# Traffic Offloading in Unlicensed Spectrum for 5G Cellular Network: A Two-Layer Game Approach
## 4 Coalition Game
### Merge-and-Split Based Coalition Formation Algorithm
: _(\(u_{WiFi}^{NES}\), \(u_{LAA}^{NES}\)) is a Nash equilibrium solution if it solves the following problem_

\[(u_{WiFi}^{NES},u_{LAA}^{NES})=\underset{\tilde{n},d_{i}}{max}\quad u_{LAA}u_{ WiFi}. \tag{21}\]

_subject to \(d_{i}\geq w_{i}\) and \(0\leq\tilde{n}\leq N\)._

Therefore, solving the multi-objective optimization problem above is equivalent to solving

\[\underset{\tilde{n},d_{i}}{max}\quad u_{LAA}u_{WiFi}. \tag{22}\]

This problem can be solved by getting the second-order derivative of Equation (22) on \(\theta_{i}\), that is

\[\frac{\partial^{2}u_{LAA}u_{WiFi}}{\partial\theta_{i}^{2}}=\frac{2 rR(N-\tilde {n})}{T_{off}}<0, \tag{23}\]

which means Equation (22) is a concave function and we can get the Nash Equilibrium solution through first-order derivative on \(\theta_{i}\) as zero, as

\[\theta_{i}^{*}=max\quad\left\{\frac{T_{off}r[R(N-\tilde{n})-R(N)]+R(N-\tilde{ n})R(N)}{2 rR(N-\tilde{n})},\,\frac{R(\tilde{n})}{r}\tilde{T}_{slot}\right\}. \tag{24}\]

In addition, the proposed LAA-WiFi auction process is described in Algorithm 3 and \(I_{max}\) denotes the set of AP coalitions with the maximum bid. ### Computational Complexity and Convergence Analysis

From analyses above we know the computational complexity of the auction game is \(L\) in the round of \(i\), because the AP will compare \(L\) coalitions and then onload data from the coalition which maximize its utility in a round. Therefore, the computational complexity of this auction game is \(LI\). The convergence of this algorithm can be obtained because the maximum point of the utility function in (24) exists, that is to say, a steady Nash equilibrium solution can be found. ## 6 Simulation Results

In this part, some MATLAB simulations to evaluate the performance of modified CSAT and the proposed two-layer game are presented. The unlicensed spectrums has the bandwidth of 20 MHz and the IEEE 802.11 ac protocol working around 5 GHz is used. Other WiFi related simulation parameters are listed in Table 1. The optimal number of stations \(n^{*}\) which could maximize the throughput of an AP is 10 by using CVX in MATLAB in the above-mentioned parameter setting environment which has been discussed in Section 2.2 from the calculation results of Equation (7). We assume that there are \(K\) APs working in the network but we mainly concern one coalition with three APs and one LAA cell which are randomly deployed in the network in 5 GHz unlicensed bands. The fixed station numbers within the coverage of AP1 is 3, which means AP1 is a light-loaded AP. In addition, there are 20 stations in AP3, that is to say, AP3 is a heavy-loaded AP. In addition, the number of stations in AP2 is ranging from 1 to 25 to evaluate the influence on system performance vs. the number of stations changing. Interferences among APs could be ignored due to the short transmission range and low transmission power of the WiFi AP. We further assume the time of a full CSAT cycle is 100 ms and the slot time in the WiFi is 2 ms. The maximum BS transmit power in 5 GHz is 20 dBm and the path loss model in unlicensed band is given as \(-15.3-50\) log\(10(\hat{d}(m))(dB)\). Figure 4 reveals the advantages after using the modified CSAT mechanism. The ratio of periodic \(T_{on}:T_{off}=1:1\) means the time of the \"on\" state period and the \"off\" state period are both 50 ms and the ratio \(T_{on}:T_{off}=1:2\) indicates the time for LAA transmission and for WiFi transmission is 33 ms and 67 ms, respectively. Assuming that on average the collision occurs half-way at the start of an LAA transmission [13]. From the figure, we can find that the throughput of both LAA and WiFi decreases with the increased number of aggregated packets [13] in a WiFi transmission due to the increased