# Dynamic RACH Partition for Massive Access of Differentiated M2M Services
## 3 System Model
### Random Access Procedures
Accordingly, \(A_{s}\) and \(A_{n}\) are expressed as follows:

\[A_{n}=\frac{\gamma A_{i}}{1+\gamma} \tag{15}\]

\[A_{s}=\frac{A_{i}}{1+\gamma} \tag{16}\]

Next, once we have solved the \(\beta^{*}\) in the proposed scheme, we would obtain the number of devices successfully completing the access attempts \(S_{N}\) in a certain slot by substituting \(\beta^{*}\) into Equation (7), described as:

\[S_{N}=A_{s}f_{s}\times e^{-\frac{A_{s}f_{s}(1+\beta^{*})}{\varepsilon^{\beta^{ *}}}}+A_{n}f_{n}\times e^{-\frac{A_{n}f_{n}(1+\beta^{*})}{c}} \tag{17}\]

Thus, the ASP in a certain slot is calculated by:

\[P_{S}=\frac{A_{s}f_{s}\times e^{-\frac{A_{s}f_{s}(1+\beta^{*})}{\varepsilon^{ \beta^{*}}}}+A_{n}f_{n}\times e^{-\frac{A_{n}f_{n}(1+\beta^{*})}{c}}}{A_{i}} \tag{18}\]

Then, substituting Equations (15) and (16) into Equation (18), the expression Equation (18) can be simplified as:

\[\frac{A_{s}f_{s}}{c}\left(1+\frac{1}{\beta}\right)-\frac{A_{nf}f_{n}}{c}(1+ \beta)+2\log\beta=\log\left(\frac{A_{s}^{2}f_{s}^{2}}{A_{nf}^{2}f_{n}^{2}}\right) \tag{10}\]0

In accordance with the simulation setup in Section 6, here we generally present the parameter settings in advance. First we devise the connection between two ac_barringfactors as: \(f_{n}=0.6+\frac{1}{3}*f_{s}\), which are explained in Section 6.1. Moreover, the ratio of the number for two clusters is defined as \(\gamma=9\). The traffic model is devised as a \(\beta\)-distribution within 10 s. In this way, from a realization example of the number of MTC arrivals within 10 s, we can see that the number of access attempts falls within the range from one to 50. It is worth noting that in our analysis, we set the lower bound for \(A_{i}\) as one instead of zero, which is the actual minimum just to avoid the denominator in Equation (18) being zero, while not affecting the analysis results. On the other hand, we set the maximum value for \(A_{i}\) as 50. Since the number of access attempts during every slot is rand, here we typically choose two extremes to present the analysis on ASP. One is the maximum number of access attempts; the other is the minimum one. Here, we depict the relationships between \(f_{s}\) and \(P_{S}\) in two extreme cases respectively in \[\frac{A_{s}f_{s}}{c}\left(1+\frac{1}{\beta}\right)-\frac{A_{nf}f_{n}}{c}(1+ \beta)+2\log\beta=\log\left(\frac{A_{s}^{2}f_{s}^{2}}{A_{nf}^{2}f_{n}^{2}}\right) \tag{10}\]1 It can be seen that the ASPs of two extreme scenarios are both rising with the increase of ac_barringfactor. Meanwhile, the larger the access load is, the worse the access condition becomes. \[\frac{A_{s}f_{s}}{c}\left(1+\frac{1}{\beta}\right)-\frac{A_{nf}f_{n}}{c}(1+ \beta)+2\log\beta=\log\left(\frac{A_{s}^{2}f_{s}^{2}}{A_{nf}^{2}f_{n}^{2}}\right) \tag{10}\]2 The access success probability (ASP) of two extreme cases _versus_\(f_{s}\) during one slot. ### Analysis of Preamble Collision Probability

Similarly, on the basis of the derivation in ASP performance, we can analyze the preamble collision probability along this way. Inspired by Equations (5) and (7), we present the PCP during one slot defined as \(P_{\text{C}}\):

\[\frac{A_{s}f_{s}}{c}\left(1+\frac{1}{\beta}\right)-\frac{A_{nf}f_{n}}{c}(1+ \beta)+2\log\beta=\log\left(\frac{A_{s}^{2}f_{s}^{2}}{A_{nf}^{2}f_{n}^{2}}\right) \tag{10}\]3

After similar arrangements and simplifications, Equation (20) is described as Equation (21):

\[\frac{A_{s}f_{s}}{c}\left(1+\frac{1}{\beta}\right)-\frac{A_{nf}f_{n}}{c}(1+ \beta)+2\log\beta=\log\left(\frac{A_{s}^{2}f_{s}^{2}}{A_{nf}^{2}f_{n}^{2}}\right) \tag{10}\]4

We depict the three-dimensional curve, which presents the variation tendency of \(P_{\text{C}}\) when \(A_{i}\) and \(f_{s}\) synchronously change in \[\frac{A_{s}f_{s}}{c}\left(1+\frac{1}{\beta}\right)-\frac{A_{nf}f_{n}}{c}(1+ \beta)+2\log\beta=\log\left(\frac{A_{s}^{2}f_{s}^{2}}{A_{nf}^{2}f_{n}^{2}}\right) \tag{10}\]5 During a certain slot, we can observe that the PCP is increasing as \(f_{s}\) raises. Furthermore, the PCP also climbs with the rising of the total attempts during one slot.