# Traffic Offloading in Unlicensed Spectrum for 5G Cellular Network: A Two-Layer Game Approach
## 4 Coalition Game
### Merge-and-Split Based Coalition Formation Algorithm
## Appendix A Derivation of Transition Probability of Markov Chain

For \(AP_{x}\), the transition probability \(P_{x}(x\in S_{L}\in\mathrm{Y}\to x\in S_{L}^{\prime}\in\mathrm{Y}^{\prime})\) in Equation (14) means the probability that \(AP_{x}\) changes from coalition \(S_{L}\in\mathrm{Y}\) to coalition \(S_{L}^{\prime}\in\mathrm{Y}^{\prime}\). Because APs form coalitions based on the Transfer Rule and Merge-and-Split Rule, both of which depend on the utility function of the player, the throughput changes of \(AP_{x}\) can clearly show the utility function variation. We assume \(AP_{x}\), which has \(n_{x}\) stations, is a lighted-loaded AP which will find overloaded APs to form coalitions. \(n_{l,a}\) and \(n_{l,a}^{\prime}\) mean the number of stations waiting to be offloaded from heavy-loaded APs in coalition \(S_{L}\) and \(S_{L}^{\prime}\), respectively. \(n_{l,s}\) and \(n_{l,s}^{\prime}\) indicate the number of stations which could handover to the light-loaded APs in coalition \(S_{L}\) and \(S_{L}^{\prime}\). In addition, \(n^{*}\) is the optimal station number which could maximize the \(AP_{x}\) throughput. Accordingly, \(\triangle=\sum_{S}n_{l,a}-\sum_{S}n_{l,s}\) and \(\triangle^{\prime}=\sum_{S^{\prime}}n_{l,a}^{\prime}-\sum_{S^{\prime}}n_{l,s}^ {\prime}\) mean the station number which are expected to be offloaded to \(AP_{x}\) after \(AP_{x}\) joining in coalition \(S_{L}\) and \(S_{L}^{\prime}\), respectively. Next, we consider four cases for the derivation of transition probability. Case 1: \(\triangle\geq n_{x}\) and \(\triangle^{\prime}\geq n_{x}\). \[\begin{split}& P_{x}(x\in S_{L}\in\mathrm{Y}\to x\in S_{L}^{ \prime}\in\mathrm



be happened in Case 2. Thus, the probability of \(AP_{x}\) transmitting from \(S_{L}\in\mathrm{Y}\) to coalition \(S_{L}^{\prime}\in\mathrm{Y}^{\prime}\) is zero in these two cases. In Case 4, if \(\triangle^{\prime}\geq\triangle\), the player \(AP_{x}\) can get more benefits to join in coalition \(S_{L}^{\prime}\). Thus, transmitting probability of \(AP_{x}\) from \(S_{L}\in\mathrm{Y}\) to \(S_{L}^{\prime}\in\mathrm{Y}^{\prime}\) is positive. In addition, in Case 1, staying in coalition \(S_{L}\) or joining in coalition \(S_{L}^{\prime}\) is comparative because the utility function of \(AP_{x}\) is equal in these two coalitions. ## References

* [PERSON] et al. (2015) [PERSON]; [PERSON]; [PERSON]; [PERSON] Coexistence of Wi-Fi and Heterogeneous Small Cell Networks Sharing Unlicensed Spectrum. _IEEE Commun. Mag._**2015**, _53_, 158-164. * [PERSON] et al. (2016) [PERSON]; [PERSON] [PERSON]; [PERSON] Optimal Strategy Design for Enabling the Coexistence of Heterogeneous Networks in TV White Space. _IEEE Trans. Veh. Technol._**2016**, _65_, 7361-7373. * [PERSON] and [PERSON] (2015) [PERSON] [PERSON]; [PERSON] [PERSON] Interference Analysis for Secondary Coexistence in TV White Space. _IEEE Commun. Lett._**2015**, _19_, 383-386. * [PERSON] et al. (2015) [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON]; [PERSON] LTE Unlicensed: The future of spectrum aggregation for cellular networks. _IEEE Wirel. Commun._**2015**, _22_, 150-159. * (5)_IEEE Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications;_ IEEE Standard 802.11 ac-2013; IEEE: Piscataway, NJ, USA, 2013. * (6) 3 GPP TR 36.889 v0.3.1. 3 rd Generation Partnership Project; Technical Specification Group Radio Access Network; Study on Licensed-Assisted Access to Unlicensed Spectrum (Release 13). Available online: [http://www.3 gpp.org/ftp/Specs/archive/36_series/36.889/36889-031.zip](http://www.3 gpp.org/ftp/Specs/archive/36_series/36.889/36889-031.zip) (accessed on 28 February 2015). * [PERSON] et al. (2012) [PERSON]; [PERSON] [PERSON]; [PERSON] [PERSON]; [PERSON]; [PERSON] [PERSON]; [PERSON] License-exempt LTE deployment in heterogeneous network [PERSON]. In Proceedings of the International Symposium on Wireless Communication Systems (ISWCS), Paris, France, 28-31 August 2012; pp. 246-250. * [PERSON] et al.