# An Optimal Resource Allocation Scheme with Carrier Aggregation in 5G Network under Unlicensed Band
## 3 System Model
### Mean Packet Transmission Delay for Delay-Sensitive UE QoS Requirement  
We model one M/G/1 queue in each AP for downlink communications to analyze the time-domain behaviors of delay-sensitive UE; then, the mean packet transmission delay in small-cell AP, \(E[D_{s}|b_{s},p_{s}^{t}]\), is shown as [18]  
\[E[D_{s}|b_{s},p_{s}^{t}]=E[S_{s}|b_{s},p_{s}^{t}]+E[W_{s}|b_{s},p_{s}^{t}], \tag{21}\]  
Figure 2: Mean ergodic achievable rate for rate-sensitive UE as a function of \(\gamma\) with Wi-Fi system performance threshold. where \(E[S_{s}|b_{s},p_{s}^{t}]\) is the mean service time. The mean waiting time for packets in the AP \(E[W_{s}|b_{s},p_{s}^{t}]\) with given \(b_{s}\) and \(p_{s}^{t}\) is shown as [34]  
\[E[W_{s}|b_{s},p_{s}^{t}]=E[L_{s}]E[S_{s}|b_{s},p_{s}^{t}]+\rho_{s}E[R_{s}|b_{s},p _{s}^{t}], \tag{22}\]  
where \(E[L_{s}]\) denotes the number of packets in small-cell APs and \(\rho_{s}=\lambda_{s}E[S_{s}]\) is the utility ratio. \(E[S_{s}|b_{s},p_{s}^{t}]\) and \(E[R_{s}|b_{s},p_{s}^{t}]\) are mean service time and residual service time, respectively. \(\rho_{s}=\lambda_{s}E[S_{s}]\) is the utility ratio and, in order to ensure the stability of the queue system, condition \(\rho_{s}\leq 1\) must be satisfied. With the Little's law and relationship between service time and residual service time [34], \(E[R_{s}|b_{s},p_{s}^{t}]=E[S_{s}^{2}|b_{s},p_{s}^{t}]/2E[S_{s}|b_{s},p_{s}^{t}]\), mean waiting time in (22) can also be represented as  
\[E[W_{s}|b_{s},p_{s}^{t}]=\frac{\rho_{s}E[S_{s}^{2}|b_{s},p_{s}^{t}]}{2(1-\rho _{s})E[S_{s}|b_{s},p_{s}^{t}]}. \tag{23}\]  
Therefore, the mean packet transmission delay in small-cell AP, \(E[D_{s}|b_{s},p_{s}^{t}]\), is shown as [18]  
\[E[D_{s}|b_{s},p_{s}^{t}]=E[S_{s}|b_{s},p_{s}^{t}]+\frac{\rho_{s}E[S_{s}^{2}|b_ {s},p_{s}^{t}]}{2(1-\rho_{s})E[S_{s}|b_{s},p_{s}^{t}]}. \tag{24}\]  
We define service time \(S_{s}\) as packet transmission processing time in the AP, which means \(E[S_{s}]=1/C_{s}\) per unit-length packet. Then, the CDF of \(S_{s}\) can be obtained as  
\[F_{S_{s}}(t|b_{s},p_{s}^{t})=Pr[g>\frac{(2^{\frac{1}{Bi}}-1)(\mathbf{Ic_{s}}+ \sigma^{2})}{p_{s}^{t}r^{-\alpha}}]. \tag{25}\]  
With the fact that \(g{\sim}exp(h)\)[27], we can obtain  
\[Pr[g>\frac{(2^{\frac{1}{Bi}}  
The mean packet transmission delay minimization problem **P2** for the delay-sensitive UE with objective function and constraints is given in (29). \[ \begin{array}{ll}\textbf{P2:}\underset{\gamma,\xi}{\text{min}}&E[D_{t}]( \gamma,\xi)\\ \text{s.t.}&\sum_{i=1}^{I}b_{i}=B,\quad i\in\{1,\dots,I\},\\ &\sum_{i=1}^{I}p_{i}=P,\quad i\in\{1,\dots,I\},\\ &\rho_{i}\leq 1,\quad i\in\{1,\dots,I\},\\ &0\leq\gamma,\xi\leq 1,\end{array}  \tag{29}\]  
where \(b_{i}\) and \(p_{i}\) are the spectrum bandwidth and power allocated to delay-sensitive UE \(i\) with \(i\in\{1,\dots,I\}\). The optimization problem can also be solved by utilizing the Gradient descent method with the Bisection method. Figure 3 shows the mean packet transmission delay for delay-sensitive UE as a function of unlicensed spectrum allocation ratio \(\gamma\) with each power allocation ratio \(\xi\). The QoS threshold of small-cell APs shown in Figure 3 guarantees the performance of delay-sensitive UE by restricting the minimum unlicensed spectrum allocation ratio \(\gamma\) for the corresponding power allocation ratio \(\xi\). In the figure, to provide the consistent mean packet transmission delay for delay-sensitive UE, more spectrum resource is needed for a smaller power allocation ratio to delay-sensitive UE, and vice versa. Figure 4 shows the mean packet transmission delay for rate-sensitive UE as a function of unlicensed spectrum allocation ratio \(\gamma\) for various power allocation ratios \(\xi\), respectively. As the figure shows, rate-sensitive UE is not sensitive to the change in packet transmission delay as the allocations of power and spectrum resources change, until the \(\gamma\) is over 0.75 due to the reserved spectrum resource.