# Dynamic RACH Partition for Massive Access of Differentiated M2M Services
## 3 System Model
### Random Access Procedures
Here, it is worth noting that there are several _singular points_ changing suddenly. According to the conceptual design discussed in Section 4, \(A_{s}f_{s}=1\) is taken as the \"critical condition\" distinguishing between two different ways of analysis, as well as solving. In Figure 5, it is verified that _singular points_ approximately appeared near \(A_{s}f_{s}=\frac{A_{s}f_{s}}{1+\gamma}=1\). ### Analysis of Average Access Delay

The access delay includes two main parts comprised of the random backoff _Tbarred_ in Equations (24) and (25) due to ac_barringfactor, as well as the fixed backoff due to collisions. Since there is a random number (defined as _rand_) ranging from zero to one in _Tbarred_, here we consider the potential upper bound and lower bound during one slot. First, in the case of the lower limit, we assume _rand = 0_ in Equations (24) and (25), and the AAD (seconds) is described as:

\[ \begin{split} D_{lower}=& A_{s}(1-f_{s})\times(0.7+ 0.6*0)*4+A_{n}(1-f_{n})\times(0.7+\frac{0.6*0}{f_{n}})*4\\ &+P_{\text{C}}\left(A_{s}f_{s}+A_{n}f_{n}\right)\times 5\times 1 0^{-3}\end{split}  \tag{22}\]

where \(5\times 10^{-3}\), denoting five milliseconds, represents the fixed backoff due to collisions. That is to say, once collision occurs, the access backs off for 5 ms.

Figure 5: Collision probability varies with the synchronously change of \(A_{i}\) and \(f_{s}\) during one slot. Then, in the case of the upper bound, we assume \(rand=1\), and similarly, the AAD (seconds) is expressed as:

\[ \begin{split} D_{upper}=& A_{s}(1-f_{s})\times(0.7+0.6 *1)*4+A_{n}(1-f_{n})\times(0.7+\frac{0.6*1}{f_{n}})*4\\ &+P_{\text{C}}\left(A_{s}f_{s}+A_{n}f_{n}\right)\times 5\times 1 0^{-3}\end{split}  \tag{23}\]

## 6 Simulation Evaluation

In this section, we present the setup of the simulation parameters and scenarios, as well as the simulation results that demonstrate the advantageous performance of the proposed scheme for solving the RAN overload problems. ### Simulation Setup

According to 3 GPP protocols [39], the detailed LTE-A simulation parameters [40], as well as the typical traffic model [34] are depicted in Table 1. In order to evaluate our proposed scheme, we need to illustrate the basic parameters in the common ACB mechanism available in the proposed scheme. (**a**)As discussed in Section 4, we first define the proportion between sensitive devices and non-sensitive devices as 1:9 due to the low incidences of sensitive devices according to our comprehensive research. (**b**) As for the two parameters broadcast in the ACB mechanism, we shall assume that the ac_barringtime of two clusters is equal to 4 s. According to [41], in case of being barred, a sensitive device re-attempts access after a delay of:

\[\mathit{Tbarred}=(0.7+0.6*rand)*ac\_barringtime \tag{24}\]

where \(rand\) represents a random number uniformly drawn from \([0,1)\). Yet, the non-sensitive device re-attempts access after a delay of:

\[\mathit{Tbarred}=(0.7+\frac{0.6*rand}{ac\_barringfactor})*ac\_barringtime \tag{25}\]

(**c**) According to the ACB mechanism, ac_barringfactor ranges from zero to one. Hence, we shall select suitable values for \(p\) and \(q\) in \(f_{n}=p+q*f_{s}\) in order to satisfy \(f_{n}\in(0,1)\) when \(f_{s}\) is given in advance. Accordingly, through simulations in Figures 6 and 7, on the basis of the considerations of ASP and AAD, we choose the linear correlation of \(f_{s}\) and \(f_{n}\) as: \(f_{n}=0.6+\frac{1}{3}*f_{s}\), which can achieve the optimal overall performance. Notably, there is a sharp decline in Figure 6 when \(f_{s}\) changes from 0.8 to 0.9 in the case of \(f_{n}=0.6+\frac{1}{3}*f_{s}\). This is because when \(f_{s}\) changes from 0.8 to 0.9, \(f_{n}\) changes from 0.87 to 0.9, which means nearly 90% of delay-tolerant devices would compete for access. In addition, due to the huge number of delay-tolerant devices, devices competing for access do not successfully access the network more easily compared to the situations for fewer devices.