# Resource Allocation in 5G Cellular IoT Systems with Early Transmissions at the Random Access Phase
## 5 Numerical Analysis
### Fixed RA/DT Resource Allocation  
We begin by reporting the performance of the system where the amount of resources allocated for the RA and DT phases is fixed and no EDT regime is utilized. To this end, we start with the delay at the RA phase illustrated in Figure 7, where only uplink packets are present. Note that here and in what follows, we show the results for the stable system conditions that are assessed by utilizing the criterion provided in Section 4. Here, we observe a typical behavior of a system with decentralized access in stable conditions--as the arrival rate of EDs generating traffic in the uplink direction, \(\lambda_{r}\), increases, the mean delay increases exponentially. We specifically attract attention to the fact that, for a wide range of values of \(\lambda_{r}\), the mean delay stays extremely low and changes insignificantly, by just 6 ms. This implicitly highlights that, even for \(\lambda_{r}=5500\) pkts/s, the RA phase remains underloaded.  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Notation** & **Description** & **Value** \\ \hline \(B\) & Bandwidth & 180 KHz \\ \hline \(\Delta\) & Frame duration & 0.01 s \\ \hline \(l\) & Number of preambles & 12 \\ \hline \(P\) & RU (packet) size in uplink direction & 128 bytes \\ \hline \(W\) & Number of RU at the DT phase & 278 RUs \\ \hline \(n_{1}\) & Number of RU at the RA phase & 14 RUs \\ \hline \(n\) & Overall number of RU in a frame & 292 RUs \\ \hline \(\lambda_{r}\) & Uplink packet arrival rate & 2–5.5K pkts/s \\ \hline \(\lambda_{d}\) & Rate of software updates & 1, 5, 10 upd/s \\ \hline \(\{\gamma_{k}\}\) & Number of updates in a slot & Poisson \\ \hline \(\{l_{j}\}\) & pmf of the software update batch & Binomial \\ \hline \(\bar{l}\) & Mean software update size in RUs & 100/500 RUs \\ \hline \(p_{S}^{(E)}\) & Early transmission probability & 0, 0.2, 0.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Default system parameters.  
We now show the full mean delay experienced by uplink packets including both the RA and DT phases illustrated in Figure 8, where the values of the software update intensity \(\lambda_{d}\) are chosen such that the offered traffic load is 0% (\(\lambda_{d}=0\) upd./s) and \(\approx\)3% of the DT utilization (\(\lambda_{d}=1\) upd./s) and \(\approx\)30% (\(\lambda_{d}=10\) upd./s). By analyzing the results in Figure 8, we see that the presence of downlink traffic at the DT phase does not significantly contribute to the overall delay of uplink traffic. In fact, the difference between \(\lambda_{d}=0\) upd./s and \(\lambda_{d}=1\) upd./s is less than 1 ms and increases by 3-4 ms for \(\lambda_{d}=10\) upd./s. Furthermore, the gap between curves corresponding to different values of \(\lambda_{d}\) becomes smaller as the intensity of uplink traffic increases. This implies that not only the DT phase in modern CloT technologies may handle the additional load but this phase is severely overprovisioned in absence of software update traffic. As a result, to make mMTC CoI technologies scalable for diverse traffic load conditions, the amount of resource allocation to the RA and DT phases needs to be dynamically adjusted.  
Figure 8: The mean delay for uplink packets.  
Figure 7: The mean delay at the RA phase.