# Preamble Design and Collision Resolution in a Massive Access IoT System
## 3 Preamble Design
### Sub-Preamble Sequence Design
The RAR message contains the TA command, uplink resource grant for MSG3, the shared sub-preamble index, the dedicated sub-preamble index and the location index \(\mu\). * Step-3: MSG3 transmission. Devices can obtain their own RAR messages by confirming the shared sub-preamble index, the dedicated sub-preamble index and the location index \(\mu\). If RAR message is decoded correctly, the corresponding device will transmit MSG3 data on the PUSCH resource blocks that are allocated at step-2. As a result, the gNB can recognize that an RA collision occurs in this case. * Step-4: Contention resolution. If the gNB successfully decodes MSG3, it will transmit a contention resolution message to the corresponding device. Such message contains the device ID that includes sub-preamble indexes and location index. Otherwise, nothing will be sent to the device. If the device receives contention resolution message, it will send back an ACK message and complete RA procedure. Otherwise, the device will restart a new RA procedure. Under the proposed RA procedure, the preamble collision occur only if the shared and dedicated sub-preamble collisions occur at the same time. Additionally, the proposed RA scheme can still work when one type of the sub-preamble collision occur. In the next section, the effectiveness of the proposed RA scheme will be proved analytically. ## 5 Performance Analysis

In this section, the RA collision probability and the RA success probability under the conventional and proposed schemes are derived analytically. ### Collision Probability

Under the proposed scheme, the preamble collision probability is associated with the probabilities of sub-preamble collisions as well as location collision of the dedicated sub-preambles. In other words, the preamble collision occurs when multiple devices select the same shared sub-preambles, the same dedicated sub-preambles as well as the same location indexes at the same time. Assuming that \(n_{k}\) devices attempt RAs with \(n_{p}\) available preambles. Consider a target device with deterministic preamble selection, the probability that another device users the same preamble holds as \(1/n_{p}\). Since the devices are independent of each other, the preamble collision probability for the target device under the conventional RA [24] scheme holds as

\[P_{collision}^{c}=1-(1-\frac{1}{n_{p}})^{n_{k}-1}. \tag{11}\]

For the proposed RA scheme, we assume that \(n_{k}\) devices attempt random access with \(n_{p,s}\) shared sub-preambles and \(n_{p,d}\) dedicated sub-preambles, and the number of available location index is \(n_{\mu}\). For the target device and another device, if the shared sub-preamble, dedicated sub-preamble and location index used by those two devices are all the same, preamble collision will occur for the target device. The probability that another device usesidentical shared sub-preamble, dedicated sub-preamble and location index at the same time holds as

Figure 2.0

Hence, the preamble collision probability for the target device under the proposed RA scheme holds as

Figure 2.1

Comparing Equation (11) with Equation (13), when the number of available preambles in conventional RA scheme is equal to the number of available sub-preambles in the proposed RA scheme, i.e., \(n_{p}=n_{p,s}\), \(n_{p}=n_{p,d}\), the proposed scheme can guarantee lower collision probability than the conventional scheme. ### RA Success Probability

During a complete RA procedure, the gNB allocates PUSCH resources for MSG3 transmission after successful preamble detection and TA capturing. For a conventional RA procedure, there are 4 cases for preamble collision and PUSCH resource allocation:

1. \(p_{1}^{c}=Pr\{\text{preamble collision, PUSCH resource allocated}\}\);
2. \(p_{2}^{c}=Pr\{\text{preamble collision, PUSCH resource unallocated}\}\);
3. \(p_{3}^{c}=Pr\{\text{no preamble collision, PUSCH resource unallocated}\}\);
4. \(p_{4}^{c}=Pr\{\text{no preamble collision, PUSCH resource allocated}\}\);

where \(p_{4}^{c}\) represents the RA success probability in conventional scheme.