# Low-Latency Short-Packet Transmission over a Large Spatial Scale
## 5 Simulation Results  
In this section, numerical and simulation results are presented to validate the theoretical analysis and demonstrate the advantages of our work. First, Figures 3 and 4 present the error probability with respect to the overall block-length and the overall energy, respectively. The two curves are drawn for the loss exponent \(\alpha=2\) and \(\alpha=2.2\). We consider the transmission of a one-hundred-bit data packet with the help of two relays, i.e., \(L=3\). The distance from the source to the destination is \(D=6\). The normalized unit distance is \(d_{0}=1\). In Figure 3, the overall normalized energy \(E^{th}\) is set as 4200 and the overall blocklength \(N^{th}\) ranges from 200 to 500. In Figure 4, the overall blocklength \(N^{th}\) is set as 450 and the overall normalized energy \(E^{th}\) ranges from 2400 to 4800. In the subfigure of Figure 3, the relative error is less than \(10^{-5}\), which is mainly caused by the rounding function of \(\bar{n}=[\frac{N^{th}}{L}]\). In Figure 4, the relative error is less than \(10^{-4}\), which is mainly caused by the approximation of \(\epsilon\) in Equation (6). It is easy to see that the theoretical results obtained via Theorem 3 match well with their corresponding simulation results. Furthermore, we can see that a larger \(\alpha\) leads to a higher error probability. Next, we present the numerical result of the optimal solution in Figures 5 and 6. The number of relays was set as \(L=5\). Figure 5 presents the overall blocklength versus the error probability curves for three average transmission powers, namely \(\bar{\gamma}=1.5\), \(\bar{\gamma}=2\), and \(\bar{\gamma}=2.5\). Figure 6 presents the overall normalized energy versus error probability curves for three average blocklengths, namely \(\bar{n}=60\), \(\bar{n}=80\), and \(\bar{n}=100\). In Figure 7, we present the optimal \(\epsilon\) curves with respect to the number of relays \(L\) when different kinds of \(E^{th}(L)\) and \(N^{th}(L)\) are given. In most cases, the error probability increases with the number of relays \(L\). However, under the circumstance where \(E^{th}(L)=L^{2}E\), \(N^{th}(L)=N\), we have an optimal \(L=2\). Finally, Figure 8 presents the optimal distance-latency-reliability tradeoff. The parameters were set as follows: the number of relays \(L=3\), the overall normalized energy \(E^{th}=4150\), and the symbol duration \(\tau=2\times 10^{-4}\). We chose the total distance \(D\in[300\) km,600 km]. The normalized distance was unitized by a factor \(c\tau\), where \(c\) is the speed of light. Figure 8a shows the reliability with respect to the distance and latency. Figure 8b-d presents the projection of the surface when the distance, latency, or reliability is fixed. Specifically, Figure 8b shows the reliability-latency tradeoff when the total distance is fixed as 360 km. Figure 8c presents the reliability-distance tradeoff when the latency is set as a constant of 10.14 ms. Figure 8d shows the latency-distance tradeoff when the error probability is \(1\times 10^{-12}\). Figure 3: The numerical result of the optimal error probability–blocklength curve. From the simulation results, we can draw some conclusions about the advantages of our model as follows. First, we consider both propagation delay and decoding delay. Our model was established to assess the performance of large-scale low-latency communication systems, which has not been included in any other previous works. We obtained a low-complexity solution for the error probability by giving an analytical solution of the relaxed problem. Furthermore, we attained the best resource allocation policy. Last but not least, we determined the theoretical limit of our model, under the condition of whether the number of relays is fixed or not. In particular, our result determines the minimum error probability of relaying communication given the latency, power allocation scheme, and the relay nodes' positions. Figure 4: The numerical result of the optimal error probability–energy curve.