A Reciprocal-Selection-Based ‘Win–Win’ Overlay Spectrum-Sharing Scheme for Device-to-Device-Enabled Cellular Network

Abstract

This paper proposes a reciprocal-selection-based ‘Win–Win’ overlay spectrum-sharing scheme for device-to-Device-enabled cellular networks to address the resource sharing between Device-to-Device devices and the cellular users by using an overlay approach. Based on the proposed scheme, the cell edge users intend to lease part of its spectrum resource to Device-to-Device transmission pairs. However, the Device-to-Device users have to provide the cooperative transmission assistance for the cell edge users in order to improve the Quality of Service of the uplink transmission from the cell edge users to the base station. Compared to the underlay spectrum-sharing scheme, overlay spectrum-sharing scheme may reduce spectrum efficiency. Hence, Non-Orthogonal Multiple Access technology is invoked at the Device-to-Device transmitter in order to improve the spectrum efficiency. The Stackelberg game is exploited to model the behaviours of the cell edge users and Device-to-Device devices. Moreover, based on matching theory, the cell edge users and Device-to-Device pairs form one-to-one matching and the stability of matching is analysed. The simulation results show that the proposed reciprocal-selection-based ‘Win–Win’ overlay spectrum-sharing scheme is capable of providing considerable rate improvements for both EUs and D2D pairs and reducing transmit power dissipated by the D2D transmitter to forward data for the EU compared with the existing methods.

1. Introduction

1.1. Background

A Device-to-Device (D2D) enabled cellular network was proposed for improving spectral efficiency and throughput of the cellular network, with the aid of direct peer-to-peer communication between the proximity devices [1,2,3]. More explicitly, in a D2D-enabledd cellular network, the closely located user devices may directly connect to each other and bypass the cellular base station (BS), for the sake of reducing the energy consumption and the end-to-end delay. However, enabling D2D communication in a cellular network generates several technical challenges. As one of the important technical challenges, the cellular spectrum-sharing scheme may be classified into two categories, namely underlay and overlay spectrum-sharing strategies [4]. According to the underlay cellular spectrum-sharing scheme, the D2D devices are capable of accessing the same spectrum with cellular users, provided that they do not substantially interfere with the transmissions of cellular users. Hence, when the cellular and D2D users are occupying the same spectrum, the interference between them must be invested for designing appropriate underlay cellular spectrum-sharing scheme [4]. Under the overlay cellular spectrum-sharing scheme, the D2D devices can only access the cellular channels which are not used by the cellular users. Compared to the underlay scheme, the spectrum efficiency of overlay cellular spectrum-sharing scheme may be limited by assigning a dedicated portion of the cellular spectrum to the D2D communication [4].

1.2. Related Work

D2D communication shares cellular spectrum resource with cellular users in two modes: underlay and overlay. Based on the underlay cellular spectrum-sharing scheme, the cellular users and the D2D devices simultaneously access the same spectrum band. Hence, the interference mitigation between D2D communication and cellular transmission is inevitable and significant [5,6,7,8], when underlay cellular spectrum-sharing scheme is exploited. Considering the interference generated by the D2D communications, a fair distribution of the cellular resources shared by the D2D transmission pairs was developed with the aid of Jain’s fairness index method [5]. Moreover, in order to minimize the interference caused by multiple D2D transmission pairs which simultaneously share a single cellular link, a power control scheme was proposed for supporting underlay D2D communications [6,7,8]. Aiming for minimizing the interference generated by multiple D2D transmission pairs which share the spectrum resource in multiple cells, a spectrum-sharing scheme was proposed in [9]. The works in the above literature [5,6,7,8,9] focused on the mitigation of the interference between the underlay D2D communication and cellular transmission. Based on the overlay spectrum-sharing scheme, the cellular users and D2D users intend to access different channels. Hence, the interference between the D2D and cellular users were not considered in [10,11,12]. Without considering the interference between users, improving spectral efficiency in overlay mode is the main goal [10,11,12].

