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\dochead{Research}

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\title{Sub-Granting Radio Resources in Overlay D2D-Based V2V Communications}

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   addressref={aff1},                   % id's of addresses, e.g. {aff1,aff2}
   corref={aff1},                       % id of corresponding address, if any
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   email={dariush.soleymani@iis.fraunhofer.de}   % email address
]{\inits{dms}\fnm{Moahammad Soleymani} \snm{Dariush}}
\author[
   addressref={aff2},
   email={mr.gholami@ericsson.com}
]{\inits{mrgh}\fnm{Gholami} \snm{Mohammad Reza}}
\author[
addressref={aff3},
email={jens.mueckenheim@hs-merseburg.de}
]{\inits{jmk}\fnm{Mueckenheim} \snm{Jens}}

\author[
addressref={aff4},
email={mitsch@tu-ilmenau.de}
]{\inits{mitsch}\fnm{Mitschele-Thiel} \snm{Andreas}}

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	\orgname{Department of Engineering and Natural Science, Merseburg University of Applied Science},
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	\city{Merseburg},
	\cny{Germany}
}
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	\orgname{Integrated Communication System Group, Ilmenau University of Technology},
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	\city{Ilmenau},
	\cny{Germany}
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\note[id=n1]{A short and preliminary
version of this work was presented in the conference
publication \cite{soleymani2019dedicated}.} % note, connected to author
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Capacity, reliability, and latency are seen as key requirements of new emerging applications, namely \ac{V2X} and \ac{MTC} in future cellular networks. \ac{D2D} communication is envisaged to be the enabler to accomplish the requirements for the aforementioned applications. Due to the scarcity of radio resources, hierarchical radio resource allocation, namely the sub-granting scheme, has been considered for the overlay~\ac{D2D} communication. In this paper, we investigate the assignment of un-utilized radio resources to~\ac{D2I} users, i.e., beneficiary user, for moving users in a dynamic environment. The sub-granting assignment problem is mathematically cast as the uplink cell throughput maximization problem. To this end, two heuristics are proposed: 1)~\ac{DSGRR} in a centralized manner, and 2)~\ac{OSGRR} in a distributed fashion. Simulation results show improved cell throughput for the \ac{OSGRR} compared with the \ac{DSGRR} yet less overhead while having reasonable tightness to the maximum achievable uplink throughput. 
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\begin{keyword}
\kwd{\ac{D2D} communication}
\kwd{Radio resource allocation}
\kwd{Sub-granting scheme}
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\section*{Introduction}
\label{intro}
Capacity, reliability, and latency are the major requirements of applications for future wireless communications. \ac{D2D} communication is foreseen as the first realization for the new emerging applications, e.g., vehicle to everything and machine type communication~\cite{schulz2017latency}.
Due to the increasing demand for spectral efficiency and the massive number of users, efficient utilization of the spectrum has attracted attention in industry and academia. Radio resource management is one of the avenues that aid in addressing the scarcity of spectral efficiency in future cellular networks. \textit{Overlay} and \textit{underlay} are two radio resource allocation mechanisms that are considered for the integration of \ac{D2D} in wireless communications. In the underlay technique, a \ac{D2I} shares radio resources with~\ac{D2D} users. This approach increases spectral efficiency; however, it causes interference between \ac{D2D} and \ac{D2I} users. One solution to mitigate this interference is to assign dedicated radio resources to the \ac{D2D} users, i.e., overlay. However, in this approach, the number of available radio resources for~\ac{D2I} users is reduced, and the allocated resources may not be sufficiently used by~\ac{D2D} users. To address this problem, authors in~\cite{cho2014spectrum} proposed a new resource allocation technique based on energy sensing and mode selection, in which every user can measure received signal strength of each radio resource configured by~\ac{eNB}. After that, the cellular user performs a~\ac{D2I}/~\ac{D2D} mode selection based on the measured received signal strength in a distributed manner. In~\cite{zhang2016research}, further study was taken wherein both centralized and distributed radio resource allocations are studied. The authors proposed to allocate radio resources based on geographical areas so to improve spectral efficiency. Results reveal the superiority of the distributed algorithm in terms of the spectral efficiency for applications with periodic traffic compared with the centralized algorithm.

Typically, many new applications are characterized by small payload, and thus current subframe granularity in~\ac{LTE} is too coarse for the traffic payload in such applications~\cite{schulz2017latency}. As a result, a part of allocated resources is wasted, especially for the overlay radio resource allocation. In~\cite{Pusc1604:Hierarchical}, a new idea of sub-granting has been proposed wherein the allocated but not fully utilized resources are granted in a finer granularity to the other nearby users, i.e., beneficiary users. Further studies have been conducted in \cite{comptti2018} and~\cite{soleymani2017implementation} to improve the efficiency of the sub-granting scheme. In~\cite{comptti2018}, the sub-granting and shortening~\ac{TTI} are compared in terms of uplink cell throughput in a scenario with the users having a small traffic payload. The results show that the uplink cell throughput degrades in the shortening \ac{TTI} scheme compared with the sub-granting scheme when the radio resources are assigned to the \ac{D2D} users in a semi-persistent manner. Inspired by this study, a new customized subframe in~\cite{soleymani2017implementation} is proposed by which better results in terms of overhead, uplink cell throughput, and latency can be achieved.
In the previous works, it was assumed that a nearby beneficiary user with the highest modulation scheme is always available, which can utilize the sub-granting resources. Note that in a dynamic environment, multiple beneficiary users with different bandwidths and modulation coding schemes exist. Thus a higher spectral efficiency can be achieved when the sub-granting is granted to a full buffer beneficiary user with the highest modulation coding scheme. With this aim, a new~\ac{DSGRR} algorithm is suggested in~\cite{soleymani2019dedicated}. Therein, the~\ac{BS} as a central controller chooses a beneficiary user for every \ac{D2D} user, i.e., sub-grant provider, based on some criteria and accordingly informs the sub-grant provider about the candidate beneficiary users. 

Consequently, the sub-grant provider disseminates the unused resources information along with the selected beneficiary user identity. Note that the~\ac{CQI} between the beneficiary and sub-grant provider users are unknown or can be measured at the cost of high signaling overhead on the cellular network. Henceforth,  a new \ac{eLA} is proposed. The \ac{eLA} is a geographical area wherein the beneficiary user can decode the sub-granting signaling reliably. In~\cite{soleymani2019dedicated}, a scenario where all users are stationary is considered. However, to have a precise eLA in a dynamic scenario, every entity should transmit the measurement information, e.g., positioning and \ac{CSI}, more frequently, which results in incurring huge overhead on the cellular network. Therefore, a distributed approach needs to be considered since the process can not be performed centrally. In this paper, we further study a new \ac{OSGRR} scheme where a sub-grant provider user openly broadcasts the sub-granting resources, and all beneficiary users become involved to select the beneficiary user candidate for a specific time interval in a cooperative manner. A short and preliminary version of this work was presented in the conference publication~\cite{soleymani2019dedicated}. The main contributions of this study are summarized as follows:
\begin{itemize}
	\item We formulate the beneficiary user selection problem for the sub-granting scheme as an optimization problem. The optimization problem aims to select the beneficiary users subject to some constraints in order to maximize the uplink cell throughput.
    \item Two new algorithms are proposed and compared in terms of uplink cell throughput, the number of selected beneficiary users, and sub-granting errors considering mobility, measurement transmission interval.     
	\item The overhead is formulated for both algorithms. We calculate the overhead of both algorithms, taking into account positioning information, \ac{CSI} measurements, sub-granting signaling, bidding information in a dynamic environment while \ac{D2D} and \ac{D2I} users are moving.
   \end {itemize}
	The remainder of this paper is organized as follows. In Section~\ref{secmodel}, the system model is described. We explain the problem formulation in Section~\ref{sec:prob}. In Section~\ref{totsubalg}, the proposed algorithms are presented. The results are discussed in Section~\ref{secresults}. Finally, in Section~\ref{secconclude}, some concluding remarks are presented.        
\section{Scenarios and System Model}
\label{secmodel}
%\underline{}
\subsection{Scenarios}
\label{scenarios}
In this section, we analyze different sub-granting scenarios considering the communication type for beneficiary users and sub-grant provider users. The sub-grant provider user and the beneficiary user could be either \ac{D2D} or \ac{D2I} communication, whereby four types of sub-granting scenarios are defined. Table~\ref{table:scenarios} illustrates the sub-granting scenarios and use cases. Generally, in a \ac{D2I} communication, the~\ac{eNB} has global knowledge of the location, signal level, and buffer status of the user based on the measurements received from the cellular users in every measurement interval. In contrast, in~\ac{D2D} communication, the~\ac{eNB} is not aware of some information, namely user buffer status report and channel measurement between two communicating, or this information can be achieved at the cost of high measurement and signaling overhead on the eNB. In the \ac{DSGRR} algorithm, the eNB selects a beneficiary user based on the available measurement information to increase the overall uplink throughput. Note that although the \ac{eNB} has initially scheduled radio resources for the \ac{D2D} users, the radio link condition and traffic buffer status of the \ac{D2D} communication may change, and thus the \ac{D2D} communication information becomes outdated quickly. This reason makes the \ac{D2D} user an inappropriate candidate for the beneficiary user in a centralized scenario.
 In contrast, the \ac{D2D} user can independently decide and grant un-utilized resources to the beneficiary user selected by the \ac{BS}, whereby the \ac{D2D} user becomes a suitable candidate as the sub-grant provider in a centralized scenario. In a decentralized approach, i.e., \ac{DSGRR}, the \ac{eNB} is not involved in the beneficiary user selection procedure. Thus the sub-grant provider user and the beneficiary user can be either a \ac{D2D} or a~\ac{D2I} user, and thus four different scenarios can be defined. 

