# CeRA-eSP: Code-Expanded Random Access to Enhance Success Probability of Massive MTC
## 3 System Model  
### Target System Model  
We considered a single cell consisting of \(N\) mMTC devices and one BS. The overview of the target system model is illustrated in Figure 1. A device becomes active when data are available for transmission. The activation patterns of devices are sporadic because the data collection of the devices differs depending on the sensing environment. This sporadicity renders it difficult to predict when and which device will be active in a single RA slot. The number of activated users at time \(t\in(0,T_{A})\) follows the beta distribution. Specifically, the beta distribution is used with the parameters \(\alpha=3\) and \(\beta=4\) of the 3 GPP standard as follows [27]:  
\[p(t)=\frac{t^{\alpha-1}(T-t)^{\beta-1}}{T_{A}^{\alpha+\beta-1}B(\alpha,\beta)}, \tag{1}\]  
where \(B(\alpha,\beta)\) denotes the beta function, \(\int_{1}^{1}t^{\alpha-1}(1-t)^{\beta-1}dt\). The beta distribution is commonly used in mMTC RA scenarios where bursty traffic occurs [24; 28].  
As shown in Figure 1, the PUSCH waiting message was used in our system model. The PUSCH waiting message is used when the PUSCH resources are insufficient to support all the preamble codewords that are transmitted in the specific RAO. In this case, some preamble codewords cannot be allocated using PUSCH resources because of the lack of PUSCH resources in the corresponding RAO. Devices that have transmitted the PUSCH resource-unallocated preamble codeword are dropped in the existing RA scheme, leading to a decrease in the RA success rate. To prevent a decrease in the RA success rate, the BS identifies the successfully received preamble codeword and broadcasts the PUSCH waiting message to the PUSCH resource-unallocated devices. The devices that correspond to the waiting preamble codeword do not perform RA retransmission in the next RAO, although they monitor the RAR message for the given PUSCH waiting time window.  
Among the existing RA schemes, the back-off scheme improves the RA success rate by distributing the access intensity to one RAO [29]. Similar to the back-off scheme, the PUSCH waiting message can improve the RA success rate by distributing high-access requestsover other RAOs. Specifically, in a situation where high access intensity continues, a large preamble codeword is set to lower the high collision rate. If the size of the transferable preamble codeword set is larger than that of the PUSCH resource, the BS transmits a PUSCH waiting message to some devices because of the lack of PUSCH resources. Devices that have received the PUSCH waiting message wait for the PUSCH waiting expiration and retransmit the preamble codeword. Therefore, the PUSCH waiting message can lower the access intensity by setting a time interval for devices with high access intensity.  
Figure 2 shows the state transition of the device during the RA. The active device had sufficient power and data required for RA, and the activation of the device indicated that the device could perform RA. After the preamble transmission, the device waits for the RAR message including the PUSCH resource. If the PUSCH resource is allocated, the device transitions to the complete state of the communication setup; otherwise, the device performs RA retrial up to the maximum number of preamble transmissions allowed. When the devices reach the maximum preamble transmission, they transition to the communication setup failure state. Our system model contained a single-shot RA case for simplicity of the analysis model. Single-shot RA means that once a device falls into a communication setup failure state, the RA reattempt for the device is no longer considered. Notably, the system model consisted of the RA retrial until it reached the maximum number of preamble transmissions without further retransmission after the communication setup failure.  
Figure 1: System model.