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| \documentclass[conference]{IEEEtran}
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\hyphenation{optical networks semiconductor}
\begin{document}
\title{Full-duplex Heterogeneous Cellular Networks}
\author{
\IEEEauthorblockN{
XXX\IEEEauthorrefmark{2}\IEEEauthorrefmark{1},
and XXX\IEEEauthorrefmark{3}} \\
\IEEEauthorblockA{
\IEEEauthorrefmark{2}
XXX\\
\IEEEauthorrefmark{1}
XXX\\
\IEEEauthorrefmark{3}
XXX\\
Emails: \{XXX\}@XXX
and XXX@XXX}
} \maketitle
\begin{abstract}
We study the joint user association and resource allocation problem in both uplink (UL) and downlink (DL) for full-duplex heterogeneous cellular networks...
\end{abstract}
\begin{IEEEkeywords}
Heterogeneous cellular networks, Full-duplex, Decoupled multiple association, Resource allocation, Matching theory.
\end{IEEEkeywords}
\section{Introduction}
\IEEEPARstart{W}{ith} the rapid proliferation of smart mobile devices and the surging requirement for limited licensed spectrum, small-cell base stations (SBSs) have been densely deploying to cooperate with traditional macro-cell base stations (MBSs), which are distinct in the transmission powers and coverage sizes~\cite{Agiwal2016,Kamel2016}. These heterogeneous cellular networks (HCNs) can effectively offload user equipments (UEs) from MBSs to SBSs, wherein the same channels can be reused and shared for high spectrum efficiency. Moreover, several new techniques can be introduced to further improve the performance. For example, full-duplex (FD) scheme, which allows simultaneous transmission and reception of wireless signals in the same frequency band~\cite{Sabharwal2014}, has been viewed as a promising potential to integrate into HCNs. And the spectral efficiency can be improved as investigated in~\cite{Thilina2015}.
The main contributions of this paper can be summarized as follows:
\begin{itemize}
\item We develop a framework for decoupled multiple association mechanism (DMA) and resource allocation for a full-duplex heterogeneous cellular network. In this framework, we jointly optimize the DMA, subchannel allocation and power allocation problem in both UL and DL to maximize the sum-rate of the overall network.
\item Extensive numerical results are provided to demonstrate the superiority of the proposed approach in terms of the overall network's sum-rate in UL and DL, respectively. By comparing with the existing user association schemes, the performance improvement of DMA is also illustrated.
\end{itemize}
The remainder of this paper is structured as follows. The related work is shown in Section \uppercase\expandafter{\romannumeral2}. Section \uppercase\expandafter{\romannumeral3} describes the system model, including the network deployment, the intercell interference and utility function formulation. In Section \uppercase\expandafter{\romannumeral4}, a joint optimization problem is formulated and the four-sided matching game approach is proposed. In Section \uppercase\expandafter{\romannumeral5}, a low-complexity four-sided stable matching algorithm is developed. Numerical results and analysis are presented in Section \uppercase\expandafter{\romannumeral6}. Finally, conclusions are drawn in Section \uppercase\expandafter{\romannumeral7}.
\section{System Model}
\subsection{Network Model}
\begin{table}
\caption{List of Key Notations}
\label{table2}
\begin{tabular}{p{45pt}p{180pt}}
\hline\noalign{\smallskip}
Notation & Definition \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$\mathcal{I} $ & the set of UEs \\
$\mathcal{J} $& the set of BSs \\
$\mathcal{J}_{s}$ & the set of SBSs\\
$\mathcal{J}_{m} $ & the set of MBSs \\
$q_{i}^{(\cdot)}$ & the quota for associating with UEs \\
$q_{j}^{(\cdot)}$ & the quota for associating to BSs\\
$\mathcal{L} $ & the set of power levels \\
$\mathcal{K} $& the set of subchannels\\
$\mathcal{P}_{i} $ & the set of the power levels of UEs \\
$\mathcal{P}_{j} $ & the set of the power levels of BSs \\
$P_{i} $ & the maximum transmit power of each UE $i$ \\
$P_{j} $ & the maximum transmit power of each BS $j$ \\
$G_{u,v,k}$ & the channel gain between transmitter $u$ and receiver $v$ on the subchannel $k$\\