Different use cases can apply the sub-granting scheme. One example is sub-granting radio resources from \ac{V2V} user to \ac{P2I}, \ac{M2M}, \ac{V2I} user, and vice versa.
In the following sections, the centralized and decentralized approaches are compared in terms of the uplink cell throughput, overhead, and the average number of the selected beneficiary user in a dynamic scenario. In this study, we consider the D2D and the D2I \ac{UE} as the sub-grant provider and the beneficiary user, respectively, in order to have a fair comparison between two algorithms.   
\begin{table*}[h!]
	\centering 
	\caption{Different scenarios of the sub-granting scheme} % title of Table
	\label{table:scenarios} 
	\resizebox{0.75\textwidth}{!}{%
		\begin{tabular*}{0.75\textwidth}{p{2cm}|p{2cm}|p{3cm}}
			\hline
			\textbf{Sub-grant}&\textbf{Beneficiary}&\textbf{Use Cases}\\ \textbf{Provider}&\textbf{User}\\  
			\hline\\ 
			\text{D2D}
			&
			\text{D2D}
			&
			\text{V2V, V2I, MTC, P2I}
			\\
			\text{D2D}
			&
			\text{D2I}
			\\
			\text{D2I}
			&
			\text{D2D}
			\\
			\text{D2I}
			&
			\text{D2I}
			\\
		\end{tabular*}
	}
\end{table*}
\subsection{System Model} 
\label{sec:model}
%to do system model and notation 
We consider a single-cell environment with $M$ D2I users (\ac{D2I}-\ac{UE}s) and $N$ D2D users (\ac{D2D}-\ac{UE}s) denoted by sets $ \mathnormal{C}=\lbrace1,...,M\rbrace$ and $\mathnormal{D}=\lbrace1,...,N\rbrace$, respectively. All users are uniformly distributed over the cell. We assume users also randomly move through the cell. Figure~\ref{fig:system_model} graphically shows an example of the network. Let us assume there are $F$~\ac{RB}s in the uplink direction for both \ac{D2D} and \ac{D2I} users. The \ac{eNB} coordinates~$L$ \ac{RB}s for \ac{D2D} pairs and the remaining~$\left(F-L\right)$ \ac{RB}s for \ac{D2I}-\ac{UE}s. The \ac{eNB} orthogonally schedules uplink radio resources for \ac{D2I} users at every scheduling time by any reasonable scheduling scheme. To avoid scheduling delays, the \ac{eNB} assigns one \ac{RB} to every \ac{D2D}-\ac{UE}s for a specific time~(See Label~(1) in Figure~\ref{fig:system_model}). We assume that the \ac{D2D}-\ac{UE}s can disseminate the signaling information indicating the allocated but unused resources, i.e., sub-granting~(See Label~(2) in Figure~\ref{fig:system_model}), to the all nearby \ac{D2I}-\ac{UE}s, i.e., beneficiary user~(See Label~(3) in Figure~\ref{fig:system_model}). Also, the \ac{D2I}-\ac{UE}s are a side-link capable user who can communicate with other users in proximity through the side-link communication~\cite{3gpp_ts_36_321}.
 \begin{figure}[h!]
 	\minipage{0.45\textwidth}
	\centering
	\captionsetup{justification=justified}
	\includegraphics[height=6cm,width=0.9\columnwidth]{Fig1_system_model.pdf}
	\caption{A typical network is consisting of one \ac{eNB}, one \ac{D2D}-\ac{UE}, and one \ac{D2I}-\ac{UE}. 
		The k-th \ac{D2D} user sub-grants the un-used radio resources to the m-th \ac{D2I} user.}	  
	\label{fig:system_model}
	\endminipage
\end{figure} 	

The further assumptions in this study are made as follows:
\begin{itemize}
	\item All \ac{D2I}-\ac{UE}s are full buffer users with best-effort traffic payload.
	\item \ac{D2D}-\ac{UE}s have small traffic payload with the same reliability and latency requirements, e.g., ultra-reliable and low-latency applications.
	\item All \ac{D2D} and \ac{D2I} users are synchronized in time and frequency from the \ac{eNB}.
	\item All users' velocities are assumed to be constant during movement. At the initiating position and the cell border, the users choose an arbitrary route on a random basis.  
	\item The processing time for decoding sub-granting signaling message and encoding data $S_\text{Min}$ is assumed to be less than the time of two symbols for the beneficiary user. 
	\item We assume the channel condition remains unchanged during the beneficiary user selection process. Besides, the amplitude of the received signal with distance is assumed to follow an exponential decay as follows:
	\begin{equation}
	\label{eqgain} 
	G={\mathrm{c}\times r^{-\alpha} |\mu|}, 	 
	\end{equation} 
	where $c$ is a constant value, and $r$ is the distance between two entities, $\alpha$ represents a path-loss exponent, and $\mu$ captures the large- and small-scale fading phenomena.
	\item We assume an open-loop power control mechanism that the transmission power of every communicating entities is controlled by the path-loss~\cite{dahlman20134g}. %The power control mechanism has performed the signal to noise ratio (SNR) drops a threshold set by the network.    
	\item We assume that the \ac{eNB} is aware of the \ac{CQI} and buffer status report of $\ac{D2I}$ users. Besides, all users are equipped with a \ac{GPS}, and thus be able to send the positioning information to the central network in every (pre-)configured time interval.
	\item To reduce the processing time due to blind decoding during the sensing procedure, it is assumed that energy-sensing is only performed on (pre-)configured radio resources by the \ac{eNB}.
	\item The notations used in this study are summarized in Table~\ref{table:notation}.
\end{itemize}
\begin{table}[h!]
	\centering 
	\caption{List of Notations} % title of Table
	\label{table:notation} 
	\begin{tabular*}{\columnwidth}{c l }
		\hline
		Notation& Interpretation \\  
		\hline\\ 
		$\mathnormal{D}$& The set of \ac{D2D} users where $d_{k}\in\mathnormal{D}$~ for $k=1,...,N$\\	
		$\mathnormal{C}$& The set of \ac{D2I} users where $c_{m}\in\mathnormal{C}$ for $m=1,...,M$ \\
		$L$& The set of allocated \ac{RB}s to \ac{D2D} communication\\
		$F$& The set of available \ac{RB}s on \ac{eNB}\\
		$B_w$& Allocated bandwidth\\
		$b$& Allocated \ac{RB}s to every entities where $b=1,...,F$\\
		$\mathbb{X}$& Allocation indicator for sub-granting\\
		$\alpha_c$& Path loss component for \ac{D2I} user\\
		$\alpha_d$& Path loss component for \ac{D2D} user\\
		$\mu_c$& Fading component for \ac{D2I} user\\
		$\mu_d$& Fading component for \ac{D2D} user\\
		$\sigma_0$& White Gaussian noise\\
		$\sigma$& Shadowing term\\
		${P}_{m}$& Transmission power of \ac{D2I} transmitter, $m=1,...,M$ \\
		${P}^C_{max}$& Power threshold limit for \ac{D2I} transmitter\\
		${P}_{k}$& Transmission power of \ac{D2D} transmitter, $k=1,...,N$\\
		$q$& Modulation and coding scheme of every entities\\ 
		$\epsilon$& General term for~\ac{BER}\\
		$\epsilon^D_{th}$& Minimum BER threshold for \ac{D2D} communication\\
		$\epsilon^{sg}_{th}$& Minimum BER threshold for sub-granting\\
		$\epsilon_{mk}$& Measured BER between \ac{D2D} and \ac{D2I} users\\
		$\mathit{T}$& Subframe transmission time\\
		$\mathit{\tau}$& Transmission duration\\
		$T_{me}$& Channel state information measurement interval\\
		$T_{pos}$& Positioning measurement interval\\
		$T_{br}$& Bids transmission interval\\
		$S_{Min}$& Processing time for all entities\\
		$\mathcal{Z}$& Unique number of a cellular user in network\\
		%$\eta$& Quality factor between beneficiary user and infrastructure\\
		%$h$& Overhead for D2I and D2D users\\
		%$O_{xx} \&T_{xx}$& Constant term and time occurrence of overhead for measurement report, positioning sub-granting and broadcast information, $xx=me, pos, sg, br$ %\\
		%$i$& Simulation Run\\
		\hline
	\end{tabular*}
\end{table}
\section{The Problem Formulation}
\label{sec:prob}
In~\cite{soleymani2019dedicated}, it is proposed to decompose the uplink cell throughput into the aggregated throughput of the \ac{D2I} and \ac{D2D} users within the cell. Therein, a scenario with stationary users was studied in which the radio link condition does not vary significantly, and thus the overhead due to the user radio link measurement, i.e., channel state information and positioning transmission, was ignored. In this study, we consider a scenario that all users are moving within the cell. Therefore, the radio link condition and the positioning information of all users need to be transmitted to the cellular network more often. The measurement transmission incurs a significant overhead on the cellular network, which results in cell throughput degradation. Considering the measurement and positioning information overhead, the uplink cell throughput stated in~\cite{soleymani2019dedicated} can be reformulated as:
\begin{equation}
\label{r_tot}
\begin{aligned}
R_{cell}\left(\epsilon,\tau\right)=&\sum_{{m}=1}^{M}\left(R_{m} \left(\epsilon,\tau\right)-\hbar_m\left(\tau\right)\right)\\ 
&+~\sum_{k=1}^{N} \left(R_{k}\left(\epsilon,\tau\right)-\hbar_k\left(\tau\right)\right),  
\end{aligned}
\end{equation} 
where $R_m$ is the achievable data rate of $\ac{D2I}$ user at every scheduling time $\tau$ for a specific bit errors rate $\epsilon$ and yields as follows~\cite{goldsmith1997capacity}:
\begin{equation}
\label{r_d2i}
R_m \left(\epsilon,\tau\right)=B_{w}\log_2\left(1+\frac{GP_m}{\sigma_0B_{w}\Gamma}\right), 
\end{equation} 
%=\frac{(1-\epsilon)qB_{w}\tau}{T}
where $G$ is the radio channel gain that is calculated from equation~$\left(\ref{eqgain}\right)$, $P_m$ is the transmission power of every \ac{D2I} users, $B_w$ and $\sigma_0$ stand for the allocated bandwidth in $Hz$ and white Gaussian noise, respectively. And, $\Gamma = \frac{-ln\left(\epsilon \right)}{1.5}$~\cite{goldsmith1997capacity}. In case of $\ac{D2D}$ user with small traffic payload, the equation~(\ref{r_d2i}) is not accurate enough, thus the achievable throughput $R_k$ for \ac{D2D} user is reformulated as follows~\cite{durisi2016toward}: 
\begin{equation}
\label{r_d2d}
R_{k} \left(\epsilon,\tau\right)=B_{w}\log_2\left(1+\frac{GP_{k}}{\sigma_0B_{w}}\right)-\sqrt{\frac{B_{w}V}{\tau}} Q^{-1}\left(\epsilon\right)\log_2(e), 
\end{equation}
where $Q^{-1}\left(.\right)$ is the inverse Gaussian Q-function, and $V$ reflects stochastic variability of the channel given by:
\begin{equation}
\label{r_d2d_v}
V =1-\frac{1}{1+\left(\frac{GP_k}{\sigma_0B_w}\right)^2}.
\end{equation}
%\subsection{Overhead in Open and Dedicated Sub-granting}
In equation~$\left(\ref{r_tot}\right)$, $\hbar_k$ and $\hbar_m$ are the overhead due to the sub-granting signaling, the positioning information and radio link measurement reports transmitted by the \ac{D2I} and~\ac{D2D} users.  
In~\cite{soleymani2019dedicated}, it has been manifested that the user throughput is proportional to $qb$ over the transmission time, $\tau$ where $q$ is the~\ac{MCS}, and captures the bit error rate $\epsilon$. Moreover, $b$ stands for the number of allocated \ac{RB}s. Consequently, the sub-granting throughput~$R_{mk}$ yields from $q_mb_k$ over the transmission duration $\left(T-\tau_k\right)$, where $b_k$ is the sub-granted \ac{RB}s from the $k$-th \ac{D2D} user and~$q_m$ is \ac{MCS} of the $m$-th $\ac{D2I}$ beneficiary user. Additionally, a binary variable of~$\mathbb{X}_{mk}$ for resource allocation from $k$-th~$\ac{D2D}$ user to the $m$-th~$\ac{D2I}$ beneficiary user is defined:
\begin{equation}
\mathbb{X}_{mk}=
\begin{cases} 
0,&\text{if } T-\tau_k \textless S_{Min}, \\
1,&\text{otherwise}.
\end{cases}
\label{lblxmk}
\end{equation}
Where $S_{\text{Min}}$ is a processing time of a UE that depends on 
equipment and the number of allocated resources in time and frequency domain~\cite{3gpp_ts_36_881}.
We now rewrite equation~(\ref{r_tot}) for the dedicating and open sub-granting considering the \ac{D2I} and \ac{D2D} overhead as follows:
\begin{equation}
\label{r_subgtot}
\begin{aligned}
R_{cell}\left(\epsilon,\tau\right)=&\sum_{m=1}^{M}\left(R_m\left(\epsilon,\tau\right)-\hbar_m\left(\tau\right)\right)\\ +&\sum_{m=1}^{M}\sum_{k=1}^{N} \mathbb{X}_{mk}R_{mk}\left(\epsilon,T-\tau_k\right)\\ 
&^+\sum_{k=1}^{N}\left(R_k\left(\epsilon,\tau\right)-\hbar_k\left(\tau\right)\right).  
\end{aligned}
\end{equation}   
Equation~(\ref{r_subgtot}) considers the case where the \ac{D2D} users are ultra-reliable and low-latency communications with absolute reliability and latency requirements, while \ac{D2I} users have the best-effort traffic. More precisely, we assume that the reliability requirements for \ac{D2D} users are satisfied if the bit error rate of \ac{D2D} communication~$\epsilon_{k}$ is smaller than the configured threshold~$~\epsilon^\mathnormal{D}_{th}$. Then, \ac{D2D} users can grant~$T-\tau$ of the allocated but unused resources in symbols basis to the \ac{D2I} users. However, in the case of the erroneous environment, the \ac{D2I} users may fail to decode the sub-granting signaling message. Therefore, we adopt the general approach initially proposed in~\cite{goldsmith1997capacity} to calculate the upper bound bit error rate~($\epsilon$) between \ac{D2D} and \ac{D2I} users as follows:
\begin{equation}
\label{eq_berd2d2i}
\epsilon \leq 0.2e^{\frac{-1.5{\delta}}{q-1}},
\end{equation}
where $\delta=\frac{GP}{\sigma_0B_w}$. 
We then proceed to maximize the sum rate of the cell by selecting the best beneficiary users. To optimize the throughput in~$\left(\ref{r_subgtot}\right)$, we only need to maximize the second term since the first and third terms are constant and have no effect on the solution. The optimization problem can be expressed as follows:    
\begin{subequations}\label{eqoptp}
	\begin{equation}
	\label{eqoptp_1}
	\begin{aligned}
	\mathop{\text{maximize}}\limits_{\left(m,k\right) \in \mathnormal{C}\times\mathnormal{D}}
	%\max_{\left(m,k\right) \in \mathnormal{C}\times\mathnormal{D}}
	&\sum_{m=1}^{M}\sum_{k=1}^{N} \mathbb{X}_{mk}R_m\left(\epsilon,T-\tau_k\right), \\ 
	\end{aligned}
	\end{equation}	
	\text{subject to}
	%\begin{equation}
	%\label{eqoptp_2}
	%\epsilon_k \textless \epsilon^\mathbb{D}_{th},~\forall~k= %1,...,N 
	%\end{equation}
	\begin{equation}
	\label{eqoptp_3}
	\epsilon_{mk} \textless \epsilon^{sg}_{th}, ~\forall~m=1,...,M, k=1,...,N,
	\end{equation}
	\begin{equation}\label{eqoptp_4}
	\mathbb{X}_{mk} \in \left\lbrace0,1\right\rbrace~\forall~m=1,...,M, k=1,...,N, 
	\end{equation}
	%\begin{equation} \label{eqoptp_5}
	%\mathbb{X}_{mk}=
	%\begin{cases}
	%0,&\text{if } T-\tau_k \textless S_{Min}, \\
	%\mathbb{X}_{mk},&\text{otherwise}
	%\end{cases}
	%\end{equation}
	\begin{equation}\label{eqoptp_6}
	\sum_{m=1}^{M} \mathbb{X}_{mk}=1,~\forall~k=1,...N,
	\end{equation}
	\begin{equation}
	\label{eqoptp_7}
	\sum_{k=1}^{N} P_{m}+ \mathbb{X}_{mk}\times \frac{\left( P^C_{max}-P_m \right)}{b_m }\textless P^C_{max},~\forall~m=1,...,M,
	\end{equation}
\end{subequations}
where~(\ref{eqoptp_3}) is constraint showing errors limit for the \ac{D2D} sub-granting signalling. Constraint~(\ref{eqoptp_4}) denotes that the available resources for sub-granting should be greater than the processing time required by the beneficiary users. 
It is assumed that only one beneficiary user is allowed to use a sub-granted resource~(constraint~(\ref{eqoptp_6})). Note that the power headroom indicates how much a beneficiary user is allowed to increase the transmission power in addition to the current allocated transmission power, i.e., $P^C_{max}-P_m$. Generally, the transmission power of every entity is proportional to the number of allocated \ac{RB}s~\cite{dahlman20134g}. Thus, the additional transmission power due to the sub-granted resources to the beneficiary users should not increase the beneficiary user transmission power beyond the power constraint $P^C_{max}$~(constraint~(\ref{eqoptp_7})).

