# Low-Latency Short-Packet Transmission over a Large Spatial Scale
## 3 Optimal Relay Deployment and Resource Allocation
### An Approximate, but Analytical Solution
Substituting the negative condition property into Equation (32), we have:  
\[\mathcal{H}_{\gamma}^{{}^{\prime\prime}} =\Bigl{(}(\mathcal{G}_{\gamma}^{{}^{\prime}})^{2}\mathcal{G}- \mathcal{G}_{\gamma}^{{}^{\prime\prime}}\Bigr{)}\exp\biggl{(}-\frac{\mathcal{ G}^{2}}{2}\biggr{)} \tag{34}\] \[>(\mathcal{G}_{\gamma}^{{}^{\prime}})^{2}\mathcal{G}\exp\biggl{(}- \frac{\mathcal{G}^{2}}{2}\biggr{)}\] \[>0.\]  
Similarly, \(\mathcal{H}_{n}^{{}^{\prime\prime}}>0\). To deal with the cross partial derivative \(\mathcal{H}_{n\gamma}^{{}^{\prime\prime}}\), we calculate the factor in the first round brackets in Equation (33) as:  
\[ \begin{split}\mathcal{G}_{n}^{{}^{\prime}}\mathcal{G}_{\gamma}^{{}^{ \prime}}-\mathcal{G}_{n\gamma}^{{}^{\prime\prime}}=& n^{-\frac{1}{2}}[\gamma(\gamma+2)]^{-\frac{3}{2}}\bigg{\{} \bigg{[}\frac{\gamma(\gamma+2)-\big{(}\ln(1+\gamma)-\frac{k\ln 2}{n}\big{)}}{2} \bigg{]}\times\\ &\bigg{[}n\bigg{(}\ln(1+\gamma)-\frac{k\ln 2}{n}\bigg{)}\bigg{(}\ln(1+ \gamma)+\frac{k\ln 2}{n}\bigg{)}-1\bigg{]}-\frac{k\ln 2}{n}\bigg{\}}.\end{split}  \tag{35}\]  
Since \(\ln(1+\gamma)>\frac{k\ln 2}{n}\), we have:  
\[ \begin{split}\Big{(}n\big{(}\ln(1+\gamma)-\frac{k\ln 2}{n} \big{)}\big{(}\ln(1+\gamma)+\frac{k\ln 2}{n}\big{)}-1\Big{)}&> \Big{(}2n\big{(}\ln(1+\gamma)-\frac{k\ln 2}{n}\big{)}\frac{k\ln 2}{n}-1 \Big{)}\\ &=2k\ln 2\big{(}\ln(1+\gamma)-\frac{k\ln 2}{n}\big{)}-1\\ &>1,\end{split}  \tag{36}\]  
which applies the condition of (30d). Thus,  
\[ \begin{split}\mathcal{G}_{n}^{{}^{\prime}}\mathcal{G}_{\gamma}^{{}^ {\prime}}-\mathcal{G}_{n\gamma}^{{}^{\prime\prime}}>& n^{-\frac{1}{2}}\big{[}\gamma(\gamma+2) \big{]}^{-\frac{3}{2}}\bigg{[}\frac{\gamma(\gamma+2)-\ln(1+\gamma)-\frac{k\ln 2}{n}}{2} \bigg{]}\\ >& n^{-\frac{1}{2}}\big{(}\gamma(\gamma+2)\big{)}^{ -\frac{3}{2}}\bigg{[}\frac{\gamma(\gamma+2)-2\ln(1+\gamma)}{2}\bigg{]}\\ >& 0.\end{split}  \tag{37}\]  
Since \(\mathcal{G}_{n}^{{}^{\prime}}\mathcal{G}_{\gamma}^{{}^{\prime}}-\mathcal{G}_ {n\gamma}^{{}^{\prime\prime}}>0\) and \(\mathcal{G}_{n\gamma}^{{}^{\prime\prime}}<0\), from Appendix A, we obtain:  
\[ \begin{split}\mathcal{H}_{n\gamma}^{{}^{\prime\prime}}\mathcal{H }_{n\gamma}^{{}^{\prime\prime}}&=(\mathcal{G}_{n}^{{}^{\prime}} \mathcal{G}_{\gamma}^{{}^{\prime}}-\mathcal{G}_{n\gamma}^{{}^{\prime\prime}}) ^{2}\exp\left(-\mathcal{G}^{2}\right)\\ &<(\mathcal{G}_{n}^{{}^{\prime}}\mathcal{G}_{\gamma}^{{}^{\prime \prime}})^{2}\exp\left(-\mathcal{G}^{2}\right)\\ &<\mathcal{H}_{n}^{{}^{\prime\prime}}\mathcal{H}_{\gamma}^{{}^{ \prime\prime}}.\end{split}  \tag{38}\]  
This yields that \(\
abla^{2}\mathcal{H}\) is positive definite. The proof is complete. Theorem 2 proves the convexity of the objective function mathematically. Furthermore, we have the optimal solution of Problem (30) as follows. Theorem 3: _The optimal solution of Problem (30) is \(LQ\Big{(}\mathcal{G}\Big{(}\frac{E^{th}}{N^{th}},\frac{N^{th}}{L}\Big{)}\Big{)}\), when we set \(n_{\iota}=\frac{N^{th}}{L}\) and \(\gamma_{\iota}=\frac{E^{th}}{N^{th}}\) for each \(\iota=1,\ldots,L\)._  
Proof of Theorem 3.: From Theorem 2, we have that \(Q(\mathcal{G}(\gamma,n))\) is a convex function. Applying [PERSON]'s inequality, we have:  
\[\epsilon\geq LQ\Bigg{(}\mathcal{G}\bigg{(}\frac{\sum_{\iota=1}^{L}\gamma_{ \iota}}{L},\frac{\sum_{\iota=1}^{L}n_{\iota}}{L}\bigg{)}\Bigg{)}=LQ\Bigg{(} \mathcal{G}\bigg{(}\frac{E^{th}}{N^{th}},\frac{N^{th}}{L}\bigg{)}\Bigg{)}. \tag{39}\]  
The proof is complete. Theorem 3 states that the system achieves the highest reliability when the transmission power and blocklength of the relays are the same. Furthermore, we obtain an analytical expression of the optimal error probability. This relay resource allocation strategy is intuitively reasonable, corresponding to the equal distance between each successive relay pair. Finally, from the discussion above, we generate a suboptimal result of the initial problem.