$h_{u,v,k}^{2}$ & the channel coefficient between transmitter $u$ and receiver $v$ on the subchannel $k$\\
$U_{i,j,k,l}^{(\cdot)}$ & the data rate of UE $i$ associating with BS $j$ on the subchannel $k$ in the UL or in the DL\\
${\rm SINR}_{i,j,k,l}^{(\cdot)}$ & the SINR in the UL or in the DL\\
$W_{i,j}$ & the bandwidth occupied by UE $i$ associating with BS $j$\\
$\varepsilon$ & signifies the self-interference cancellation capability of the UE or BS \\
$\sigma$ & the noise \\
\hline
\end{tabular}
\end{table}
We consider a heterogeneous cellular network composed of an MBS, multiple SBSs and UEs. The sets of UEs and BSs are denoted as $\mathcal{I} = \{1,2,3...,I\}$ and $\mathcal{J} = \{1,2,3...,J\}$, respectively. The SBS is denoted as $j_{s} \in \mathcal{J}_{s}$, and the MBS is denoted as $ j_{m}\in \mathcal{J}_{m}$, $\mathcal{J} = \mathcal{J}_{s} \cup \mathcal{J}_{m}$. A UE can be associated with multiple BSs in the UL and DL, respectively. And each BS $ j $ (an SBS or an MBS) is given the quotas for associating with UEs which are represented by $q_j^{(\cdot)}$. When transmitting in the UL, a UE can choose at most $q_i^u$ BSs to associate. Similarly, when transmitting in the DL, a UE can choose at most $q_i^d$ BSs to associate. Meanwhile, the shortened communication distance among nodes also create the conditions for the application of FD communication in the considered model. For spectrum reuse, we consider that both SBSs and UEs are capable of performing FD communication. For example, UE$_{1}$ and BS$_{4}$ shown in Fig.~\ref{system}, simultaneous transmissions in UL and DL are allowed. Note that the communication between the MBS and the UE still applies half-duplex mode for avoiding the relatively large self-interference. Major notations used in this paper are given in Table \uppercase\expandafter{\romannumeral2}.
\begin{figure}[!t]
\centering
\includegraphics[width=3.3in]{eps/system.eps}
\caption{A UE can be associated with multiple BSs, one subchannel and one power level.}
\label{system}
\end{figure}
\begin{equation}
\begin{split}
F_{j}^{self}=\frac{p_{j,k}}{\varepsilon}\beta_{j}.
\end{split}
\label{Fj-self}
\end{equation}
where $\theta_{i',j',k}^{(\cdot)}\in \{0,1\}$ if $\theta_{i',j',k}^{(\cdot)}=1$ which means that a subchannl $k$ is allocated for UE $i'$ by BS $j'$ and $\theta_{i',j',k}^{(\cdot)}=0$ otherwise. $\beta_{j}=0$ if $j \in \mathcal{J}_{m}$\footnote{Since the MBS is restricted in half-duplex mode, the self-interference is absent from each MBS in the same frequency band.}, $\beta_{j}=1$ if $ j \in \mathcal{J}_{s}$, and $\varepsilon$ is the self-interference cancellation capability of the UE or BS.
\subsection{Problem Formulation}
In this part, we describe the formulation of joint decoupled multiple association, subchannel allocation and power allocation problem. The purpose is to maximize the sum-rate of the UEs in UL and DL by appropriately associating to BSs, allocating subchannels and power levels. Accordingly, an optimization problem is formulated as follows:
\begin{equation}
\begin{split}
\textup{max} \quad & \sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{K}\sum_{l=1}^{L}x_{i,j}^{u}\theta_{i,j,k}^u\ U_{i,j,k,l}^u + x_{i,j}^{d}\theta_{i,j,k}^d\ U_{i,j,k,l}^d\\
\textup{s.t.} \quad C1: & \sum_{i=1}^{I}\theta_{i,j,k}^u\leq \ 1 \ ,\sum_{i=1}^{I} \theta_{i,j,k}^d\leq \ 1 \ ,\forall \ j \in \mathcal{J}_{s}, k , l,\\
C2: & \sum_{i=1}^{I}(\theta_{i,j,k}^u + \theta_{i,j,k}^d)\leq \ 1 \ ,\forall \ j \in \mathcal{J}_{m}, k, l,\\
C3: &\sum_{j=1}^{J} \theta_{i,j,k}^u\leq \ 1 \ , \sum_{j=1}^{J} \theta_{i,j,k}^d\leq
\ 1 \ , \forall \ i, k,l,\\
C4: &\sum_{i=1}^{I} x_{i,j}^{u}\leq \ q_{j}^{u} , \sum_{i=1}^{I} x_{i,j}^d\leq
\ q_{j}^{d} \ , \forall \ j, \\
C5: & \sum_{j=1}^{J} x_{i,j}^{u}\leq \ q_{i}^{u}, \sum_{j=1}^{J} x_{i,j}^d\leq
\ q_{i}^{d} , \forall \ i, \\
C6: &\ \theta_{i,j,k,l}^{(\cdot )}\in \{ 0,1\}, x_{i,j}^{(\cdot )} \in \{ 0,1\} \ , \forall \ i, j, k,l,\\
C7: &\ p_{i,k}=\sum_{j=1}^{J}\sum_{l=1}^{L}p_{i,l}\theta_{i,j,k,l}^{u}\ ,p_{j,k}=\sum_{j=1}^{J}\sum_{l=1}^{L}p_{j,l}\theta_{i,j,k,l}^{d} ,\ \\
C8: &\sum_{k=1}^{K} p_{i,k}\leq \ P_{i} \ , \sum_{k=1}^{K} p_{j,k}\leq
\ P_{j} \ , \forall \ i, j.
