# Resource Allocation in 5G Cellular IoT Systems with Early Transmissions at the Random Access Phase
## 5 Numerical Analysis
### Performance with EDT Enabled  
Finally, we assess the performance when the EDT regime is enabled. We begin with the delay of packets in the uplink direction \(w_{r}\). To this end, Figure 10 illustrates the study metric as a function of the rate of software updates in the downlink direction, where \(T_{S}\) is set to 0, that is, all uplink packets require service at both the RA and DT phases for \(\lambda_{d}=1,5,10\) upd/s. As one may observe, the least delay is attained for the lowest load in the downlink direction, i.e., \(\lambda_{d}=1\) upd/s. When this rate increases, the delay increases linearly. The rationale is that uplink packets share the same DT phase as downlink packets. It can be observed that the delay also increases as \(\lambda_{r}\) increases but the difference between the different curves becomes smaller. The rationale is that, for high values of \(\lambda_{r}\), more load is imposed in the DT phase and the impact of the RA phase on the overall delay of the uplink packets becomes smaller.  
Figure 9: Fraction of resources allocated for the DT phase.  
Furthermore, Figure 11 shows the packet delay in the uplink direction as a function of the packet size threshold, \(p_{S}^{(E)}\), for a fixed value of \(\lambda_{d}=5\) upd/s. Here, we can see that the impact of \(p_{S}^{(E)}\) is significant. Specifically, when half of the packets can already be transmitted in the RA phase, the delay decreases by 20-40%. This gap also decreases as \(\lambda_{d}\) increases but is still noticeable. By comparing these results to those shown in Figure 10, we see that the system with \(p_{S}^{(E)}=0.5\) and \(\lambda_{d}=5\) upd/s performs twice better than the one with \(p_{S}^{(E)}=0\) and \(\lambda_{d}=10\) upd/s. This behavior shows that early transmission is a useful mechanism for mMTC systems.  
In previous illustrations, we have seen that both the rate of software updates in the downlink direction \(\lambda_{d}\) and the packet size threshold \(p_{S}^{(E)}\) significantly affect uplink packet delay. However, these illustrations were prepared by assuming a fixed distribution of resources between the RA and DT phases. Now, we demonstrate the optimal allocation of these resources as a function of the parameters \(\lambda_{d}\) and \(p_{S}^{(E)}\) in  
Figure 11: The mean delay for uplink packets as a function of \(p_{S}^{(E)}\).  
Figure 10: The mean delay for uplink packets as a function of \(\lambda_{d}\).  
respectively. Starting with the former, plotted for different \(\lambda_{d}\) and fixed \(p_{S}^{(E)}=0\), we observe that the optimal fraction of resources allocated for the DT phase varies drastically, that is, the range of values is 0.75-0.95, which deviates significantly from the default division. Getting deeper into details, we highlight that the joint impact of \(\lambda_{d}\) and \(\lambda_{r}\) is important. Specifically, for small values of \(\lambda_{d}\), the difference between the fraction of resources that must be allocated for the DT phase between \(\lambda_{d}=1\) upd/s and \(\lambda_{d}=10\) upd/s is just 0.04. However, for \(\lambda_{r}=5000\), it was already 0.13.  
The impact of different values of \(p_{S}^{(E)}\) on the optimal fraction of resources allocated to the DT phase shown in Figure 13 for \(\lambda_{d}=5\) upd/s is more straightforward. Here, we logically see that as \(\lambda_{r}\) increases, the study metric also decreases linearly. The rationale is that early transmission affects the RA phase significantly unloading the DT phase, which heavily contributes to the delay under high traffic conditions. In general, the presented results show that the optimal performance of 5G mMTC technologies with or without early transmission mode can only be attained if dynamic resource allocation between the RA and DT phases is feasible.  
Figure 12: Fraction of resources allocated for the DT phase.  
Figure 13: Fraction of resources allocated for the DT phase.