The optimization problem in equation~(\ref{eqoptp}) aims to find a list of beneficiary users that maximizes cell throughput. This problem can be defined as a~\ac{MWM} problem in bipartite graphs with some non-linear constraints. When there exists a large number of \ac{D2I} and \ac{D2D} users, an exhaustive search becomes intractable due to its high computational complexity. To avoid drawbacks in using an exhaustive search solution, in the following sections, two algorithms are suggested in centralized and distributed fashions to address the beneficiary user selection problem in the sub-granting scheme. 
  
\section{Sub-Granting Radio Resource Algorithms}
\label{totsubalg}
In this section, first, the operational functionality of the beneficiary user and the sub-grant provider user in the proposed heuristic algorithms is briefly explained. Then, we describe some basics in the bipartite graph whose maximum weighted matching problem of the beneficiary user selection in the sub-granting scheme is simpler to solve in the bipartite graph. Finally, two heuristic algorithms that, in our opinion, can address the beneficiary selection problem in the sub-granting scheme are discussed.

 Figures~\ref{figstatemachine} demonstrates the operational state-machine of a beneficiary user in the centralized algorithm, i.e., dedicated sub-granting. In the dedicated sub-granting algorithm, every UE provides the \ac{BS} with their actual position information and \ac{CQI} towards the \ac{eNB} over every selected time interval. Afterward, the \ac{eNB} considers a hypothetical geographical area around every sub-grant provider users and seeks for a beneficiary user within this area to increase the overall cell throughput and informs every selected beneficiary users about the paired sub-grant provider users. 
 The beneficiary user monitors the sub-granting signaling at every selected time interval and then utilizes the sub-granting resources until the next selection time interval. 

 In a mobile scenario, the positioning and~\ac{CSI} information need to be transmitted more frequently, resulting in additional overhead on the cellular network. To reduce the overhead arising from the positioning and~\ac{CSI} information transmission, a distributed sub-granting algorithm, i.e., open sub-granting, is suggested. Figure~\ref{figodsm} shows the state-machine of the beneficiary user functionality for the open sub-granting algorithm. In general, each beneficiary user assesses on the received signal from the nearby sub-grant provider users to calculate a bid value based on the measured received signal strength from the sub-grant provider and the eNB. The calculated bid value, along with the desired sub-grant provider identity, is shared with other nearby beneficiary users. Then the beneficiary user who offers the highest value among nearby beneficiary users is allowed to transmit on the sub-granting resource for a selected time interval. 

In both algorithms, the sub-grant provider only informs the nearby beneficiary users about the unused resources and plays no role in the beneficiary user selection process. To be more specific, in the dedicated sub-granting, the selected beneficiary user identity and the number of free symbols are conveyed by the sub-granting signaling. Whereas in the open dedicated algorithm, only the number of free symbols and the sub-grant provider identity are transmitted. Figure~\ref{figsbgsm} depicts the functionality of the sub-grant provider in the dedicated and open sub-granting algorithms. 

Many problems can be cast as a matching problem in a bipartite graph. For example, in radio resource allocation, the relation between the users and the resources is modeled by a bipartite graph in order to maximize the number of allocated resources~\cite{mei2018joint}. Similarly, in this study, the relation between the sub-grant provider users and the beneficiary users is first modeled by a time-varying bipartite graph wherein the edge of the graph is weighted differently based on the beneficiary user selection algorithm, i.e., dedicated or open sub-granting algorithm, at every selection time instant. In the sequel, the concept of the weighted bipartite graph is introduced first.  
\begin{figure}[h!]
	\centering
	%\captionsetup{justification=justified}
	%\fbox{
	%begin{minipage}{\columnwidth}
		%\begin{subfigure}[]
		%\centering
		%\captionsetup{justification=centering}
		%\begin{subfigure}[c]{\columnwidth}
		\includegraphics[height=5cm,width=0.9\columnwidth]{Fig2_dd_sm.pdf}
		%\caption{Dedicated sub-granting algorithm in beneficiary user.}
		%	\label{subfigddsm}
		%\end{subfigure}
		%		   \begin{subfigure}[]
		%		   	\centering
		%		   	%\captionsetup{justification=centering}
		%		   	%\begin{subfigure}[c]{\columnwidth}
		%		   	\includegraphics[height=5cm,width=0.9\columnwidth]{images/Fig_op_sm.pdf}
		%		   	%\caption{Open sub-granting algorithm in beneficiary user.}
		%		   	\label{subfigodsm}
		%		   \end{subfigure}
		%	   
		%	       \begin{subfigure}[]
		%	       	\centering
		%	       	%\captionsetup{justification=centering}
		%	       	%\begin{subfigure}[c]{\columnwidth}
		%	       	\includegraphics[height=5cm,width=0.9\columnwidth]{images/Fig_sgp_sm.pdf}
		%	       	%\caption{Sub-granting scheme in sub-grant provider user.}
		%	       	\label{subfigspsm}
		%	       \end{subfigure}
		%		   
		%		\label{figddsm}
	%\end{minipage}
	%	}
	\caption{State-machine diagram of a beneficiary user \\functionality on the dedicated sub-granting algorithm.}
	\label{figstatemachine}
\end{figure} 

\begin{figure}[h!]
	\centering
	%\captionsetup{justification=justified}
	%	\fbox{
	%\begin{minipage}{\columnwidth}
		%\begin{subfigure}[]
		%\centering
		%\captionsetup{justification=centering}
		\includegraphics[height=5cm,width=0.9\columnwidth]{Fig3_op_sm.pdf}
		
