## 4 Analysis Model  
In this section, we derive an analytical model for evaluating the performance of the proposed scheme. We use a Markov chain to analyze each state of the random access slot. The state transition diagram is depicted in Figure 3. The state of the Markov chain represents the number of DS devices right before the start of an RA slot, i.e., the number of active devices in a slot. In this model, state \(M\) means that there are greater than or equal to \(M\) active devices arrive at the slot. Since we are considering the simultaneous DS devices' access environment, the setting value of \(M\) is far greater than \(L_{avail\_DS}\) under the background of LTE-A system. In order to obtain average access delay, average preamble utilization rate and average number of active devices in each slot, etc. We need to calculate the probability of each state by steady-state equations. Therefore, the first thing we need to do is calculating the steady-transition matrix \(P\). Let \(P_{m,n}\) denote the transition probability that the state transferred from \(m\) to \(n\). We can derive:  
\[P_{0,n}=A_{n},\text{ for }n\in(0,M-1) \tag{14}\]  
\[P_{0,M}=A_{\geq M} \tag{15}\]  
where \(A_{n}\) denotes the probability that there are \(n\) devices arrived at the access slot, \(A_{\geq M}\) as denotes the probability that more than \(M\) devices arrive at the slot. We define \(B_{m,s}\) and \(F_{m,s,t}\) as the probabilities that \(s\) devices are blocked by ACB mechanism among \(m\) devices and \(t\) devices are collided in the preamble transmission section among \(m-s\) contending devices. Then the state transition probability \(P_{m,n}\) can be represented as follows:  
\[P_{m,n}=\sum_{s=0}^{m}B_{m,s}\cdot\left(\sum_{t=0}^{m-s}F_{m,s,t}\cdot A_{n-s -t}\right),\text{ for }m\in(1,M-1),\ n\in(m,M-1) \tag{16}\]  
\[P_{m,n}=\sum_{s=0}^{m}B_{m,s}\cdot\left(\sum_{t=0}^{m-s}F_{m,s,t}\cdot A_{n-s -t}\right),\text{ for }m\in(1,M),\ n\in(m-\min(m,L_{avail}),m-1) \tag{17}\]  
\[P_{m,M}=\sum_{s=0}^{m}B_{m,s}\cdot\left(\sum_{t=0}^{m-s}F_{m,s,t}\cdot A_{ \geq n-s-t}\right),\text{ for }m\in(1,M) \tag{18}\]  
In our paper, we use Poisson distribution as our arrival model, which is widely used to analyze a slotted ALOHA [32]. The arrival rate of DS devices in our paper is \(\lambda\), thus we can obtain:  
\[A_{n}=\frac{\left(\lambda T\right)^{n}}{n!}e^{-\lambda T},n=0,1,2,... \tag{19}\]  
Figure 3: Markov chain for the number of access requests at the beginning of a slot. \[A_{\geq M}=1-\sum\limits_{n=0}^{M-1}A_{n} \tag{20}\]  
Let \(p_{ACB,m}^{*}\) denote the optimal ACB factor derived from Algorithm 1 when the number of active devices is \(m\), then \(B_{m,s}\) can be written as:  
\[B_{m,s}=\left( \begin{array}{c}m\\ s\end{array} \right)\bigl{(}1-p_{ACB,m}^{*}\bigr{)}^{s}\bigl{(}p_{ACB,m}^{*} \bigr{)}^{m-s} \tag{21}\]  
In addition, we can obtain the expression of \(F_{m,s,t}\) as:  
\[F_{m,s,t}=\left( \begin{array}{c}m-s\\ t\end{array} \right)\Bigl{(}p_{fail,m,s}\Bigr{)}^{t}\Bigl{(}1-p_{fail,m,s} \Bigr{)}^{m-s-t} \tag{22}\]  
where \(p_{fail,m,s}\) denotes the probability that a DS device fails to access the network because of contention of PRACH resources with other \((m-s-1)\) devices. Knowing the number of allocated preambles, we can further obtain the expression of \(p_{fail,m,s}\):  
\[p_{fail,m,s}=\sum\limits_{r=1}^{m-s-1}\left( \begin{array}{c}m-s-1\\ r\end{array} \right)\bigl{(}L_{DS,m}^{*}\bigr{)}^{r}\bigl{(}1-L_{DS,m}^{*} \bigr{)}^{m-s-1-r} \tag{23}\]  
here \(L_{DS,m}^{*}\) denotes the calculated optimal number of \(L_{DS}\) obtained by Algorithm 1 when the number of active devices is \(m\). By substituting Equations (19)-(23) into Equations (14)-(18), we can get the state transition matrix:  
\[\mathrm{P}=\left[ \begin{array}{ccccc}P_{0,0}&P_{0,1}&\cdots&P_{0,m}&\cdots& P_{0,M}\\ P_{1,0}&P_{1,1}&\cdots&P_{1,m}&\cdots&P_{1,M}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ P_{m,0}&P_{m,1}&\cdots&P_{m,m}&\cdots&P_{m,M}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ P_{M,0}&P_{M,1}&\cdots&P_{M,m}&\cdots&P_{M,M}\end{array}