# Enhanced Matching Game for Decoupled Uplink Downlink Context-Aware Handover
## 3 System Model and Problem Formulation
### Problem Formulation  
To associate UEs to BSs, each UE try to recognize the greatest arrangement of BSs for which it can reach the particular QoS prerequisites. In order to display a UE and BS association issue, we determine an adjusted setting mindful utility capacity for a \(UE_{n}\) (\(n\in N\)) served by \(BS_{m}\) (\(m\in M\)). We manage an adjusted context-aware optimization issue which includes finding pointers \(x_{ij}^{(. )}\) relating to the association. The pointers variable upholds exceptional association, which is combinatorial. Along these lines, we concentrate only on the examination of how the UEs should be related to rates \(r_{ij}^{(. )}\) in order to enhance network throughput and admit fairness. Specifically, the logarithmic function is an exceptionally popular alternative to the utility function; this logarithm is concave and thus has unavoidable diminishing statements. This property promotes load balancing. This is consonant with association theory in concrete systems, where a relationship for a more overloaded BS is viewed as a low advantage. Accordingly, in the rest of this work, we employ a logarithmic utility function. Our goal is to boost the network sum rate with SINR requirements, and we define the UE association issue as follows:  
\[ \begin{array}{l}\max\limits_{x}\sum\limits_{m\in M}\sum\limits_{n\in N}\log (x_{mn}^{(.)}r_{mn}^{(.)}),\\ s.t\sum\limits_{n\in N}x_{mn}^{(.)}=1,\forall m\in M,\\ x_{mn}^{(.)}\in\{0,1\},\forall n\in N\ and\ \forall m\in M.\end{array}  \tag{5}\]  
Besides, the association optimization issue defined by Equation (5) will be redefined as Equation (6), since \(\sum\limits_{n\in N}x_{mn}^{(. )}=1,\forall m\in M\) and \(x_{mn}^{(. )}\in\{0,1\},\forall n\in N\) and \(\forall m\in M.\) \begin{table}
\begin{tabular}{l l} \hline
**Notation** & **Description** \\ \hline \(\mathfrak{M}\) & Set of macro BSs and femto BSs. \\ \(\mathfrak{N}\) & Set of UEs. \\ \(G_{m}\) & The subset of UEs served by the \(BS_{m}\). \\ \(x^{(.)}\) & Association matrix for a UE. \\ \(\gamma_{mn}^{(.)}\) & Signal-to-interference-plus-noise ratio of \(UE_{n}\) associated with \(BS_{m}\) in uplink or downlink. \\ \(K_{i}\) & Effective load of \(BS_{i}\). \\ \(r_{mn}^{(.)}\) & The \(UE_{n}\)’ s data rate associated with \(BS_{m}\). \\ \(\mu\) & Result of the UE–BS matching game. \\ \(\succ_{n}\) & Preference relations of every \(UE_{n}\). \\ \(\succ_{m}\) & Preference relations of every \(BS_{m}\). \\ \(b_{in}\) & Benefit for the \(UE_{n}\) to be served by \(BS_{i}\). \\ \(a_{i}\) & Scaling effects factor of \(UE_{n}\). \\ \(\phi_{th,n}\) & Capacity threshold determined by the QoS requirements of \(UE_{n}\). \\ \(h_{ij}\) & Assignment matrix describes when \(UE_{n}\) is served by \(BS_{i}\) for uplink and \(BS_{j}\) for downlink \\ \hline \end{tabular}
\end{table} 
Table 2: Mathematical notations. \[ \begin{array}{l}\max\limits_{x}\sum\limits_{m\in M}\sum\limits_{n\in N}x_{mn}^{( )}\log(r_{mn}^{(.)}),\\ s.t\sum\limits_{n\in N}x_{mn}^{(.)}=1,\forall m\in M,\\ x_{mn}^{(.)}\in\{0,1\},\forall n\in N\textit{ and }\forall m\in M.\end{array}  \tag{6}\]  
It is effectively perceived that the foregoing enhancement issue is a mixed-integer non-linear programming problem (MINLP), which is frequently NP-hard and engenders a complexity which increases exponentially with the included quantity of a BS and a UE. This unpredictability joined with the self-coordination necessarily requires the consideration of a disseminated arrangement where every UE exclusively picks the best BS to relate, in light of its nearby estimate. Then, we build up a distributed BS selection process for the support of comparable and reasonably fair throughput. Along these lines, we apply matching theory, which allows a distributed process for the combinatorial issue that groups players in two sets in view of their individual capabilities. In this manner, the involved UE affiliation issue positively belongs to the matching system.