	%\end{minipage}
	%}
	\caption{State-machine diagram of a beneficiary user\\ functionality on the open sub-granting algorithm.}
	\label{figodsm}
\end{figure} 

\begin{figure}[h!]
	\centering
	\captionsetup{justification=justified}
	%\fbox{
	%\begin{minipage}{\columnwidth}
		%\begin{subfigure}[]
		%\centering
		%\captionsetup{justification=centering}
		\includegraphics[height=5cm,width=0.9\columnwidth]{Fig4_sgp_sm.pdf}
	%\end{minipage}
	%}
	\caption{State-machine diagram of sub-grant provider \\ functionality.}
	\label{figsbgsm}
\end{figure} 
\begin{defn}
	The bipartite graph is a graph with two independent disjoint vertices $U$ and $V$ such that every edge $E$ connects a vertex in $U$ to one in $V$. We denote a graph with $G=(U,V,E)$.
\end{defn}

\begin{defn}
	Two edges of a bipartite graph are said to be independent when they have no common end vertex and loop. A matching is a set of independent pair edges of a graph. A matching with maximum cardinality is called maximum matching. 
\end{defn}
Figure~\ref{figgraph} is an illustration of the system model in the form of a graph model, where the edge of the graph is being updated in every selection time instant. The sub-grant provider $\mathnormal{D}$ and beneficiary users~$\mathnormal{C}$ construct two vertices of the graph. Every user in $\mathnormal{D}$ is connected to the users in $\mathnormal{C}$. In the following sections, we discuss two heuristic algorithms to achieve the maximum matching to address the sub-granting problem. 
General background on maximum weighted matching and bipartite graphs can be found in~\cite{goemans2001approximation}.
\subsection{Dedicated Sub-Granting Radio Resource ~(DSGRR) Algorithm}
\label{sec:calg}
In this section, we discuss the centralized dedicated sub-granting radio resource algorithm to address the beneficiary user selection problem of the sub-granting scheme indicated in equation~(\ref{eqoptp}). The optimization problem is decomposed into two stages. In the first stage, a hypothetical geographical area, an error- limited area, for every sub-grant provider is calculated wherein the sub-granting signaling can be reliably received. In the second stage, edges of the constructed bipartite graph are weighted by the beneficiary user data rate, and are being updated in every beneficiary user selection time interval.
The beneficiary selection problem is then solved using the proposed algorithm, and the maximum number of the beneficiary users to achieve the highest cell throughput is obtained.
\subsubsection{error-Limited Area (eLA)}
\label{ela_sec} 
As previously discussed, \ac{CQI} between the sub-grant provider and the beneficiary user is not known or, at least can be achieved at the cost of additional signaling overhead on the cellular network. To avoid such an overhead, a hypothetical circle around every sub-grant users based on the maximum error probability criterion, i.e.,~$\epsilon^{sg}_{th}$, is calculated. To this end, we use equation~(\ref{eq_berd2d2i}) to calculate the signal to noise level $\delta^{sg}_{th}$ related to~$\epsilon^{sg}_{th}$ on the margin of hypothetical circle. Then, considering equation~$\left(\ref{eqgain}\right)$ and the channel model parameters for \ac{D2D} communication in~\cite{3gpp_ts_36_814}, the \ac{eLA} $\left(r_{eLA}\right)$ is bounded as:
\begin{equation}
\label{ela_r}
\mid r_{eLA}\mid\leq\left(\frac{cP_k \mid \mu_d \mid}{\sigma_0B_w\delta^{sg}_{th}}\right)^{\alpha^{-1}_d},~ k=1,...,N. 
\end{equation}
Algorithm~$\ref{dsgalg1}$ explains the dedicated beneficiary user selection procedure. When the beneficiary user is inside the hypothetical circle of the sub-grant provider, i.e., \ac{eLA}. An edge $e \in E$ is weighted with $q_m.b_k$, if there exists at least one vertex $c_m \in \mathnormal{C}$ inside the \ac{eLA}~(see lines~(1)~to~(10)) in Algorithm~$\ref{dsgalg1}$. Additionally, we take the power constraint~(\ref{eqoptp_7}) into consideration. Next, the algorithm chooses the beneficiary user~$c_m$ with maximum weighted edge~$e_{mk}$ associated to every sub-grant providers $d_{k}$ in a greedy manner. Then, it is iteratively run and ended when all beneficiary users $c_{m}$ are successfully selected. Also, in every iteration in order to find the maximum matching, the allocated edge is removed from all the sub-grant provider vertices,~$d_{k}~\in~\mathnormal{D}$. Finally, every sub-grant provider users are informed about the selected beneficiary users~$\mathbb{X}$.
\begin{figure}[h!]
	%\fbox{
	%\begin{minipage}{\columnwidth}
		\centering
		\captionsetup{justification=centering}
		\begin{psfrags}
		%\captionsetup{justification=centering}
		\psfrag{C}{$\mathnormal{C}$}
		\psfrag{D}{$\mathnormal{D}$}
        %\psfragfig[height=5cm,width=0.9\columnwidth]{images/graphmodel.eps}}
		\includegraphics[height=5cm,width=0.9\columnwidth]{Fig5_graphmodel.eps}
	 	\end{psfrags}
	%\end{minipage}
	%	}
	\caption{An Illustrative example of graph model.}
\label{figgraph}
\end{figure} 
\begin{algorithm}
	\caption{Dedicated Sub-Granting Radio Resource}
	\label{dsgalg1}
	\begin{algorithmic}[1]
		\Procedure{Beneficiary user selection}{}
		\Statex $\textbf{Input:}$ $d_k~\in~\mathnormal{D}^{1\times N},c_m~\in~\mathnormal{C}^{1\times M},$ $\text{$\epsilon^{sg}_{th}$} $
		\Statex $\textbf{Initialization:}$
		\State \quad $\text{Calculate error limited area radius for}$\; 
		\Statex \qquad $d_k~\in~\mathnormal{D}$,  $k=1,...,N$~$\left(\text{Equation}~\left(\ref{ela_r}\right)\right)$\;
		\State \qquad $E\leftarrow\phi$ 
		\For {$d_k~\in~\mathnormal{D}$} 
		\For {$c_m~\in~\mathnormal{C}$} 
		\If{$\left(c_m~\in {eLA}_k\right) \&  \left(\text{Constraint~$\left(\ref{eqoptp_7}\right)$}\right)$} 
		\State $e_{mk}=q_m.b_k $
		\State $\textit{$E \leftarrow \left(E\cup e_{mk}\right)$ }$
		%\Else
		%\State$\textit{$e_{mk}$=0}$
		\EndIf
		\State $\textbf{endif}$
		\EndFor
		\State $\textbf{endfor}$
		\EndFor
		\State $\textbf{endfor}$
		\Statex $\textbf{Repeat:}$
		\State $\mathbb{X}\leftarrow\phi$
		\For {$d_k\in \mathnormal{D}$}
		\State $\text{Find $c_m~\in~\mathnormal{C}$ with Maximum $e_{mk}$}$
		\State $\mathbb{X}_{mk}=1~\&~\textit{$\mathbb{X}\leftarrow \mathbb{X}\cup \mathbb{X}_{mk}$}$
		\State $\textit{$E\leftarrow E-\cup_{k=1}^{N} e_{mk}$}$
		\EndFor
		\State $\textbf{endfor}$
		\Statex $\textbf{Output:}$
		$\textit{Transmit $\mathbb{X}_{mk}$ to every $d_{k}$}$
		\EndProcedure
		
	\end{algorithmic}
	
\end{algorithm}
\subsubsection{Overhead }
Due to the mobility of all users, the positioning and measurement information of users should be transmitted in a shorter time interval, which results in the additional overhead. As previously discussed in section~\ref{sec:prob}, the beneficiary user overhead is shown by~$\hbar_m$ and yields.
\begin{equation}
\label{overhead_d2i}
\hbar_m\left(\tau\right)=\frac{\mathbb{X}_{me} \times O_{me}+\mathbb{X}_{pos} \times O_{pos}}{T},
\end{equation}
where~$O_{me}$ and~$O_{pos}$ are the constant overhead values due to the channel state measurement and positioning information. $\mathbb{X}_{me}$ and $\mathbb{X}_{pos}$ are the measurement time interval $T_{me}$ and positioning information time interval $T_{pos}$. 
Similarly, the sub-granting provider transmits the positioning information to the BS and disseminates the unused radio resources, which incurs additional overhead the cellular network. In Section~(\ref{sec:prob}), this overhead is shown by ~$\hbar_k$  and yields:
\begin{equation}
\label{overhead_d2d_sg}
\hbar_k\left(\tau\right)=\frac{\sum_{m=1}^{M}\mathbb{X}_{mk}\times O_{sg}+\mathbb{X}_{pos} \times O_{pos}}{T-\tau},
\end{equation}
where $O_{sg}$ is the sub-granting signalling overhead in the every sub-granting occurrences~$\mathbb{X}_{mk}$ as explained in equation~$\left(\ref{lblxmk}\right)$ and constraint~$\left(\ref{eqoptp_6}\right)$. Also, the denominator term  $T-\tau$ stands for the remaining transmission time.  
\subsubsection{Time complexity}
In the proposed algorithm, nested loops are considered where the sub-grant provider and beneficiary users are the outer and inner loops within the algorithm, respectively. Thus, the central controller requires to run the algorithm~$\mathcal{O}\left(N\times M \right)$ operations to complete the beneficiary users' selection process.
\subsection{ Open Sub-Granting Radio Resource~(OSGRR) Algorithm}
\label{sec:oalg}
Recently the auction theory, which initially developed in the economy, has attracted many scholars' attention and has been applied to various problems in engineering. The essence of an auction environment consists of auctioneers or sellers, bidders, commodities to be sold, and a set of rules which give rise to the game among all the bidders. In some auctions, there exists one seller that can perform the role of auctioneer. As a result, auctioneer and seller terms can be used interchangeably. 
An auction theory, a sub-field of economics, is a useful tool to model and optimize radio resource allocation in wireless communication wherein radio resources can be allocated among different users, following some rules regulated in the market. One well-known auction is the Vickrey-Clarke-Groves (VCG) auction~\cite{auc2002}, which requires gathering global information from all entities and performing centralized computations. % here i need to add some distributed papers 

In this study, we consider one-shot open-cry auction in which the bidders advertise their offers at once and openly based on a bidding strategy in a distributed manner. Let~$\mathnormal{D}$ be a set of distinct objects which offer some commodities, say sub-granting resources, for sale. Moreover, $\mathnormal{C}$ be a set of buyers wherein each buyer, say beneficiary user, is assumed to assign a valuation $\mathbf{s}_{mk}$ to each seller, i.e., sub-grant provider user, where $k \in \mathnormal{D}$ and $m \in \mathnormal{C}$. Every beneficiary user monitors other bids and advertises a selected sub-grant provider after the exclusion of the assigned sub-grant provider users indicated in the broadcast bid. Note that every bid contains information that indicates the preferred sub-grant provider user of every beneficiary user. In this study, it is assumed that a sub-grant provider does not ask for any cost from the beneficiary user on the sub-granting radio resources. Furthermore, the achieved throughput of every beneficiary user from the sub-granting resources is reflected in a bid generated employing the strategy function. Recall that symmetric equilibrium wherein all players use the same bidding strategy function~\cite{auc2002}, the strategy function in every beneficiary users $\mathit{s_{mk}}$ yields:

\begin{equation}
\resizebox{0.9\columnwidth}{!}{$
\label{eqstrategy_d2i}
\mathit{s_{mk}}=\frac{\beta q_{m}b_k+(1-\beta) q_{mk}}{q_{max}}+\gamma_m,~\forall m=1,...,M, k=1,...,N, 
$}
\end{equation}
where $q_{m}$ and $q_{mk}$ are modulation and coding scheme of $m$-th the beneficiary user towards the BS and the sub-grant provider user, which are normalized to maximum modulation and coding rate $q_{max}$. The number of sub-granted resources from the sub-grant provider to the beneficiary user is denoted by $b$. The term $\beta$ takes a value between 0 and 1 that shows the impact of the multiplied terms in the bidding strategy function and is configured by the BS. In the first term of the equation, we consider two factors, the first factor guarantees the sub-granting gain, and the latter ensures to choose a sub-grant provider with the higher signal strength to reduce the probability of the sub-granting errors.

