) is the number of RUs in the UORA trigger frame, and \(W_{j}\) is the size of the UORA CW for the given retransmission index \(j\).  
To determine the average delay, we must obtain the average number of stages for each STA.  
\[E[N_{j}]=\sum\limits_{N=0}^{\left\lfloor\frac{W_{j}-2}{r}\right\rfloor}NP[N_{ j}] \tag{25}\]  
The intervals of the UORA trigger frames can be described as a random variable as well. We use a random variable \(I\) to represent it and describe the delay as below.  
\[E[T_{overall}]=E[T_{Initial}]+[\sum\limits_{i=0}^{M}\sum\limits_{j=0}^{i}E[T_{ j}]P[R=i]] \tag{26}\]  
\[E[T_{overall}]=E[T_{Initial}]+[E[I]\sum\limits_{i=0}^{M}\sum\limits_{j=0}^{i}E[N _{j}]P[R=i]] \tag{27}\]  
where \(R\) is the number of retransmissions with the distribution:  
\[\begin{array}{ll}P[R=j]=(1-p)p^{j}&j\in(0,M-1)\\ P[R=M]=1-\sum\limits_{k=0}^{M-1}(1-p)p^{k}=p^{M}\end{array} \tag{28}\]  
The probability \(p\) is an identical parameter we used in the Markov chain model. Not only can we control \(p\) value to model UORA channel contention directly, but we can also set the \(p\) value to the value we obtained from Equation (13). If we assume that W\(j\) and \(r\) are constant values over every \(j\), \(E[N_{j}]\) will have a constant value \(N\) from Equation (24). In this case, Equation (27) can be re-expressed as follows.  
\[E[T_{overall}]=E[T_{Initial}]+[E[I]N\sum\limits_{i=0}^{M}(i+1)P[R=i]] \tag{29}\]\[E[T_{overall}]=E[T_{Initial}]+(E[I]N\frac{1-p^{M+1}}{1-p}) \tag{30}\]  
If we assume that the interval \(I\) is deterministic and the initial frame generation time is random, Equation (30) can be expressed as  
\[E[T_{overall}]=\frac{I_{c}}{2}+I_{c}N\frac{1-p^{M+1}}{1-p} \tag{31}\]  
where \(I_{c}\) is a constant value of the UORA trigger interval. If the interval follows an exponential distribution and the interval \(I\) has an average value \(I_{p}\), then  
\[E[T_{overall}]=I_{p}+I_{p}N\frac{1-p^{M+1}}{1-p} \tag{32}\]  
The delays shown in Equations (31) and (32) are special cases as we referred. To simplify the equation, we assumed a constant size of CWs along with a constant size of RUs. If we have more complex patterns of CW size, the distribution of \(N_{j}\) will change and hence we need to calculate Equation (27) with appropriate calculation methods. Basically, computational software packages can calculate it easily even though \(N_{j}\) has a complex distribution. As a result, the delay can be calculated by using Equation (27), where (24), (25), and (28) are for the general UORA delay case.