## 3 Dynamic Resource Allocation and ACB Scheme for Delay-Sensitive Devices
### Proposed Optimization Scheme
Substituting Equations (4)-(6), access delay can be determined as:  
\[T_{delay}=\frac{T_{slot}}{p_{ACB}\cdot\left(1-\frac{1}{L_{DS}}\right)^{N_{pa}-1}} \tag{7}\]Now, let us investigate the optimization strategy for DS devices. Considering the characteristics of DS services, we mainly think about two optimization parameters:  
1. Average access delay of delay-sensitive devices. 2. Preamble utilization rate for delay-sensitive devices. The significance for discussing the above two parameters are as follows: firstly, by studying the average access delay, we can effectively meet the basic QoS requirements of DS devices; secondly, by studying the preamble utilization rate, we can achieve the most efficient use of PRACH resources for DS devices, thus leaving more preambles for Pool 2. There have been a lot of articles about estimating the network load [17; 18; 28; 29; 30; 31], the number of attempting devices can be estimated using the number of idle preambles or the number of collision preambles. Therefore, we do not discuss how to estimate network load in our paper. In the following discussion, we assume that the eNodeB knows the actual number of DS and DT devices that attempt to access the network. Our focus is on the joint optimization of the ACB mechanism and the preamble allocation scheme for DS devices. It has been mentioned in the first part of the paper that \(P_{L\_\text{succ}}\) is maximized if \(L_{DS}\) is equal to \(N_{pa}\). However, we must consider the characteristics of the LTE-A system, i.e., the maximum number of available preambles for DS devices. In order to maintain high preamble utilization rate of DS devices, the number of contending devices which can be controlled by ACB mechanism needs to be minimized, as the number of contending devices increases, more preambles are required to maximize the preamble utilization rate. In addition, a reasonable setting of the value of \(L_{DS}\) and \(p_{ACB}\) is needed to increase the delay performance. We have defined \(L_{avail\_DS}\) as the maximal number of available preambles for DS devices in Section 3.1. According to the number of active devices (\(N\)) in the slot, the investigation of our proposal can be divided into two cases: one is \(N\leq L_{avail\_DS}\) and the other is \(N>L_{avail\_DS}\). By theoretical analysis, we can obtain the optimal values of \(N\) and \(p_{ACB}\) for each case. We discuss these two situations separately:  
1. \(N\leq L_{avail\_DS}\)  
In this situation, ACB mechanism is not necessary as we have sufficient preamble resources for DS devices. Therefore, our principle is to deal with as many DS devices as possible for each slot. For the purpose of enabling more DS devices to transmit preambles, we set \(p_{ACB}\) to the maximum value \(p_{ACB}^{*}=1\), where \(p_{ACB}^{*}\) denotes the optimal value of \(p_{ACB}\). Our problem can be formulated as:  
\[ \begin{split} L_{DS}^{*}&=\arg_{0\leq L_{DS}\leq L _{avail\_DS}}P_{L\_succ}\\ s.t.&:T_{delay}\leq D_{req}\end{split}  \tag{8}\]  
where \(L_{DS}^{*}\) is defined as the optimal value of \(L_{DS}\). \(D_{req}\) is defined as the delay requirement of DS devices. Considering that \(N_{pa}=N\) when \(p_{ACB}\) is set to 1, by substituting \(N_{pa}=N\), \(p_{ACB}=1\) into Equation (7), the set of \(L_{DS}\) to meet the delay requirement can be determined as:  
\[L_{set}=\left\{L_{DS}|\frac{T_{slot}}{\left(1-\frac{1}{L_{DS}}\right)^{N-1}} \leq D_{req},0<L_{DS}\leq L_{avail\_DS}\right\} \tag{9}\]  
reviewing Equation (8), we can subsequently obtain the optimal value of \(L_{DS}\) as:  
\[L_{DS}^{*}=\arg_{L_{DS}\in L_{set}}P_{L\_succ} \tag{10}\]  
For the special case that \(L_{set}\) calculated from Equation (9) is \(\varnothing\), we take \(L_{DS}^{*}=N\) referring to the conclusion we have mentioned before that \(P_{L\_succ}\) can achieve its maximal value when \(L_{DS}=N_{pa}\).