 Note that if two beneficiary users have the same~\ac{MCS}, the first term of the equation may return the same value resulted in a collision between two beneficiary users due to transmission on the same sub-granting resource. To avoid a tie situation in the equation, a small value of $\gamma_m$ is added to the first term of bidding strategy function, calculated from the reverse of a unique cellular user-specific number $\mathcal{Z}_m$, say, a~\ac{TMSI}~\cite{3gpp_23_3}. %Consequently, the highest cell throughput is achieved when the sub-granting resources are granted to the beneficiary users offering the highest bid.  

Figure~\ref{figbidstg} shows an illustrative example of the equilibrium bidding value of the strategy function, considering a specific \ac{TMSI} value for every beneficiary user.
\begin{remark}
The highest cell throughput is achieved when the sub-granting resources are granted to the beneficiary users offering the highest bid value.  
\end{remark}
\begin{figure}[h!]
	%\begin{minipage}{\columnwidth}
		\centering
		\captionsetup{justification=centering}
		\includegraphics[height=5cm,width=0.9\columnwidth]{Fig6_Bidding_strategy.eps}
		\caption{An Illustrative example of bidding strategy function.}
		\label{figbidstg}
	%\end{minipage}
\end{figure} 
A bipartite graph is used to model the beneficiary user selection problem in the sub-granting radio resource wherein the beneficiary users, bidders, and the sub-grant provider users, sellers, or auctioneers, are two vertices of a graph as illustrated in figure~\ref{figgraph}. The edges of the graph are weighted by bidding values obtained from the bidding strategy function. This way, the problem is transformed into the maximum matching in the bipartite graph. Now we propose a closed-form heuristic algorithm, say, open sub-granting radio resource to address the beneficiary user selection problem in the sub-granting radio resource as stated in Equation~$\left(\ref{eqoptp}\right)$. 

Algorithm $\ref{opsgalg}$ shows the principle of operation of the open sub-granting radio resource. The beneficiary user's unique value $\gamma_m$, $\beta$, and bid transmission start time are configured by the BS. 
Where the bid transmission start time is a time value between two consecutive selection time instances in which every beneficiary user is allowed to transmits the bid value. 
Also, this value ensures that two beneficiary users do not start transmission at the same time. Thus any possible collision due to a half-duplex communication in the D2D communication is avoided.
Then, every beneficiary user calculates a bid value $s_{mk}$ associated to every sub-grant provider users using the equation $\left(\ref{eqstrategy_d2i}\right)$ considering the bit errors rate and power head room stated in constraints~$\left(\ref{eqoptp_3}\right)$ and $\left(\ref{eqoptp_7}\right)$ in equation $\left(\ref{eqoptp}\right)$. Note that every beneficiary user chooses a maximum bid value $S_{mk}$ associated to the sub-grant provider user and disseminates the bid value along with the corresponding sub-grant provider user identity. 
The beneficiary user informs the nearby users about the bid value~$s_{mk}$~through D2D communication on the scheduled uplink radio resource at the configured bid transmission start time. 
Next, the edges of the graph are updated based on its bid value, and other monitored beneficiary users bid values~$($see Lines $(1)$ to $(15)$ in the Algorithm~$\ref{opsgalg})$.    
Finally, every beneficiary user constructs a list of maximum bid values corresponding to the monitored beneficiary users and the associated sub-grant provider users, i.e., matching list $\mathbb{X}_{mk}$. Note that, a beneficiary user having the biggest bid value on the matching list is allowed to transmit on the sub-granting radio resources over the beneficiary user-selection time interval configured by the \ac{eNB} $($see Lines $(16)$ to $(20)$ in Algorithm~$\ref{opsgalg})$.

\begin{algorithm}
	\caption{Open Sub-Granting Radio Resource}
	\label{opsgalg}
	\begin{algorithmic}[1]
		\Procedure{Beneficiary user selection}{}
		\Statex $\textbf{Input:}$  $\text{Configure $\beta$ value used in Equation~$\left(\ref{eqstrategy_d2i}\right)$}$
		\Statex $\textbf{Initialization:}$
		%\For {$c_m\in~\mathnormal{C}$} 
		\State $\text{Calculate $\gamma_m=\frac{1}{\mathcal{Z}_m}$},~m=1,...,M$
		\State $\text{$E\leftarrow \phi$}$
		\For{$d_k\in~\mathnormal{D}$} 
		\If{$ \text{Constraints~$\left(\ref{eqoptp_3} \right)$} ~\& ~\text{$\left(\ref{eqoptp_7}\right)$}$} 
		\State $\text{Calculate bid value $s_{mk}$ from Equation~$\left(\ref{eqstrategy_d2i}\right)$}$
		\State $\text{ Update edge value of graph, $E \leftarrow \left(E\cup s_{mk}\right)$ }$
		\EndIf
		\State $\textbf{endif}$
		\EndFor
		\State $\textbf{endfor}$
		\State $\text{Select maximum bid value $s_{mk}$ and  broadcast}$
		\State $\text{Update edge value of graph, $E \leftarrow E-\cup_{j=1}^{N} s_{mj} , \forall j~\neq k$}$
		\For {$c_{m_{-1}}\in~\mathnormal{C}$}
		\State $\text{Monitor bid value $s_{m_{-1}k}$ of other beneficiary user}$
		\State $\text{Update edge value of graph, $E \leftarrow \left(E~\cup s_{m{-1}k}~\right)$}$
		\EndFor
		\State $\textbf{endfor}$
		\Statex $\textbf{Selection:}$
		\State $\mathbb{X}\leftarrow\phi$
		\For {$d_k\in \mathnormal{D}$}
		%\For {$c_m\in~\mathnormal{C}$}
		\State $\text{Select $c_m~\in \mathnormal{C}~$with maximum bid value~$s_{mk}$}$
		\State $\text{$\mathbb{X}_{mk}=1$ and~$\mathbb{X}\leftarrow \mathbb{X}\cup \mathbb{X}_{mk}$}$
		%\EndFor
		%\State $\textbf{endfor}$
		\EndFor
		\State $\textbf{endfor}$
		\Statex $\textbf{Output:}$~$\textit{Macthing List $\mathbb{X}$}$
		\EndProcedure
	\end{algorithmic}
\end{algorithm}
\subsubsection{Overhead}
The overhead in the open sub-granting algorithm is mainly due to bidding messages that are exchanged among the beneficiary users and also sub-granting signaling messages. Therefore, the imposed overhead on the beneficiary users~$\hbar_m$ yields: 
\begin{equation}
\label{openoverhead_d2i_sg}
\hbar_m\left(\tau\right)=\frac{\mathbb{X}_{br}\times O_{br}}{T},
\end{equation}
where $O_{br}$ stands for the overhead value owing to bidding message exchanged among the beneficiary users, and $\mathbb{X}_{br}$ is a value that is set to 1 at every broadcast time interval $T_{obr}$. Moreover, in case of the sub-granting scheme overhead on the \ac{D2D} communication $h_k$, positioning information is not transmitted to the BS, and thus equation~$\left(\ref{overhead_d2d_sg}\right)$ can be rewritten as follows:
\begin{equation}
\label{openoverhead_d2d_sg}
\hbar_k\left(\tau\right)=\frac{\sum_{m=1}^{M}\mathbb{X}_{mk}\times O_{sg}}{T-\tau},
\end{equation}
\subsubsection{Time complexity}
This section explains the steps required to execute the algorithm. The $\ac{OSGRR}$ algorithm includes two terms, which each runs in~$O(N)$ and~$O(M)$~time, respectively. Therefore, the algorithm takes about $O(N+M)$ to find a match list. %Furthermore, although the algorithm can be done in~$O(M+N)$, it can be thought of as~$O(N)$ (assuming~$N>M$) or ~$O(M)$~(if $M>N$).
For $N>>M$ or $M>>N$, the complexity is simply $O(N)$ or $O(M)$ respectively.
\section{Simulation Parameters and Performance Metrics}
\label{secresults}
%\subsection{Simulation setup and parameters}
We assume a single cell system with a carrier frequency centered at 2.6 GHz. There are 100 \ac{RB}s available, and 40 \ac{RB}s are allocated to 40 \ac{D2D} users so that each \ac{D2D} user is assigned an RB in a semi-persistent manner. The remaining \ac{RB}s are scheduled among 60 \ac{D2I} users equally. In this topology, the cell radius is 300 meters, and all users are uniformly distributed within the cell and move in random directions with constant speeds. At the cell border, the \ac{UE}s select a random direction towards inside the cell and continue moving inside the cell. The channel models in~\cite{3gpp_ts_36_814} are used for the path-loss and large-scale fading, i.e., shadowing effects. More specifically, the indoor hot-spot non-line-of-sight (InH-NLOS) and the urban micro hexagonal cell layout non-line-of-sight (UMi-NLOS) models are regarded as channel gains for the \ac{D2D} and \ac{D2I} communications, respectively~\cite{3gpp_ts_36_814}. Besides, we consider the Rayleigh and Rician fading models to capture the small-scale fading effects, but without loss of generality, we assume that the channel conditions do not vary during the sub-granting signaling and bid information transmission. For both \ac{D2D} and \ac{D2I} communications, \ac{LTE} open loop power control is assumed~\cite{dahlman20134g}. The transmission power distribution of~\ac{D2I} users is shown in Figure~\ref{figpowdist}. In this paper, a traffic model based on the requirements given in~\cite{schulz2019network} is considered~(See Table~\ref{tab:parameters}). Figure~\ref{figtraf} illustrates the distribution of traffic payload generated by \ac{D2D} users over the simulation run.

