# Unlicensed Spectrum Allocation for LTE and Wi-Fi Coexistence with HAP
## 4 The Proposed Scheme
### Lower Bound and Upper Bound of \(\delta\)  
According to the utility function \(U(\delta)\) defined in (1), we would like to find the lower and upper bounds of \(\delta\), which is the ratio of transmission duration for WiFi stations and LTE users. As described in the previous sub-section, we can get the \(Throughput_{th}^{WiFi}\) dynamically in response to the loading change of LTE. After the threshold is obtained, the lower bound of \(\delta\) will be calculated. First, considering the constraints (1b) and (2), we can substitute (2) into the constraint (1b) and obtain one of the lower bound conditions of \(\delta\), as in (7), which is deduced as follows:  
\[ \begin{array}{l}Throughput_{i}^{WiFi}=\delta\cdot L\cdot P\big{(}d<D|\ N_{STA} \big{)}\cdot\mathbb{E}\big{(}s_{i}^{WiFi}\big{)}\geq Throughput_{th}^{WiFi}\\ \Rightarrow\delta\geq\frac{Throughput_{th}^{WiFi}}{L\cdot P\big{(}d<D|\ N_{ STA}\big{)}\cdot\mathbb{E}\big{(}s_{1}^{WiFi}\big{)}}\end{array}  \tag{7}\]  
According to the explanation in Appendix A, we can get a \(D^{*}\), which is the minimum duration satisfying the successful transmission constraint (1d). That is, the contention period, \(D=\delta\times L\), must be greater than or equal to \(D^{*}\) to satisfy the threshold of successful transmission probability. We now have the following second bound:  
\[\delta\geq\frac{D^{*}}{L} \tag{8}\]  
Moreover, \(0\leq\delta\leq 1\), we finally have the lower bound of \(\delta\), given in (9). \[\delta\geq min\Bigg{\{}1,max\Bigg{\{}\frac{Throughput_{th}^{WiFi}}{L\cdot P \big{(}d<D|\ N_{STA}\big{)}\cdot\mathbb{E}\big{(}s_{i}^{WiFi}\big{)}}\,\ \frac{D^{*}}{L}\Bigg{\}}\Bigg{\}} \tag{9}\]  
Under the same concept, the upper bound of the time interval ratio \(\delta\) can be derived. The constraint on LTE throughput is given in (1c), that is, \(T_{j}^{LTE}\geq\frac{s_{th}^{LTE}}{\mathbb{E}\big{(}s_{j}^{LTE}\big{)}}\). During the contention-free period of a repetition interval, the sub-period at the front is for the QoS-STAs (QSTAs) to access the channel resources. Assume that there are \(N_{QSTA}\) QSTAs in the environment. The sum of the polling time for each QSTA plus the longest transmission time is \(T_{QSTA}\). Therefore, \(T_{QSTA}\cdot N_{QSTA}\) is the longest duration that the channel is occupied by the Wi-Fi PCF mechanism. In each repetition interval, for all LTE users, the total time to access the unlicensed band is the competition-free period subtracted by the time interval occupied by the PCF mechanism, as follows:  
\[\sum_{j=1}^{N_{UE}}T_{j}^{LTE}=(1-\delta)\cdot L-T_{QSTA}\cdot N_{QSTA} \tag{10}\]  
The upper bound of \(\delta\) can be obtained by using (10) and satisfying (1c) as shown in (11). \[\delta\leq 1-\Bigg{(}\sum_{j=1}^{N_{UE}}\frac{s_{th}^{LTE}}{\mathbb{E}\Big{(}s _{j}^{LTE}\Big{)}}+T_{QSTA}\cdot N_{QSTA}\Bigg{)}\times\frac{1}{L} \tag{11}\]  
Since the upper bound of \(\delta\) varies with the number of LTE users, the LTE user filtering mechanism described in the following subsection is used to limit the number of LTE users. Finally, we can combine the calculated upper and lower bound of \(\delta\), i.e., Equations (9) and (11), to obtain the interval of the feasible solution of the time ratio \(\delta\), as shown in (12). \[ \begin{split} min\bigg{\{}1,max\bigg{\{}&\frac{Throughput _{i}^{WIFI}}{L\cdot P\big{(}d<D\big{|}N_{STA}\big{)}\cdot\mathbb{E}_{\big{(}S_{1}^{ filter}\big{)}}^{WIFI}},\frac{P}{L}\bigg{\}}\bigg{\}}\leq\delta\\ &\leq 1-\bigg{(}\frac{N_{UE}}{j=1}\frac{s_{UE}^{STE}}{\mathbb{E}_{ \big{(}s_{1}^{ITE}\big{)}}^{ITE}}+T_{QSTA}\cdot N_{QSTA}\bigg{)}\times\frac{1}{ L}\end{split}  \tag{12}\]  
In the previous subsection, we calculated the lower bound of \(\delta\) by using Wi-Fi stations' successful transmission probability and throughput requirement. Therefore, the length of CFP in a repetition interval should not be less than this limit. We also use the minimum demands of QSTA users to calculate the upper bound of \(\delta\).