\end{split}
\label{maximize}
\end{equation}
It can be observed that the optimization problem given in ($\ref{maximize}$) is a combinatorial $0$-$1$ integer non-linear programming problem which is NP-complete. And for DMA scheme in full-duplex networks, it is difficult to obtain the exact solution of the original problem in reasonable time due to the complicated association process and combinatorial characteristics. In this regard, we adopt matching theory to develop an efficient approach for obtaining a suboptimal solution which will be shown in the next section.
\section{Solutions and Implementation Issues}
In previous many-to-many matching games, there are only two distinct types of players, which is obviously not suitable for joint user association and resource allocation issue. And the UEs, the BSs, the subchannels and power levels are four sets of players to be matched with each other to achieve the maximum sum-rate, which involves a multivariate matching process.
In this regard, we first propose a novel four-sided matching game to model the many-to-many matching among four types of players (i.e., the UEs, BSs, subchannels and power levels). Then the stable matching is investigated and viewed as the optimal solution to the formulated problem. To obtain the optimal matching, a low-complexity four-sided stable matching algorithm is developed to enable the players to make decisions for DMA and resource allocation. Finally, some implementation issues of the proposed algorithm are analyzed.
\begin{myDef}
A four-sided many-to-many matching $\Phi^{(\cdot)} = (UE_{i}, BS_{j}, SC_{k},PL_{l})$ of the UARA can be defined as a function from the set $ \mathcal{I}\cup \mathcal{J} \cup \mathcal{K} \cup \mathcal{L}$ mapped into the set $ \mathcal{I}\cup \mathcal{J} \cup \mathcal{K} \cup \mathcal{L}$, and satisfy:.....
\end{myDef}
The conditions 1)-6) in Definition 1 correspond to the constraints in ($\ref{maximize}$).
\begin{enumerate}
\item The $UE_{i}$ connect to the new match $D(UE_{i})$ on the basis of original match, makes the $UE_{i}$ has increased overall utility value in the uplink and downlink, $UE_{i} \in \Phi _{block} ^ {(\cdot)}$;
\item The $UE_{i}$ delete the one of the matching combination from original match $\Phi ^ {(\cdot)} (UE_{i})$ and be associated with the new match $D(UE_{i})$, namely swap matching, makes the $UE_{i}$ has increased overall utility value in the uplink and downlink, $UE_{i} \in \Phi _{block} ^ {(\cdot)}$;
\item All matches in the network after new match or swap match meet the constraints of channel orthogonality and power in definition 1, and the total utility value of all users in the network are increased.
\end{enumerate}
\begin{figure}[!t]
\centering
\includegraphics[width=3.6in]{eps/block.eps}
\caption{An illustrative example of blocking matching.}
\label{block}
\end{figure}
\begin{myDef}
When there is no blocking matching in the system, that is, the total utility value of users in the system can be increased by adding new matching or exchanging matching under the premise of satisfying the constraint conditions, then the system is stable and achieves stable matching.
