# Preamble Design and Collision Resolution in a Massive Access IoT System
## 3 Preamble Design
### Sub-Preamble Sequence Design
Typically, the RA success probability in the conventional scheme is defined as the probability that a device uses an exclusive preamble and is allocated an exclusive PUSCH resource. Besides, \(p_{1}^{c}\) represents the probability that the PUSCH resources are wasted. Moreover, the RA failure probability in conventional scheme is equal to \(p_{1}^{c}+p_{2}^{c}+p_{3}^{c}\). Consequently, the RA success probability under conventional scheme can be expressed as

\[ \begin{split} P_{suc}^{c}&=p_{4}^{c}\\ &=(1-P_{collision}^{c})\cdot P_{res}^{c}\\ &=(1-\frac{1}{n_{p}})^{n_{k}-1}\cdot P_{res}^{c}\end{split}  \tag{14}\]

where \(P_{res}^{c}\) denotes the successful resource allocation rate for RA-step 3 under conventional scheme. Specifically, the PUSCH resources will be allocated to the IoT device in the RA-step 3 if its preamble is detected by the gNB sucessfully. Therefore, the successful resource allocation rate for RA-step 3 under the conventional scheme holds as

\[P_{res}^{c}=\min\{1,\frac{n_{PUSCH}}{Pr\{\xi_{m}\geq 1\}\cdot\min\{n_{k},n_{p} \}}\}, \tag{15}\]

where \(n_{PUSCH}\) denotes the number of available PUSCH resources, \(\xi_{m}\) denotes the number of devices selecting the \(m\)th preamble to perform access request. Since the devices are independent of each other, \(\xi_{m}\) follows a Binomial distribution. Hence, we have

\[Pr\{\xi_{m}\geq 1\}=1-Pr\{\xi_{m}=0\}=\binom{n_{k}}{0}(1-\frac{1}{n_{p}})^{n_ {k}}=(1-\frac{1}{n_{p}})^{n_{k}}. \tag{16}\]As a result, the RA success probability under conventional scheme can be obtained as

\[ \begin{split} P_{suc}^{c}&=p_{4}^{c}\\ &=(1-\frac{1}{n_{p}})^{n_{k}-1}\cdot P_{res}^{c}\\ &=(1-\frac{1}{n_{p}})^{n_{k}-1}\min\{1,\frac{n_{PUSCH}}{(1-\frac{ 1}{n_{p}})^{n_{k}}\cdot\min\{n_{k},n_{p}\}}\}\end{split}  \tag{17}\]

Similarly, there are 6 cases for PA collision and PUSCH resource allocation for the proposed RA procedure:

(1) \(p_{1}=Pr\)[no shared preamble collision, PUSCH resource allocated];

(2) \(p_{2}=Pr\)[no shared preamble collision, PUSCH resource unallocated];

(3) \(p_{3}=Pr\)[shared preamble collision, no dedicated preamble collision, PUSCH resource allocated];

(4) \(p_{4}=Pr\)[shared preamble collision, no dedicated preamble collision, PUSCH resource unallocated];

(5) \(p_{5}=Pr\)[shared preamble collision, dedicated preamble collision, PUSCH resource allocated];

(6) \(p_{6}=Pr\)[shared preamble collision, dedicated preamble collision, PUSCH resource unallocated]. It is easily verified that the RA success probability under the proposed scheme is equal to \(p_{1}+p_{3}\). On the contrary, the RA failure probability of the proposed scheme is equal to \(p_{2}+p_{4}+p_{5}+p_{6}\). Besides, \(p_{5}\) represents the PUSCH resources waste probability. Therefore, the RA success probability under the proposed scheme holds as

\[ \begin{split} P_{suc}&=p_{1}+p_{3}\\ &=(1-P_{collision})\cdot P_{res}\\ &=(1-\frac{1}{n_{p,s}n_{p,d}n_{\mu}})^{n_{k}-1}\cdot P_{res}\end{split}  \tag{18}\]

where \(P_{res}\) denotes the successful resource allocation rate under the proposed scheme. Additionally, let \(\xi_{s,m}\) denote the number of devices selecting the \(m\)th shared sub-preamble, \(\xi_{d,m}\) denote the number of devices selecting the \(m\)th dedicated sub-preamble, and \(\xi_{\mu}\) denote the number of devices selecting the \(\mu\)th index location.