Based on cellular spectrum-sharing scheme, the benefits of the D2D-enabledd cellular network may be improved with the aid of cooperative communications, where the relay node (RN) forwards the source’s data. More explicitly, the cell edge user (EU) may suffer from the poor link quality due to high path loss and deep path fading. However, the D2D transmitter (DT) may be capable of improving the quality of service (QoS) for the cellular users by providing cooperation assistance [13,14]. Based on the underlay cellular spectrum-sharing scheme, many meritorious contributions have been made for improving the performance of the D2D-enabled cellular network and simultaneously reducing the interference generated by the underlay D2D communication with the aid of cooperative communications [4,15,16,17,18]. However, few existing works focus on the performance of the D2D-enabled cellular network supporting overlay cellular spectrum-sharing strategies and cooperative transmission [19,20,21]. Based on the overlay spectrum-sharing scheme, the cellular users and D2D users intend to access different channels. Hence, the interference between the D2D and cellular users were not considered in [13,14,19,20,21].

Aiming for maximizing system capacity, an overlay spectrum-sharing scheme was proposed for supporting millimeter wave transmitting and D2D communication in [19]. The work in [20] jointly considered incremental relay technology and overlay spectrum-sharing scheme to allow the cellular users to share their transmission resources with the D2D users, instead of treating D2D as interference. Based on the overlay spectrum-sharing scheme, a cooperative D2D communication framework was proposed in [21] for the sake of maximizing transmission rate. However, in order to support the combination of the overlay scheme and cooperative communication, the works in [13,14,19,20,21] performed the D2D transmissions and the cooperative transmissions within different time slots. Compared to the underlay scheme, the spectrum efficiency of the cooperative overlay schemes in [13,14,19,20,21] may be reduced. Hence, more efforts are required to improve the spectrum efficiency of the cellular network supporting overlay D2D communication and cooperative communication.

Aiming for improving spectral efficiency, Non-Orthogonal Multiple Access (NOMA) superposes multiple users in the power domain [22,23]. Furthermore, the Successive Interference Cancellation (SIC) technique may be exploited by the receiver for detecting multiple users [22,23]. Hence, an improved spectral efficiency may be achieved when the NOMA is employed by the D2D pairs in order to simultaneously forward the cellular uesrs’ data and convey its own data [24,25]. As we know, NOMA technology has not been considered in overlay D2D cooperation communications in [13,14,19,20,21]. Considering the combination of NOMA technology and cooperative overlay D2D communication, the transmission power allocated by the DT for the cooperative communication and the D2D transmission may directly affect the transmission rate and the energy consumption of the D2D devices. blueIn order to achieve high energy efficiency, a new framework which aims to find the appropriate data centre is proposed [26] and how to locate the lowest energy consumption route between the user and the designated data centre is addressed [27,28]. Moreover, energy efficiency is achieved by integrating least possible services to satisfy the user requirements given the large number of resources available [29]. In D2D-enabled cellular network, both D2D communications and concurrent transmissions were utilized to achieve high energy efficiency [30]. NOMA is used for concurrent transmissions which is D2D links and cellular links. However, how to allocate power for DT and EU is significant. More explicitly, assuming the transmission power of the DT is limited, if the DT consumes lower transmitter power to forward EU’s data, while conveys itself data at higher power, the D2D transmission may achieve an increased rate and consume reduced energy. However, the QoS requirements of EU may not be satisfied. Conversely, if the DT assigns higher power for cooperative transmission and lower power for the D2D communication, the QoS requirement of EU may be guaranteed at the cost of reducing the rate achieved by the D2D communication and increasing the DT’s energy consumption for the cooperative communication. Therefore, a NOMA-based optimal power allocation strategy may be required for satisfying the QoS requirement of both the EU and D2D when both the NOMA technology and the cooperative overlay sharing scheme are exploited.

1.3. Contributions and Paper Structure

Against the above backcloth, there are some problems to solve here.

• To improve QoS of EUs, EUs ask help from D2D pairs. EUs intend to share spectrum resource with D2D pairs in exchange of D2D pairs’ relaying data for them. However, how to allocate the spectrum resource between EUs and D2D pairs in overlay mode to achieve Win–Win?
• Compared to the underlay spectrum-sharing scheme, overlay spectrum-sharing scheme does not consider interference among users but reduces spectrum efficiency. Hence, how to improve spectrum efficiency in overlay spectrum-sharing scheme?