To avoid any non-uniformity in user distribution due to mobility inside the cell and have more realistic outcomes, the simulator is run ten times in which the simulation duration is 4000ms, and then the results are averaged over every simulation run. Simulation parameters are summarized in Table~\ref{tab:parameters}. 
\begin{table}[h!]
	\renewcommand{\arraystretch}{1.2}
	\centering
	\caption{Network parameters used in the simulator.}
	\label{tab:parameters}
	\vspace*{0.1cm}	
	\resizebox{\columnwidth}{!}{%
		\begin{tabular}{|p{1.0in}|p{2.0in}|} \hline 
			\textbf{Parameter} & \textbf{Value} \\ \hline 
			Frequency, \textit{fc} and BW & 2.6~GHz, 20~MHz \\ \hline 
			Number of users & 60~D2I, 40~D2D \\ \hline 
			Cell Radius & 300~m \\ \hline 
			D2D Distance~(\textit{d})& 20~m\\ \hline 
			Channel Model~\cite{3gpp_ts_36_814}
			& UMi-NLOS\newline
			$\alpha_c$: 3.67\newline
			$\mu_c:\sigma=4~dB$,~Rayleigh\newline
			InH-NLOS\newline
			$\alpha_d$: 4.33\newline
			$\mu_d:\sigma=4~dB$,~Rician, K=20~dB
			\\ \hline 
			User Velocity & 30Kmph
			\\ \hline 
			Traffic Model & Packet size=10B, $\lambda$=1ms, $\epsilon^{\mathbb{D}}_{th}$=$10^{-5}$
			\cite{schulz2017latency}
			\\ \hline 
			Power Control & Open Loop Power Control\newline $D2D$~($P0$ = -90~dBm/RB)\newline
			$D2I$~($P0$ = -107~dBm/RB)\\ \hline 
			Noise Power Density ($\sigma_0$)  & -174~dBm/Hz\\ \hline 
			Noise Figure &5~dB \\ \hline
			MAX UE TX Power & 23~dBm \\ \hline
			D2D and D2I Antenna Gain & 1~dB \\ \hline
			eNB Antenna Gain & 10~dB \\ \hline
			$\epsilon^{sg}_{th} $&$10^{-3}$\\ \hline
			Simulation Runs~$\left(i\right)$& 4000 \\ \hline
			$T_{me}$,~$T_{pos}$,~$T_{br}$& 480ms \\ \hline 
			$\beta$&0.9\\ \hline
		\end{tabular}
	}
\end{table}

\begin{figure*}[h!]
	\centering
	\captionsetup{justification=justified}
	\begin{minipage}[b]{.4\textwidth}
		\includegraphics[scale=0.35]{Fig7_traffi_dist.pdf}
		\caption{D2D Traffic Distribution.}\label{figtraf}
	\end{minipage}\qquad
	\begin{minipage}[b]{.4\textwidth}
		\includegraphics[scale=0.33]{Fig8_powerdist.pdf}
		\caption{D2I Transmission Power Distribution.}\label{figpowdist}
	\end{minipage}\qquad
	\begin{minipage}[b]{.4\textwidth}
		\includegraphics[scale=0.4]{Fig9_ela.pdf}
		\caption{Impact of P0 on eLA. P0 is the desired received signal in the open-loop power control equation~\cite{dahlman20134g}.}\label{figela}
	\end{minipage}
\end{figure*}
\subsection{Performance Metrics}
The following metrics are studied in our study:
\begin{itemize}
	\item Uplink cell throughput.
	\item Average throughput of beneficiary user.
	\item Number of selected beneficiary users.
	\item Sub-granting signalling errors rate.
	\item Overhead rate.
\end{itemize}
\section{Results and Discussion}
 As indicated in the \ac{DSGRR} algorithm, for every sub-grant provider, a geographical area~$\ac{eLA}$ is specified, wherein the sub-granting signaling message can be reliably decoded. Figure~\ref{figela} shows the relationship between the $\ac{eLA}$ and desired received power, $P0$, in the open-loop power control equation. Considering a specified signaling error value (i.e., $\epsilon^{sg}_{th} $), a bigger \ac{eLA} area is achieved at the cost of higher \ac{D2D} transmission power~(higher~$P0$). Despite the circular~\ac{eLA} shape shown in Figure~\ref{figela}, the realistic spatial geometry of the \ac{eLA} is amorphous rather than circular. The reason is because of large-scale fading phenomena, i.e., shadowing, employed in the Equation~\ref{ela_r}, different signal power around the sub-grant provider is received. Thus the spatial geometry of the \ac{eLA} area is distorted. In our analysis, we assume an identical shadowing around a sub-grant provider in every \ac{eLA} estimation interval whereby a circular \ac{eLA} is formed.
 
% the value of each bid is twofold, which are combined in proportion to $\beta$ and complementary~$\beta$. These two parts indicate the achieved gain from the sub-granting radio resources and the received signal strength towards the sub-grant provider users. Therefore,
 
 As discussed in Algorithm~$\left(\ref{opsgalg}\right)$, the value $\beta$ should be set in a way to achieve the maximum gain from the sub-granting resources. 
   Figure~\ref{figalphaimpact} illustrates the impact of $\beta$ value on the uplink cell throughput. When $\beta$ value is set to 0.1, the achieved throughput is 3.5\% less compared with the $\beta$ value of 0.9. It is because, in the latter one, the beneficiary users with better \ac{CQI} value towards the \ac{BS} are selected. Although the difference between the achieved throughput with $\beta$ values of 0.9 and 0.5 is marginal, the results show the slightly higher uplink throughput when~$\beta$ value is set at 0.9.
\begin{figure}[h!]
	%\begin{minipage}{\columnwidth}
	\minipage{0.45\textwidth}
		\centering
		\captionsetup{justification=justified}
		\includegraphics[height=4.8cm,width=0.85\columnwidth]{Fig10_betaimp.pdf}
		\caption{Impact of beta value on cell throughput for the \ac{OSGRR}. The highest cell throughput is achieved at a value of $\beta$ 0.9.}
		\label{figalphaimpact}
	%\end{minipage}
	\endminipage
\end{figure} 

Although using a central approach could achieve an optimal solution for the beneficiary users' selection problem, it will increase the burden of overhead arise from the measurement. Therefore, an \ac{eLA} based beneficiary selection algorithm, i.e., \ac{DSGRR}, was proposed in~\cite{soleymani2019dedicated} where the overhead is reduced. However, in the \ac{DSGRR} algorithm, a large-scale fading can only be estimated, whereas, in the \ac{OSGRR} algorithm, both large- and small-scale are captured in the measurement signal from the sub-grant provider users. Due to the small-scale fading in the \ac{OSGRR}, the probability of receiving signal of the sub-grant provider is higher, and thus, the average coverage radius of a sub-grant provider might be larger in the \ac{OSGRR} than the \ac{eLA} area in the \ac{DSGRR}. Figure~\ref{figcomparisionelame} shows an illustrative example of a coverage area for the \ac{OSGRR} and \ac{DSGRR} algorithms. 
\begin{figure}[h!]
	%\begin{minipage}{\columnwidth}
	\minipage{0.45\textwidth}
	\centering
	\captionsetup{justification=justified}
	\includegraphics[height=4.8cm,width=0.85\columnwidth]{Fig11_eLA_ME.pdf}
	\caption{An Illustrative example of the sub-grant provider coverage area for measurement-and eLA-based approach.}
	\label{figcomparisionelame}
	\endminipage
	%\end{minipage}
\end{figure} 
Given the above explanation, more beneficiary users may receive the sub-grant provider signal, through which the number of candidate beneficiary users increases in the case of the \ac{OSGRR} algorithm.

 The advantage of the measurement on the algorithm is approved by a ten-times simulation experiment in which the number of the beneficiary users around the sub-grant provider users for both algorithms are averaged.
 As shown in Figure~\ref{figavgbeneficiary}, the number of candidate beneficiary users is higher in the \ac{OSGRR} compared with the \ac{DSGRR}. The reason is that the measurement-based method, i.e., \ac{OSGRR}, would increase the probability of receiving the sub-granting provider signal. It is worth noting that relaxing the uniform large-scale fading assumption in the \ac{eLA} computation, may contribute to having less number of candidate beneficiary users in the \ac{DSGRR} which leads to further performance degradation in the \ac{DSGRR} algorithm. 
\begin{figure}[h!]
\minipage{0.45\textwidth}
	%\begin{minipage}{\columnwidth}
	\centering
	\captionsetup{justification=justified}
	\includegraphics[height=4.8cm,width=0.85\columnwidth]{Fig12_average_beneficiary_users_cndidates.eps}
	\caption{An Illustrative example of the average number of candidate beneficiary users for every sub-grant provider.}
	\label{figavgbeneficiary}
	%\end{minipage}
\endminipage
\end{figure} 

Further investigation is conducted aiming to show the performance of both algorithms when the effect of the small-scale fading and overhead for both algorithms is relaxed. The simulation is run for one-thousand milliseconds, and the results are averaged over ten-times simulation run. The results show that both algorithms achieve almost the same uplink cell throughput in the studied scenario~(See Figure\ref{figcdfnooverhead} ). Note that the marginal difference is due to the stochastic essence of the large-scale fading in both algorithms, whereby the different number of beneficiary users may be selected. 

\begin{figure}[h!]
	%\begin{minipage}{\columnwidth}
   \minipage{0.45\textwidth}
	\centering
	\captionsetup{justification=justified}
	\includegraphics[height=4.8cm,width=0.85\columnwidth]{Fig13_comparison_velocity_zero_no_channel.eps}
	\caption{An Illustrative example of comparison of the uplink cell throughput for both algorithms in a scenario w/o considering overhead and small-scale fading for the stationary users.}
	\label{figcdfnooverhead}
	\endminipage%\end{minipage}
\end{figure} 

In the dedicated sub-granting algorithm, i.e., DSGRR, every entity transmits CSI and positioning measurements to the BS at a time interval of 480ms. After that, the BS transmits control information indicating the beneficiary user for every sub-grant provider user. These measurements and control information is carried on the uplink/downlink LTE physical layer control channels. In this study, we assume bandwidth one resource block and modulation coding scheme QPSK-1/2 to carry the control information and measurement information in downlink and uplink, respectively. For example, the overhead due to the uplink measurement, is calculated by (sub-carrier) * (OFDMA symbols) * (modulation order) * (code rate) = (12*14*2*1/2):8 = 21 bytes and considering 3 bytes as~\ac{CRC}, total overhead has amounted to 24 bytes~\cite{dahlman20134g}.

The BS informs the sub-grant providers about the selected beneficiary users through downlink signaling information. Similarly, in the downlink, the overhead is $(12*3*2*(1/2)):8 = 5$ bytes, and 3 bytes is added as \ac{CRC} resulted in 8 bytes overhead for the beneficiary user selection signaling.
 
Also, assuming a global navigation satellite system provides timing and positioning information for every cellular user, every entity can provide the cellular network with location information~(e.g., location estimate, pseudo-range, velocity) together with time information. To have a fair comparison, we only consider the uplink user-assisted information, which is required by the location server in order to estimate the exact position of users (c.f. section 6.5.2.5 in~\cite{3gpp_ts_36_555}). Also, the overhead caused by the transmission of messages between the server and the base station is not taken into consideration. Considering the Mac layer and physical layer overhead, about 40 bytes are required to acquire the positioning information of every cellular user. We assume that the network can obtain a sufficiently accurate position of ever users by employing user-assisted information, and the position error in the algorithm is negligible.


 In the open sub-granting algorithm, i.e., \ac{OSGRR}, every beneficiary user offers a bid value on the advertised sub-granting resources, and a beneficiary user with the highest offer can utilize the sub-granting resource for an interval time of 480ms. Therefore, the overhead scales up by increasing the number of beneficiary users exchanging signaling messages. Considering the bid value formulated in the \ac{OSGRR} algorithm, one byte is needed to indicate the \ac{CQI} values, and about 4 bytes are used to address the beneficiary user's unique number~$\gamma_m $. In other words, only 5 bytes are required to address different beneficiary users' bid. Consequently, 6 bytes are needed to capture the physical and the \ac{MAC} layer overhead~\cite{dahlman20134g}, which resulted in 11 bytes overhead in the \ac{OSGRR} algorithm. Note that the sub-granting signaling imposes the same overhead on both algorithms, which amounted to 1 byte~\cite{Pusc1604:Hierarchical}. Table \ref{tab:overhead} illustrates components and size of the overhead in both algorithms.    

