# Low-Latency Short-Packet Transmission over a Large Spatial Scale
## 4 Asymptotic Performance Analysis for Long-Distance Short-Packet Transmission
### Analysis of Relay Gains  
In this part, we express error probability \(\epsilon\) as a function of the number of relays \(L\). In this case, the maximal normalized energy \(E^{th}\) and the overall blocklength \(N^{th}\) are a function of \(L\). For instance, in practice, a longer distance means a more severe path loss, and in those cases, we should allocate more energy. We denote by \(E^{th}(L)\) the maximal normalized energy and by \(N^{th}(L)\) the overall blocklength, respectively. Certain types of \(E^{th}(L)\) and \(N^{th}(L)\) are now discussed. Define \(E=E^{th}(1)\) and \(N=N^{th}(1)\). We still use the same assumption in the last part to approximate \(\epsilon\). First, we let \(N^{th}(L)=N\) and \(E^{th}(L)=E\). The constraints do not change regardless of the number of relays. From Equation (40), we express \(\epsilon\) in terms of the number of relays \(L\), i.e.,  
\[\epsilon\approx\mu_{3}L^{\frac{3}{2}}\exp\biggl{(}-\frac{\theta_{3}}{L}\biggr{)}, \tag{47}\]  
where \(\mu_{3}=\frac{1}{\sqrt{2\pi N\ln\left(1+\frac{E}{N}\right)}}\) and \(\theta_{3}=\frac{N\ln\left(1+\frac{E}{N}\right)}{2}\). From Equation (47), the error probability increases with \(L\). This is because both communication resources, i.e., the blocklength and transmission power, are fixed. The resources allocated to each relay decreases with \(L\). Thus, the error probability of each hop increases with \(L\). Next, we let \(E^{th}(L)\) increase linearly with \(L\), i.e., \(E^{th}(L)=LE\). Moreover, we let \(N^{th}(L)=N\) be a fixed value. Under this assumption, we have:  
\[\epsilon\approx\frac{L^{\frac{3}{2}}\exp\biggl{(}-N\frac{\ln^{2}\left(1+\frac {LE}{N}\right)}{2L}\biggr{)}}{\sqrt{2\pi N}\ln(1+\frac{LE}{N})}. \tag{48}\]  
We denote by \(\epsilon=\mathcal{F}(L)\) the function of \(\epsilon\) and \(L\) in Equation (48). By solving \(\mathcal{F}^{\prime}(L)=0\), there is a local minimum point \(L_{0}\). (Actually, \(L_{0}\) may not be an integer. However, we do not use this fact in any essential way.) \(\mathcal{F}(L)\geq\mathcal{F}(L_{0})\) holds for all \(L\). Thus, there are an optimal number of relays \(L=L_{0}\) to minimize the error probability \(\epsilon\). When the number of relays \(L\) increases substantially, the error probability goes to infinity because there is a limited blocklength to be allocated. The discussion enlightens us that there is an optimal number of relays to choose in practice. Then, we let \(E^{th}(L)=LE\) and \(N^{th}(L)=LN\). In this case, we assume that the maximal tolerable latency is proportional to \(L\). It is a practical scenario when the total distance increases linearly with \(L\). Under this consumption, we have Equation (49), i.e.,  
\[\epsilon\approx\frac{L\exp\biggl{(}-N\frac{\ln^{2}\left(1+\frac{E}{N}\right)}{ 2}\biggr{)}}{\sqrt{2\pi N}\ln(1+\frac{E}{N})}, \tag{49}\]  
In this case, \(\epsilon\) increases linearly with \(L\). Last but not least, we let \(E^{th}(L)=L^{a}E\) and \(N^{th}(L)=N\). In this case, we increase the number of relays in a fixed total distance \(D\) from the source to the destination. The distance between each node is given by \(\frac{D}{L}\). Thus, the received SNR of each relay is proportional to \(L^{a}\), i.e., \(E^{th}(L)=L^{a}E\). The error probability is given by:  
\[\epsilon\approx\frac{L^{\frac{3}{2}}\exp\left(-N\frac{\ln^{2}(1+\frac{L^{a}E} {N})}{2L}\right)}{\sqrt{2\pi N}\ln(1+\frac{L^{a}E}{N})}. \tag{50}\]  
In Equation (50), there is also an optimal \(L\) to minimize \(\epsilon\). To summarize, we investigate four cases in this part as \(L\) changes in Equations (47)-(50). More generally, we let \(E^{th}(L),N^{th}(L)\) be any function of \(L\). We have Equation (51) for the general case.