\end{myDef}
\section{Proposed Algorithm and Performance Analysis}
\begin{algorithm}[t]
\caption{Four-sided Matching Algorithm}
\label{alg:Framwork}
\begin{algorithmic}[1]
\STATE{\bf Phase I - Initialization:}
\STATE {Initialize data: $I,J,K,L,P_{i},P_{j},q_{i}^{u},q_{i}^{d},\sigma,W,\xi,\varepsilon$.}
\STATE {Each UE $i, i\in\mathcal{I}$, decides its own UL cell and DL\\DL cell. Each UE associates with $q_i^u$ BSs $j, j\in\mathcal{J}$, with the smallest $s_{i,j}$ in the UL cell and associates with $q_i^d$ BSs with the largest ${\rm SINR}_{i,j}$ in the DL cell.}
\STATE {The preference list of unit $\mathcal{T}$ is constructed for UE and BS respectively, by calculating \\ ~~~~~~$U_{i,j,k,l}=\gamma_{i,j}^{u}\theta_{i,j,k,l}^u\ U_{i,j,k,l}^u + \gamma_{i,j}^{d}\theta_{i,j,k,l}^d\ U_{i,j,k,l}^d$.}
\FORALL{ $t\in\mathcal{T}$}
\IF{ $C1,C2,C3\&C9$}
\IF{$C5\&C6$}
\STATE{Output the matching: \\
~~~~$\phi_{t}=({\rm UE}_{i}, {\rm BS}_{j}, {\rm SC}_{k}, {\rm PL}_{l}),\phi_{t}\in\Phi$;}
\ELSE
\STATE{Discard the UE with a lower $U_{i,j,k,l}$;}
\ENDIF
\ELSE
\STATE{$t=t+1$;}
\ENDIF
\ENDFOR
\STATE {Output the matching set $\Phi$.}
\STATE{\bf Phase II - Update the Matching:}
\STATE {Input the matching list $\Phi$ and the reject list $\Gamma$.}
\FORALL {$i\in\mathcal{I}$}
\FORALL{ $\phi_{i} \in \Phi$}
\WHILE{ $\phi_{block}^{(\cdot)}$ exists and is not in $\Gamma_i$ of UE $i$}
\STATE {Add a new match or swaps a match for UE $i$, updating the match as $\phi'_i$.}
\IF{$C5\&C6$}
\STATE{UE $i$ holds his current matching;}
\ELSE
\STATE{Discard the UE with the lowest $U_{i,j,k,l}$;}
\STATE{Add the $\phi'$ into $\Gamma_i$;}
\STATE {Re-match to the next most best choice from his preference list $\mathcal{T}_i$ until it is empty;}
\ENDIF
\ENDWHILE
\ENDFOR
\STATE {Update the preference list $\mathcal{T}'_i$ for UE $i$ in terms of the current match $\phi'_i$.}
\ENDFOR
\label{code:recentEnd}
\end{algorithmic}
\end{algorithm}
\begin{figure}[!t]
\centering
\includegraphics[width=3.6in]{eps/algorithm.eps}
\caption{The flow chart of the proposed algorithm.}
\label{algorithm}
\end{figure}
In order to evaluate the proposed algorithm, we analyze its performance from four aspects of stability, convergence, effectiveness and complexity.
\begin{prop}
\emph{Stability.}The four-sided algorithm is guaranteed to converge to a stable matching.
\label{stability}
\end{prop}
\begin{IEEEproof}
This proposition can be proved by the method of contradiction. From a contrary standpoint, we assume that, for UE $i$, there exists a matching $\Phi (i,j,k,l)$ produced by the proposed algorithm when converges and satisfying:
\begin{itemize}
\item[1)] $\exists j'\in\mathcal{J}, k'\in\mathcal{K}$ and $l'\in\mathcal{L}, U(\Phi^A_i) \geq U(\Phi_i) $ , and
\item[2)] $\exists j'\in\mathcal{J}, k'\in\mathcal{K}$ and $l'\in\mathcal{L}, U(\Phi^S_i) \geq U(\Phi_i) $.
\end{itemize}
Evidently it is a blocking matching.
By Definition \ref{block}, it is observed that if the proposed algorithm is not blocked by an individual, the existing matching $\Phi_i$ is four-sided stable. However, the existing of $U(\Phi^S_i) \geq U(\Phi_i) $ in matching result implies that the UE $i$ prefers$(j', k', l')$ to $(j, k, l)$ and rejects the request of $(j, k, l)$, and thereby, the multivariate of $i,j,k,l$ are unmatched. Besides, by referring to Phase II in Algorithm 1, the algorithm will not stop until all blocking matchings are excluded. Consequently, the analysis results contradicts with the assumption above, which means that the blocking matching $\Phi ^B_i$ for UE $i$ does not exis. Hence, the four-sided algorithm is proved to converge to a stable matching.
\end{IEEEproof}
\section{Simulation Results and Analysis}
In the simulation experiment, we consider...
\subsubsection{Convergence Analysis}
\
\noindent
The result concerning the convergence of Algorithm $1$ is investigated
%$&$
\subsubsection{Analysis of Performance}
\
\noindent
In this subsection, we first compare our proposed
\begin{figure}[!t]
\centering
\includegraphics[width=3.6in]{eps/diffQuota.eps}
\caption{diffQuota.}
\label{diffQuota}
\end{figure}
\section{Conclusions}
In this paper, we solve the problem of joint optimization
\bibliographystyle{IEEEtran}
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\end{document}
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