% Please add the following required packages to your document preamble:
% \usepackage{multirow}
% Please add the following required packages to your document preamble:
% \usepackage{multirow}
%\begin{table}[]
%	\centering
%	\caption{An Illustration of overhead components and size in the OSGRR and DSGRR.}
%	\label{tab:overhead-table}
%	%\resizebox{0.5\textwidth}{!}{%
%	  \begin{adjustbox}{width=0.5\textwidth}
%			
%	\begin{tabular}{|c|c|c|}
%		\hline
%		\multicolumn{1}{|l|}{Selection Algorithm} & \multicolumn{1}{l|}{Overhead Component} & \multicolumn{1}{r|}{Size (Byte)} \\ \hline
%		\multirow{3}{*}{DSGRR} & Positioning Information                                     & 40 \\ \cline{2-3} 
%		& Measurement Information (e.g., buffer, power headroom, CSI) & 24 \\ \cline{2-3} 
%		& Beneficiary user selection signaling                        & 8  \\ \hline
%		OSGRR                  & Bidding Information                                         & 11 \\ \hline
%	\end{tabular}
%%}
%  \end{adjustbox}
%
%\end{table}


\begin{table}[h!]
	\large
	\renewcommand{\arraystretch}{1.2}
	\centering
	\caption{An Illustration of overhead components and size in the OSGRR and DSGRR.}
	\label{tab:overhead}
	\vspace*{0.1cm}	
	 	\resizebox{\columnwidth}{!}{%
		\begin{tabular}{|p{1.2in}|p{2.5in}|p{1in}|} \hline 
			\textbf{Selection Algorithm} & \textbf{Overhead Components}&\textbf{Size(Byte)} \\ \hline 
			DSGRR & Positioning Information\newline
			Measurement Information (e.g., buffer status, power head room) \newline
			Beneficiary user selection signaling\newline Sub-grant signaling & 40\newline 24 \newline \newline 8	\newline 1
			
			 \\ \hline 
			OSGRR & Bidding Information \newline Sub-grant signaling & 11\newline 1
			
			\\ \hline 
		\end{tabular}
	}
\end{table}



% Figure overhead
\begin{figure}[h!]
\minipage{0.45\textwidth}
	\centering
	\captionsetup{justification=justified}
	%\fbox{
	%\begin{minipage}{\columnwidth}
		\captionsetup{justification=justified}
		\includegraphics[height=4.8cm,width=0.85\columnwidth]{Fig14_New_overhead_figure_2020_01_26.eps}
		\caption{Overhead comparison between open sub-granting and dedicated sub-granting algorithms.}
		\label{figovh}
	%\end{minipage}    
	%}
\endminipage %\hfill
\end{figure} 

Figure~\ref{figovh} compares the overall overhead rate between the \ac{OSGRR} and the \ac{DSGRR} algorithms. Besides, the figure illustrates the impact of measurement and positioning interval on the \ac{DSGRR} algorithm. It can be seen that the overhead on the \ac{DSGRR} algorithm is higher than the \ac{OSGRR} algorithm when the measurement and bid transmission time interval are equal for both algorithms. The reason is that in the beneficiary user selection process, the volume of information exchanged between users and the base station in the \ac{DSGRR} algorithm is higher than the data transmitted between users in the \ac{OSGRR} algorithm.

 The overhead on the \ac{DSGRR} is reduced when the measurement transmission interval increases from 480ms to~8~$\times$~480ms; however, the result still shows less overhead in the \ac{OSGRR} algorithm compared with the \ac{DSGRR}. The reason is that the \ac{OSGRR} needs a few bytes to broadcast the bids and does not impose any~\ac{CSI} and positioning measurement overhead on the~\ac{BS}. Although in the \ac{DSGRR}, the overhead can be further reduced by incrementing measurement transmission interval, the performance will deteriorate as the outdated measurement information is used for the beneficiary user selection. 

Figure~\ref{figseluser} demonstrates the number of selected beneficiary users for both algorithms. The results show that the average number of selected beneficiary users in the \ac{OSGRR} algorithm is about 10\% higher than that of the \ac{DSGRR} algorithm. As previously discussed, the \ac{OSGRR} has a higher number of beneficiary user candidates for every sub-grant provider compared with the \ac{DSGRR} due to a measurement-based selection. For example, in the \ac{DSGRR}, if a beneficiary user is the only candidate at the border of the overlap \ac{eLA} area of two sub-grant providers, the beneficiary user can be selected by one of the sub-grant providers. In contrast, in the~\ac{OSGRR}, farther beneficiary user candidates might be able to receive a sub-grant provider without any candidate resulted in increasing the number of beneficiary users selected by the sub-grant providers. Also, the results confirm that the \ac{OSGRR} algorithm can serve a higher number of beneficiary users than the \ac{DSGRR} algorithm.

\begin{figure}[h!]
	
	\minipage{0.45\textwidth}
	%			
	\centering
	\captionsetup{justification=justified}
	\includegraphics[height=4.8cm,width=0.9\columnwidth]{Fig15_selection.pdf}
	\caption{Comparison of the average number of selected beneficiary users between the \ac{DSGRR} and \ac{OSGRR} algorithms.}
	\label{figseluser}
	\endminipage
	%}
\end{figure} 

Figure~\ref{figsgerror} compares the sub-granting error rate for the \ac{OSGRR} and~\ac{DSGRR} algorithms. 
Also, the impact of positioning and measurement interval is illustrated in the figure. In general, the~\ac{DSGRR} shows a lower error rate compared with the~\ac{OSGRR} when both algorithms use the same transmission time interval for positioning, \ac{CSI} measurement, and bid information. The reason is that in the \ac{OSGRR}, farther beneficiary users are selected, which increases the probability of not decoding the sub-granting information owing to the fading between the sub-grant provider and the beneficiary user. In the~\ac{DSGRR}, when~\ac{CSI} and positioning measurement time interval increases by eight times, the sub-granting signaling error rate increase by slightly more than two times due to using the outdated~\ac{eLA} information during the beneficiary selection process. 

\begin{figure}[h!]
	\minipage{0.45\textwidth}
	\centering
	\captionsetup{justification=justified}
	\includegraphics[height=4.8cm,width=0.9\columnwidth]{Fig16_sgerr.pdf}
	\caption{Sub-granting errors rate comparison between the \ac{DSGRR} and \ac{OSGRR} algorithms.}
	\label{figsgerror}
	
	\endminipage %\hfill
\end{figure}

% Figure CEll throughput
Figure~\ref{figcellthr} shows the impact of the measurement and positioning interval on the \ac{DSGRR} throughput and compares both \ac{DSGRR} and \ac{OSGRR} algorithms in terms of cell throughput. Also, the closeness of both algorithms to the maximum achievable cell throughput is evaluated. To this end, the cell throughput of the \ac{DSGRR} and \ac{OSGRR} algorithms are compared with the case w/o any sub-granting algorithm and the maximum achievable cell throughput when all the allocated radio resources are fully utilized. Considering the same transmission interval and speed, as shown in Table~\ref{tab:parameters} for both algorithms, the \ac{OSGRR} shows slightly better results than the \ac{DSGRR}. 
The overhead and the number of selected beneficiary users are two factors that contribute to the cell throughput reduction in the \ac{DSGRR} compared with the \ac{OSGRR}. As indicated in Figure \ref{figovh}, the overhead contributes only to about~5\% of the cell throughput reduction in the \ac{DSGRR}. The remaining reduction is due to the number of selected beneficiary users, which resulted in a higher throughput in favor of the \ac{OSGRR}.

 When the measurement transmission interval increases by eight times, the cell throughput in the \ac{DSGRR} gradually decreases over the simulation runs. The results show about a 10$\%$ reduction on the cell throughput compared with the one with shorter measurement interval. In contrast, the cell throughput remains almost constant in the \ac{OSGRR} over the simulation runs. The results show that the \ac{OSGRR} could achieve around 85\% of the maximum cell throughput. The remaining 15\% reduction is mainly due to the sub-granting signaling overhead and lack of a beneficiary user candidate or an optimum beneficiary user. It is noteworthy to mention that 35\% of the allocated radio resources are wasted w/o sub-granting scheme~(See Figure~\ref{figcellthr}). 
\begin{figure}[h!]
	\centering
		\minipage{0.45\textwidth}
	%\begin{minipage}{\columnwidth}
		\captionsetup{justification=justified}
		\includegraphics[height=4.8cm,width=0.85\columnwidth]{Fig17_cellthr.pdf}
		\caption{Impact of measurement/positioning transmission interval on cell throughput.}
		\label{figcellthr}
	\endminipage %\hfill
\end{figure}

%Figure~\ref{figvcim} shows user   

%\begin{figure}[h!]
%	\centering
%	\captionsetup{justification=centering}
%	%\begin{minipage}{\columnwidth}
%				\captionsetup{justification=centering}
%		\includegraphics[height=5cm,width=0.9\columnwidth]{images/cdf_cell_througput_new1.eps}
%		\caption{Commutative distribution function\\ of the uplink cell throughput.}
%		\label{figvcim}
%%	\end{minipage}
%	%}
%\end{figure} 
%\begin{figure}[t]
%	\fbox{
%	\begin{minipage}{\columnwidth}
%	\centering
%\captionsetup{justification=centering}
%	\includegraphics[height=5cm,width=0.9\columnwidth]{images/Fig_d2ithr.pdf}
%	\caption{Average D2I throughput}
%	\label{figd2ithr}
%\end{minipage}
%}
%\end{figure} 

Figure \ref{figd2ithr} depicts the average throughput of the beneficiary user for both algorithms considering the same transmission time interval. Both algorithms show about 55\% increase compared with no~\ac{SGRR}. This is due to the re-utilization of the unused resources in both algorithms. Although the \ac{OSGRR} shows higher signaling errors compared with the \ac{DSGRR} (See Figure\ref{figsgerror}), the average beneficiary user throughput is slightly higher than that achieved by the \ac{DSGRR} algorithm. The reason is mainly that more beneficiary users are selected in the \ac{OSGRR} compared with the \ac{DSGRR}, and thus the more beneficiary users can re-utilize the sub-granting radio resources resulted in achieving a higher throughput in the \ac{OSGRR}.

\begin{figure}[h!]
	\minipage{0.45\textwidth}
	\centering
	\captionsetup{justification=justified}
	\includegraphics[height=4.8cm,width=0.85\linewidth]{Fig18_d2ithr.pdf}
	\caption{Comparison of the average uplink throughput of a beneficiary user.}
	\label{figd2ithr}
	\endminipage %\hfill
\end{figure}

%The need for moving UEs positioning and measurement information synchronization typical of beneficiary selection algorithm is 
%a major drawback for the DSGRR algorithm as associated signalling overhead scales up with the number of UEs.
%Given that positioning information accounts for 90bytes and  

