# Traffic Offloading in Unlicensed Spectrum for 5G Cellular Network: A Two-Layer Game Approach
## 2 System Model
### System Performance Analysis  
We consider a scenario with one LAA Base Station (BS) and \(K\) APs coexisting in unlicensed band. The throughput of LTE in unlicensed spectrum can be expressed as  
\[C=B_{LAA}\log_{2}(1+SINR)\frac{T_{on}}{T_{on}+T_{off}}. \tag{1}\]  
where \(B_{LAA}\) is the bandwidth of LAA, \(SINR\) is the Signal to Interference and Noise Ratio, \(T_{on}\) and \(T_{off}\) mean the time for LAA \"on\" and \"off\" periods, respectively. According to [21] and the principle of the CSAT scheme, the per-user saturation throughput of the WiFi network can be expressed as  
\[R=\frac{P_{tr}P_{S}E\left[P\right]}{(1-P_{tr}T_{\sigma})+P_{tr}P_{S}T_{S}+P_{ tr}(1-P_{S})T_{c}}\frac{T_{off}}{T_{on}+T_{off}}, \tag{2}\]  
where \(T_{S}\) means the average time the channel is detected occupied because successful transmission occurs, \(T_{c}\) denotes the average time the channel is sensed busy by each station during a collision, \(T_{\sigma}\) is the duration of an idle slot time and \(E[P]\) represents the average packet size. Let \(P_{tr}\) and \(P_{S}\) be the probability that there is at least one transmission in the considered slot time and the probability of successful transmission, respectively; they can be expressed as  
\[P_{tr}=1-(1-\tau)^{n}, \tag{3}\]  
and  
\[P_{S}=\frac{n\tau(1-\tau)^{n}}{P_{tr}}=\frac{n\tau(1-\tau)^{n-1}}{1-(1-\tau)^{ n}}, \tag{4}\]  
where \(n\) is the stations number contending on the channel and each station transmits with probability \(\tau\). \(\tau\) only depends on the network size and on the system parameters of maximum backoff stage and the size of the contention window. Figure 1: System Model. However, too many stations will increase the collision probability and decrease the capacity [23]. Conversely, too few stations will not make full use of the channel spectrum resource. Thus, the number of stations in one AP should be carefully determined to maximize the capacity of the WiFi system. To best utilize the spectrum resources and maximize the system capacity, a coalition game among WiFi APs in the first layer of our CAGT optimization scheme is designed, and our goal is to optimize the station number \(n^{*}\) in one AP coalition to maximize the throughput of the WiFi system. Accordingly, we reshape (2) and the optimization problem can be formulated as  
\[\underset{n\in N^{+}}{max}R=\frac{an\tau(1-\tau)^{n-1}E[P]}{(T_{\sigma}-T_{c}) (1-\tau)^{n}+(T_{S}-T_{c})n\tau(1-\tau)^{n-1}+T_{c}}, \tag{5}\]  
where \(a=\frac{T_{off}}{T_{on}+T_{off}}\). Define \(f(n)=an\tau(1-\tau)^{n-1}E[P]\) and \(g(n)=(T_{\sigma}-T_{c})(1-\tau)^{n}+(T_{S}-T_{c})n\tau(1-\tau)^{n-1}+T_{c}\). Thus, Equation (5) can be expressed as \(R=\frac{f(n)}{g(n)}=\frac{-f(n)}{-g(n)}\), where \(-f(n)\) is concave and \(-g(n)\) is convex, which are judged by the Hessian Matrix. Then, according to [26], the function \(R\) is quasi-concave. Thus, the optimization problem can be expressed as  
\[\underset{n\in N^{+}}{\eta(n)}=\underset{n\in N^{+}}{max}R=\frac{f(n)}{g(n)}. \tag{6}\]  
As mentioned in [27], any optimization problem in fractional form can be transformed into an equivalent optimization problem in a subtractive form. Hence, the non-linear fractional optimization problem in Equation (6) can be transformed into a parameterized function and the optimal solution can be determined by finding the root to the \(U(n)\) as shown in Equation (7) using various root finding methods [28; 29],  
\[U(n)=\underset{n\in max\{[R^{+}],[R^{+}]\}}{max}\left\{-f(n)-\eta(-g(n)) \right\}=\underset{n\in\{[R^{+}],[R^{+}]\}}{max}\left\{-f(n)+\eta g(n)\right\}. \tag{7}\]  
The solution to Equation (7) could be formulated as an iterative two-layer solution combining Dinkelbach type method and Lagrangian dual decomposition approach. In addition, this problem can be solved by mathematical tools such as CVX or other build-in functions in MATLAB [29].