# Enhanced Matching Game for Decoupled Uplink Downlink Context-Aware Handover
## 3 System Model and Problem Formulation
### System Model  
We consider a two-tier system where a macro BS is surrounded by open-access femto BSs. In any case, this work is substantial for a general \(N\)-tier network where the BSs have distinctive coverage and capacity. Let \(\mathfrak{M}\) denote the set of macro BSs and femto BSs. The UEs are arbitrarily assigned to the two-tier network and denoted by \(\mathfrak{N}=1,2,3,...,N\). For \(m\in M,G_{m}\in\mathfrak{N}\) is the subset of UEs served by the BS\({}_{m}\).  
To model the UE association, we outline an association matrix \(x\) as:  
\[x^{(.)}=\left\{\begin{array}{ll}1,\;\text{if}\;\;UE_{n}\;\text{is associated with}\;\;BS_{m}\;\text{for}\;uplink\;\text{or}\; downlink\\ 0,\;\text{otherwise},\end{array}\right. \tag{1}\]  
where \((.)=u\) for uplink and \((.)=d\) for downlink. \(\gamma_{mn}^{(.)}\) is the SINR of \(UE_{n}\) associated with \(BS_{m}\) in uplink or downlink, and is calculated as follows:  
\[\gamma_{mn}^{(d)}=\frac{P_{m}^{d}g_{mn}}{\sum\limits_{i=1,i\
eq m}^{M}P_{i}g_{ in}+\sigma^{2}}, \tag{2}\]  
\[\gamma_{mn}^{(u)}=\frac{P_{mm}^{u}g_{mn}}{\sum\limits_{j=1}^{N}\sum\limits_{i= 1,i\
eq m}^{M}(x_{ij}P_{ij}^{u}/K_{i})g_{mj}+\sigma^{2}}, \tag{3}\]  
where \(P_{m}^{d}\) is the transmit power of \(BS_{m}\), \(\sigma^{2}\) is the noise power level, and \(g_{mn}\) is the channel power gain between \(UE_{n}\) and \(BS_{m}\), and the path-loss and shadowing are considered. Let \(K_{i}\) be the effective  
\begin{table}
\begin{tabular}{p{142.3 pt} p{142.3 pt}} \hline \hline
**Issues of Past Work** & **Main Commitment to Our Strategy** \\ \hline The matching game is applied for network selection and does not take into account a mobile environment. & We adopt the matching game as a handover process to offer an aspect of intelligence and self-organization. \\ \hline The externalities considered was the impact of the new association on the existing UE preferences solved by swap-matching. & Since we are dealing with a mobile UE, the former externalities treated in most research is no longer valid. We take into consideration the influence of the up-coming association in the preferences of the player. \\ \hline The most of the research working with the matching game realizes BS agents based on all BS available in the region as a player & In our case, we use an assignment step to maintain only the beneficial BS based on a QoS requirements of a UE. \\ \hline Most of the research in network selection was confirmed by an analytical study & In our approach to evaluate the model we use a simulation scenario on network simulator-3 (NS-3). \\ \hline \hline \end{tabular}
\end{table}
Table 1: The improvement brought by our method.  
load of \(BS_{i}\), characterized as the number of UEs associated \(K_{i}=\sum\
olimits_{n=1}^{N}x_{in}\). At each association with \(BS_{m}\), the UE gets a fraction \(1/\sum\
olimits_{n=1}^{N}x_{mn}\) of all the radio spectrum; Accordingly, the \(UE_{n}\)' s data rate associated with \(BS_{m}\) can be achieved by [PERSON]'s capacity formula:  
\[r_{mn}^{(.)}=(\sum\limits_{n=1}^{N}x_{mn}^{(.)})^{-1}W\log_{2}(1+\gamma_{mn}^{ (.)}), \tag{4}\]  
where W is the operating frequency band. A list of key mathematical notation used in the paper is given in Table 2.