# Resource Allocation of Multiple Base Stations for Throughput Enhancement in UAV Relay Networks
## 3 Simultaneous Transmissions from Multiple Base Stations in the URN (Multi-BSs-URN)  
We assume that mBS and sBS can transmit the same data to the UAV in the first time slot. Therefore, the SINR of a wireless backhaul link can be improved by simultaneous transmissions from both mBS and sBS when UAV moves inside mBS and sBS service ranges. When UAV receives signals from both BSs, the MRC is applied; hence, \(R_{xBS,UAV}^{1 st}\) can be obtained as \(R_{(mBS+sBS),UAV}^{1 st}=\log_{2}(1+SINR_{mBS,UAV}^{1 st}+SINR_{sBS,UAV}^{1 st})\). Based on the positions of mBS, sBS, UAV, and \(U_{m}\), two channel conditions can be assumed: (1) the channel condition between mBS and \(U_{m}\) is better than that between mBS and UAV (i.e., \(g_{mBS,U_{m}}>g_{mBS,UAV}\)), and (2) the channel condition between sBS and UAV is better than that between mBS and UAV (i.e., \(g_{sBS,U_{UAV}}>g_{mBS,UAV}\)). Therefore, \(SINR_{mBS,UAV}^{1 st}\) with \(P_{sBS}=0\) is used to calculate \(R_{(mBS+sBS),UAV}^{1 st}\) since UAV can eliminate interference from the sBS-to-UAV link when decoding data from the mBS-to-UAV link. In the second time slot,no transmission from sBS is assumed; therefore, \(P_{sBS}=0\) is in the denominators of (4) and (5). With the conditions that \(P_{sBS}=0\) in the denominator of (1),  
\[prob\Big{(}R_{\underline{i}_{L_{o}}}^{(1)}\geq\eta\Big{)} \tag{9a}\] \[=prob(\frac{P_{mBS,UAV}^{1 st}S_{mBS,UAV}}{P_{mBS,U_{m}}^{1 st}S_{mBS,UAV}+\sigma_{UAV}^{2}}+\frac{P_{sBS}\xi_{sBS,UAV}}{P_{mBS}S_{mBS,UAV}+\sigma_ {UAV}^{2}}\geq 2^{\omega^{+}-}-1)\] (9b) \[=E_{h_{mBS,UAV}}\left[prob\Big{(}h_{sBS,UAV}\geq\psi^{BS_{(m+s)}}( h_{mBS,UAV})\Big{)}\right]\] (9c) \[\stackrel{{(a)}}{{=}}E_{h_{mBS,UAV}}\left[1-\frac{ \gamma\Big{(}k_{sBS,UAV},k_{sBS,UAV}\psi^{BS_{(m+s)}}(h_{mBS,UAV})\Big{)}}{ \Gamma(k_{sBS,UAV})}\right]\] (9d) \[\stackrel{{(b)}}{{=}}1-\!\!\!\int\limits_{0}^{x_{ max}}\!\!\!\gamma\frac{\gamma\Big{(}k_{sBS,UAV},k_{sBS,UAV}\psi^{BS_{(m+s)}}(x) \Big{)}}{\Gamma(k_{sBS,UAV})}f_{h_{mBS,UAV}}(x)dx, \tag{9e}\]  
where  
\[\omega^{+}=\frac{2\eta}{1+\lambda},\] \[\psi^{BS_{(m+s)}}(h_{mBS,UAV})=\frac{(2^{\omega^{+}}-1)\sigma_{ UAV}^{4}-A_{1}h_{mBS,UAV}^{2}-A_{3}h_{mBS,UAV}}{A_{2}h_{mBS,UAV}+A_{4}},\] \[A_{1}=(P_{mBS,UAV}^{1 st}-1)P_{mBS,U_{m}}^{1 st}(P_{mBS,U_{m}}^{1 st }+P_{mBS,UAV}^{1 st})_{mBS,UAV}^{2},\] \[A_{2}=P_{mBS,U_{m}}^{1 st}P_{sBS,UAV}P_{sBS,UAV},\] \[A_{3}=\sigma_{UAV}^{2}|h_{mBS,UAV}(P_{mBS,UAV}^{1 st}-(2^{\omega^ {+}-1})(P_{mBS,U_{m}}^{1 st}+P_{mBS})),\] \[=prob(\frac{P_{mBS,UAV}^{1 st}S_{mBS,UAV}}{P_{mBS,U_{m}}^{1 st}S_{mBS,UAV}+\sigma_{UAV}^{2}}+\frac{P_{sBS}\xi_{sBS,UAV}}{P_{mBS}S_{mBS,UAV}+\sigma_ {UAV}^{2}}\geq 2^{\omega^{+}-}-1)\]0 \[=prob(\frac{P_{mBS,UAV}^{1 st}S_{mBS,UAV}}{P_{mBS,U_{m}}^{1 st}S_{mBS,UAV}+\sigma_{UAV}^{2}}+\frac{P_{sBS}\xi_{sBS,UAV}}{P_{mBS}S_{mBS,UAV}+\sigma_ {UAV}^{2}}\geq 2^{\omega^{+}-}-1)\]1  
\((a)\) in (9d) takes the expectation with respect to \(h_{mBS,UAV}\), and \((b)\) in (9e) follows that \(h_{mBS,UAV}\) has a \(Gamma(k_{mBS,UAV},1/k_{mBS,UAV})\) with a gamma function \(\Gamma(t)=\int\limits_{0}^{\infty}\!\!z^{t-1}e^{-z}dz\), and a lower incomplete gamma function \(\gamma(s,x)=\int\limits_{0}^{x}\!\!z^{s-1}e^{-z}dz\). The integral in (9e) should guarantee \(\psi^{BS_{(m+s)}}(x)\geq 0\); therefore, the integral interval needs to be selected carefully. Based on the fact that \(\psi^{BS_{(m+s)}}(0)\geq 0\), \(A_{2}\geq 0\), and \(A_{4}\geq 0\), it guarantees that \(\psi^{BS_{(m+s)}}(x)\geq 0\), \(\forall x\geq 0\) if \(A_{1}<0\). Otherwise, \(\psi^{BS_{(m+s)}}(x)\geq 0\), only with a proper upper limit of \(x_{max}\),  
\[x_{max}\!=\!\left\{ \begin{array}{ll}-A_{3}+\sqrt{A_{3}^{2}+4A_{1}\big{(}2^{ \omega^{+}-1}\big{)}\sigma_{UAV}^{4}}&\text{if $A_{1}\geq 0$,}\\ \infty&\text{if $A_{1}<0$.}\end{array} \right.