## 3 Dynamic Resource Allocation and ACB Scheme for Delay-Sensitive Devices
### Proposed Optimization Scheme  
In this paper, we assume that random access requests are all initialed at the beginning of a slot. In order to express our proposed scheme clearly, we present the concept of active device. Here, an active device is defined as a DS device which has a packet to send to eNodeB at the beginning of an RA slot. The active devices consist of two parts, i.e., devices which newly arrived in the current slot and devices which are barred and collided in the previous slot. In addition, due to the function of ACB mechanism, only part of the active devices can transmit preambles, we define this part of devices as contending devices. Let \(N\) denote the number of active devices that arrived in an access slot, \(N_{pa}\) denote the expected number of DS devices that pass through the ACB mechanism. Reviewing the implementation process of ACB scheme which has been introduced in Section 2.1, we obtain the expression of \(N_{pa}\) as:  
\[N_{pa}=N\cdot p_{ACB} \tag{1}\]  
Figure 2: The diagram of conceptual design. where \(p_{ACB}\) denotes the ACB factor for DS devices. Let \(S_{l}=0\), \(S_{l}=1\) and \(S_{l}=c\) respectively denote the cases that a random selected preamble \(l\) is idle (i.e., is selected by none of the users), is successfully transmitted (i.e., is selected by exactly one user) and is in conflict (i.e., is selected by more than one user). The probability that only one user among \(N_{pa}\) contending devices selects preamble \(l\) is:  
\[P(s_{l}=1)=\left( \begin{array}{c}N_{pa}\\ 1\end{array} \right)\frac{1}{L_{DS}}\left(1-\frac{1}{L_{DS}}\right)^{N_{pa}-1} \tag{2}\]  
here \(L_{DS}\) indicates the number of preambles allocated to DS devices, i.e., the number of preambles in Pool 1. It should be noted that preamble utilization rate \(P_{L\_{\text{succ}}}\), which represents the ratio of the number of successfully transmitted preambles for DS devices to the number of total preambles allocated to DS devices, is the same as the probability that a preamble is selected by exactly one user as shown in Equation (2). Thus, we can derive the expression of \(P_{L\_{\text{succ}}}\) as:  
\[P_{L\_{\text{succ}}}=P(s_{l}=1)=\left( \begin{array}{c}N_{pa}\\ 1\end{array} \right)\frac{1}{L_{DS}}\left(1-\frac{1}{L_{DS}}\right)^{N_{pa}-1} \tag{3}\]  
Through derivation we can judge that \(P_{L\_{\text{succ}}}\) is a \(\cap\) shape of \(L_{DS}\) for a fixed \(N_{pa}\), \(P_{L\_{\text{succ}}}\) can get its maximum value when \(L_{DS}=N_{pa}\)[27]. We define \(P_{D\_succ}\) as the probability of a DS device successfully accessing the network, it can be derived as follows:  
\[P_{D\_succ}=p_{ACB}\cdot P_{access} \tag{4}\]  
where \(P_{access}\) denotes the probability that a contending DS device which passes through the ACB mechanism can successfully access the network, i.e., the preamble chosen by the device is not selected by any other UEs. We can obtain \(P_{access}\) as:  
\[P_{access}=\frac{\left( \begin{array}{c}N_{pa}\\ 1\end{array} \right)\cdot\left(1-\frac{1}{L_{DS}}\right)^{N_{pa}-1}}{N_{pa}}= \left(1-\frac{1}{L_{DS}}\right)^{N_{pa}-1} \tag{5}\]  
We define \(T_{delay}\) as the access delay, i.e., the delay between the first access attempt and the completion of a successfully preamble transmission for a DS device. In legacy RA process, each device blocked by ACB mechanism and preamble collision will reattempt access after a random backoff. In our paper, we assume that all the devices blocked in a certain RA slot will launch a new access request in the next coming RA slot and there is no retry limit for random access, thus the average value of access delay \(T_{delay}\) can be derived as:  
\[T_{delay}=\sum_{r=1}^{\infty}r\cdot P_{D\_succ}\cdot\left(1-P_{D\_succ} \right)^{r-1}\cdot T_{slot}=\frac{T_{slot}}{P_{D\_succ}} \tag{6}\]  
where \(T_{slot}\) denotes the length of a random access slot, \(r\) denotes the number of access attempts initiated by the DS device before the preamble selected by the device is successfully transmitted.