# Dynamic RACH Partition for Massive Access of Differentiated M2M Services
## 3 System Model
### Random Access Procedures
First, we determine the feasible domain described as:\[D_{1}=\left\{\beta\left|\beta\geqslant\frac{A_{s}f_{s}}{c*\log(A_{s}f_{s})-A_{s}f_{ s}}\right.\right\} \tag{11}\]

and then determine the monotonicity of the objective function in the feasible domain. Since the objective function is monotonically decreasing within the feasible domain, the maximum value of the objective function is gained at the boundary of the feasible region, namely:

\[\beta^{*}=\frac{A_{s}f_{s}}{c*\log(A_{s}f_{s})-A_{s}f_{s}} \tag{12}\]

**Case 3:**\(A_{s}f_{s}\in[3,+\infty)\)

Similar to **Case 2**, the feasible domain under this case is illustrated as:

\[D_{2}=\left\{\beta\left|\beta\geqslant\frac{A_{s}f_{s}}{c*\log(A_{s}f_{s}/2)-A_ {s}f_{s}}\right.\right\} \tag{13}\]

Similar to the analysis in **Case 2**, the optimal \(\beta^{*}\) is expressed as:

\[\beta^{*}=\frac{A_{s}f_{s}}{c*\log(A_{s}f_{s}/2)-A_{s}f_{s}} \tag{14}\]

In summary, according to the conceptual design discussed above in a certain slot, once we have known the \(A_{s},A_{n},f_{s},f_{n}\), the eNB would obtain the optimal preamble partition based on the proposed scheme. ### Access Class Barring for Two Clusters Respectively

After obtaining the preamble partition, the devices belonging to different clusters adopt the ACB scheme respectively to access the network. According to the ACB mechanism, we propose that the ac_barringtime of two clusters is configured as the same. In order to simplify our proposed architecture, the connection between two ac_barringfactors would be devised as the following linear correlation: \(f_{n}=p+q*f_{s}\) where \(f_{s}\in(0,1)\) and the value of \(p\) and \(q\) should satisfy \(f_{n}\in(0,1)\). In Section 6, we will show the parameter settings and illustrate the proofs in detail. ## 5 Analysis of the Resource Partition Scheme

In this part, we mainly address the theoretical performance analysis of the proposed scheme. Before that, the estimation of the access loads should be taken into consideration, since we assume that the traffic loads during one slot are pre-known in the proposed scheme. As previously mentioned, once obtaining \(\beta^{*}\) in a slot, the eNB would allocate the corresponding number of preambles to the two clusters based on the proposed preamble partition scheme. Based on the number of allocated preambles and the collision status in each slot, we can apply a Markov-based approach, which was developed in [38], to well estimate the traffic load of each cluster dynamically. Next, before analyzing the performance of the proposed scheme, we consider the following performance indexes:

(**a**) _Access success probability (ASP)_, defined as the ratio between total devices completing the RA procedures and the total access attempts within the same slot. (**b**) _Preamble collision probability (PCP)_, defined as the ratio between the number of preamble collision and the total preambles transmitted within the same slot. (**c**) _Average access delay (AAD)_, defined as the average delay between the first attempt and the completion of all RA procedures for the devices that successfully access the network. ### Analysis of Access Success Probability

Under the assumptions and notations defined in Section 4, we will subsequently analyze the performance from three aspects. Additionally, so as to compare with the simulation results in Section 6, we suppose that the ratio of the number for two clusters is denoted as \(\gamma\). Then, we can get \(A_{n}/A_{s}=\gamma\), as well as \(A_{s}+A_{n}=A_{i}\).