%\begin{figure}[t]
%	
%	%\fbox{
%		\begin{minipage}{\columnwidth}
%			
%	\centering
%	%\captionsetup{justification=centering}
%	\includegraphics[height=5cm,width=0.9\columnwidth]{images/Fig_selection.pdf}
%	\caption{Average D2I user selection}
%	\label{figseluser}
%\end{minipage}
%%}
%\end{figure} 
\section{Conclusions}
\label{secconclude}
This paper investigates the sub-granting radio resource allocation for \ac{D2D} communication whose radio resources are allocated in a semi-persistent manner in an overlay mode. Next, the sub-granting radio resource allocation is formulated as a maximum weighted matching in a bipartite graph problem. Also inspired from auction theory, an \ac{OSGRR} in a distributed manner is proposed. The overhead is formulated mathematically for the proposed algorithms and calculated based on some assumptions. The simulation results and analyses show the superiority of the distributed algorithm over the centralized one. To show the efficiency of both algorithms, the tightness of both algorithms to the maximum achievable uplink cell throughput is examined. Developing a new machine learning-based algorithm considering new parameters for the beneficiary user selection in a multiple cells scenario is worth investigating in future works. 
%\section*{Section title}
%Text for this section \ldots
%\subsection*{Sub-heading for section}
%Text for this sub-heading \ldots
%\subsubsection*{Sub-sub heading for section}
%Text for this sub-sub-heading \ldots
%\paragraph*{Sub-sub-sub heading for section}
%Text for this sub-sub-sub-heading \ldots
%In this section we examine the growth rate of the mean of $Z_0$, $Z_1$ and $Z_2$. In
%addition, we examine a common modeling assumption and note the
%importance of considering the tails of the extinction time $T_x$ in
%studies of escape dynamics.
%We will first consider the expected resistant population at $vT_x$ for
%some $v>0$, (and temporarily assume $\alpha=0$)
%%
%\[
% E \bigl[Z_1(vT_x) \bigr]= E
%\biggl[\mu T_x\int_0^{v\wedge
%1}Z_0(uT_x)
%\exp \bigl(\lambda_1T_x(v-u) \bigr)\,du \biggr].
%\]
%%
%If we assume that sensitive cells follow a deterministic decay
%$Z_0(t)=xe^{\lambda_0 t}$ and approximate their extinction time as
%$T_x\approx-\frac{1}{\lambda_0}\log x$, then we can heuristically
%estimate the expected value as
%%
%\begin{eqnarray}\label{eqexpmuts}
%E\bigl[Z_1(vT_x)\bigr] &=& \frac{\mu}{r}\log x
%\int_0^{v\wedge1}x^{1-u}x^{({\lambda_1}/{r})(v-u)}\,du
%\nonumber\\
%&=& \frac{\mu}{r}x^{1-{\lambda_1}/{\lambda_0}v}\log x\int_0^{v\wedge
%1}x^{-u(1+{\lambda_1}/{r})}\,du
%\nonumber\\
%&=& \frac{\mu}{\lambda_1-\lambda_0}x^{1+{\lambda_1}/{r}v} \biggl(1-\exp \biggl[-(v\wedge1) \biggl(1+
%\frac{\lambda_1}{r}\biggr)\log x \biggr] \biggr).
%\end{eqnarray}
%%
%Thus we observe that this expected value is finite for all $v>0$ (also see \cite{koon,khar,zvai,xjon,marg}).
%\nocite{oreg,schn,pond,smith,marg,hunn,advi,koha,mouse}

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\begin{backmatter}
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%\clearpage % Start a new page

%\lhead{\emph{List of Acronyms}} 
%\addcontentsline{toc}{chapter}{List of Acronyms}
%\addtotoc{List of Acronyms} 
\section*{Availability of data and materials}
All results are included in this published article; the results raw output is available from the corresponding author on reasonable request.
\section*{Abbreviations}
\begin{acronym}
	\acro{2G}{Second Generation}
	\acro{3G}{Third Generation}
	\acro{4G}{Fourth Generation}
	\acro{5G}{Fifth Generation}
	\acro{3GPP}{Third Generation Partnership Project}
	\acro{BS}{Base Station} 
	\acro{E-UTRA}{Evolved Universal Terrestrial Radio Access}
	\acro{E-UTRAN}{Evolved Universal Terrestrial Radio Access Network}
	\acro{GPS}{Global Positioning System}
	\acro{GSM}{Global System for Mobile communication}
	\acro{HARQ}{Hybrid Automatic Repeat Request}
	\acro{LTE}{Long Term Evolution}
	\acro{LTE-A}{LTE Advanced}
	\acro{MBSFN}{Multicast-broadcast single-frequency network}
	\acro{MAC}{Media Access Control}
	\acro{MAS}{Multi-Agent System}
	\acro{MDP}{Markov Decision Process}
	\acro{MIMO}{Multiple Input Multiple Output}
	\acro{PRB}{Physical Resource Block}
	\acro{QoS}{Quality of Service}
	\acro{QoE}{Quality of Experience}
	\acro{RAN}{Radio Access Network}
	\acro{RAT}{Radio Access Technology}
	\acro{RF}{Radio Frequency}
	\acro{RRM}{Radio Resource Management}
	\acro{RSRP}{Reference Signal Received Power}
	\acro{RSS}{Received Signal Strength}
	\acro{RSSI}{Received Signal Strength Indicator}
	\acro{SC-FDMA}{Single Carrier Frequency Division Multiple Access}
	\acro{SINR}{Signal to Interference and Noise Ratio}
	\acro{SMIC}{SINR Maximizing Interference Coordination}
	\acro{SNR}{Signal to Noise Ratio}
	\acro{TTT}{Time To Trigger}
	\acro{Tx}{Transmit} 
	\acro{UC}{Use Case}
	\acro{UE}{User Equipment}
	\acro{MTC}{Machine Type Communication}
	\acro{uRLLC}{ultra Reliable Low Latency Communication}
	\acro{D2D}{Device-to-Device}
	\acro{RRM}{Radio Resource Management}
	\acro{D2I}{Device-to-Infrastructure}
	\acro{OFDMA}{Orthogonal Frequency Division Multiple Access}
	\acro{NR}{Next Radio}
	\acro{uRLLC}{ultra Reliable Low Latency Communication}
	\acro{SCFDMA}{Single Carrier Frequency Division Multiple Access}
	\acro{IoT}{Internet of Things}
	\acro{3GPP}{3rd Generation Partnership Project }
	\acro{HARQ}{Hybrid Automatic Repeat reQuest}
	\acro{CSI}{Channel State Information}
	\acro{DMRS}{DeModulation Referece Signal}
	\acro{V2X}{Vehicle-to-Everthings}
	\acro{NR}{Next Radio}
	\acro{V2I}{Vehicle-to-Infrastructure}
	\acro{V2P}{Vehicle-to-Pedestrain}
	\acro{V2N}{Vehicle-to-Network}
	\acro{V2V}{Vehicle-to-Vehicle}
	\acro{ProSe}{Proximity Service}
	\acro{SC-FDMA}{Single Carrier Frequency Division Multiple Access}
	\acro{ILA}{Interference Limited Area}
	\acro{IoT}{Internet of Things}
	\acro{TTI}{Transmision Time Interval}
	\acro{RSSI}{Received Signal Strength Indicator}
	\acro{VUE}{Vehicular User Equipment}
	\acro{MCS}{Modulation and Coding Scheme}
	\acro{M2M}{Machine-to-Machine}
	\acro{RRM}{Radio Resource Management}
	\acro{PRB}{Physical Resource Block}
	\acro{P2I}{Pedestrian to Infrastructure}
	\acro{DMRS}{DeModulation Reference Signal}
	\acro{PSCCH}{Physical Sidelink Control Channel}
	\acro{eNB}{evolved Node B}
	\acro{TB}{Transport Block}
	\acro{MAC}{Medium Access Control}
	\acro{RB}{Resource Block}
	\acro{BER}{Bit Errors Rate}
	\acro{TBS}{Transport Block Size}
	\acro{CQI}{Channel Quality Index}
	\acro{SGRR}{Sub-Granting Radio Resource}
	\acro{OSGRR}{Open Sub-Granting Radio Resource}
	\acro{DSGRR}{Dedicated Sub-Granting Radio Resource}
	\acro{eLA}{error-Limited Area}
	\acro{NR}{Next Radio}
	\acro{RAN}{Radio Access Network}
	\acro{gNB}{gNodeB}
	\acro{MIMO}{Multiple Input Multiple Output}
	\acro{MRC}{Maximum Ratio Combining}
	\acro{MWM}{Maximum Weighted Matching}
    \acro{TMSI}{Temporary Mobile Subscriber Identity}
    \acro{CRC}{Cyclic Redundancy Check }
\end{acronym}
%\clearpage % Start a new page

%\printglossary[type=\acronymtype,title=Abbreviations]


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\section*{Figure Title and Legends}
\textbf{Figure 1:} System Model.

A typical network is consisting of one \ac{eNB}, one \ac{D2D}-\ac{UE}, and one \ac{D2I}-\ac{UE}. The k-th~\ac{D2D} user sub-grants the un-used radio resources to the m-th~\ac{D2I} user.

\textbf{Figure 2:}

State-machine diagram of a beneficiary user functionality on the dedicated sub-granting algorithm.

\textbf{Figure 3:}

State-machine diagram of a beneficiary user functionality on the open sub-granting algorithm.

\textbf{Figure 4:}

State-machine diagram of sub-grant provider functionality.

\textbf{Figure 5:}

An Illustrative example of graph model.
 
\textbf{Figure 6:} 

 An Illustrative example of bidding strategy function.
 
\textbf{Figure 7:} 
 
 D2D Traffic Distribution.
 
\textbf{Figure 8:} 
  
 D2I Transmission Power Distribution.

 \textbf{Figure 9:} 
   
 Impact of P0 on eLA. P0 is the desired received signal in the open-loop power control equation.
 
\textbf{Figure 10:} 
  
 Impact of beta value on cell throughput for the \ac{OSGRR}. The highest cell throughput is achieved at a value of $\beta$ 0.9.
 
\textbf{Figure 11:} 

 An Illustrative example of the sub-grant provider coverage area for measurement-and eLA-based approach.
 
 \textbf{Figure 12:}
 
 An Illustrative example of the average number of candidate beneficiary users for every sub-grant provider.
 
 \textbf{Figure 13:} 
 
 An Illustrative example of comparison of the uplink cell throughput for both algorithms in a scenario w/o considering overhead and small-scale fading for the stationary users.
 
 \textbf{Figure 14:}
 
 Overhead comparison between open sub-granting and dedicated sub-granting algorithms.
 
\textbf{Figure 15:}
 
 Comparison of the average number of selected beneficiary users between the \ac{DSGRR} and \ac{OSGRR} algorithms.
 
\textbf{Figure 16:}
 
 Sub-granting errors rate comparison between the \ac{DSGRR} and \ac{OSGRR} algorithms.
 
\textbf{Figure 17:}    
 
 Impact of measurement/positioning transmission interval on cell throughput.
 
\textbf{Figure 18:}    
 
 Comparison of the average uplink throughput of a beneficiary user.
 
  
\section*{Competing interests}
The authors declare that they have no competing interests.
\section*{Acknowledgments}
Not applicable.
\section*{Funding}
This research was supported by Fraunhofer IIS in collaboration with the Technical University of Ilmenau.% \ldots
\section*{Author's contributions}
All authors have reviewed and edited the manuscript and have approved the final manuscript.
%\ldots



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\end{backmatter}
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