To solve the above problems, this paper proposes a reciprocal-selection-based ‘Win–Win’ overlay spectrum-sharing scheme (RSWW-OSSS) for the D2D-enabled cellular network supporting overlay spectrum shring scheme and cooperative transmission among multiple EUs and multiple D2D devices. More explicitly, we developed the following contributions:

• A RSWW-OSSS is designed for a D2D-enabled cellular network in order to provide a better connective experience for EUs with the aid of cooperative communication. More explicitly, according to the proposed RSWW-OSSS, EUs are capable of achieving an increased rate with the aid of cooperation assistance provided by the D2D devices. Furthermore, in order to perform D2D transmission, the D2D devices intend to forward data for the EUs in exchange for accessing the part of cellular spectrum resource which is not used by the EUs.
• In order to improve the spectral efficiency, DTs employ the NOMA technology to simultaneously convey EUs’ and their own data at different transmit power. SIC is invoked at both the BS and D2D receivers (DRs) for separating the EUs’ and DTs’ data. As we know, NOMA technology is not introduced in overlay D2D cooperation communications in [19,20,21].
• Aiming for maximizing the utilities of both EU and D2D device, their behaviours are modelled as a two-stage Stackelberg game [31,32,33]. More explicitly, as the leader, EUs calculate the optimal time allocation coefficient in order to maximize its transmit rate. Furthermore, DTs are modelled as the followers which determine the optimal transmit powers for the cooperative transmission and D2D communication, in order to improve the achievable rate of D2D communication and simultaneously reduce the energy consumed for forwarding EUs’ data. The backward induction [34] is invoked for analysing the game and deriving the optimal strategies of EUs and DTs.
• For the sake of achieving a stable matching between multiple EUs and multiple D2D devices, we analyse the algorithmic stability of the proposed RSWW-OSSS with the aid of matching theory [35].

The rest of this paper is organized as follows. The system model is introduced in Section 2. Section 3 describes the proposed RSWW-OSSS. In Section 4, the attainable performance of our scheme is evaluated. Finally, we conclude in Section 5.

2. System Model

2.1. Construction and Assumption

We consider a D2D-enabled cellular network hosting M EUs in the set $m\in \{1,2,3,\dots ,M\}$ and N D2D transmission pairs in the set $(D{T}_n,D{R}_n)\in \{(D{T}_1,D{R}_1),(D{T}_2,D{R}_2),(D{T}_3,D{R}_3),\dots ,(D{T}_N,D{R}_N)\}$ as well as one BS that assigns the spectrum allocation to the cellular users, as seen in Figure 1. The BS assigns the spectrum band into $C=M$ orthogonal channel with equal bandwidth and each EU is preassigned an orthogonal channel for its data transmission. Hence, the interference among EUs is avoided [36,37]. Based on the considered D2D-enabledd cellular network, the DTs are capable of bypassing the BS and directly communicating with the proximity device, namely DRs. The overlay spectrum-sharing scheme is employed for the sake of sharing the spectrum between D2D links and cellular links. More explicitly, the EU with unfavorable channel condition may share part of its spectral resources to a D2D transmission pair in exchange for cooperative transmission assistance. A D2D transmission pair $(D{T}_n,D{R}_n)$ intends to provide cooperative assistance for a EU $E{U}_m$ in order to convey its own tele-traffic by using the cellular spectrum assigned to the $E{U}_m$. The EUs cooperative with D2D pairs in overlay mode which cellular link and D2D link occupy different cellular spectrum [38]. Hence, the interference of EUs and D2D pairs is avoided as well.

Both the EUs and DTs are assumed to be limited by the same maximum transmit power $P_{MAX}$. The channel gain is assumed to be modeled by [39]:

$$\begin{array}{c}\hfill h_{ij}=L{H}^2{d}_{ij}^{-\eta }\end{array}$$

(1)

where L denotes the pathloss constant, while H represents the fast Rayleigh fading channel. Furthermore, $\eta$ represents pathloss exponent and $d_{ij}$ denotes the distance between node i and node j. The notations of the system are listed in

Table 1. Table of notation.

Notation	Physical Meaning
$P_m$	$E{U}_m$’s transmit power
$P_n$	$D{T}_n$’s transmit power
$P_{MAX}$	EUs’ and DTs’ maximum transmit power
$h_{mn}$	channel gain between $E{U}_m$ and $D{T}_n$
$h_{bm}$	channel gain between $E{U}_m$ and BS
$h_{nn}$	channel gain between $D{T}_n$ and $D{R}_n$
$h_{bn}$	channel gain between $D{T}_n$ and BS
$N_0$	power of AWGN

.

2.2. Communication Model

2.2.1. Non-Cooperative Communication Model

Based on the non-cooperative communication model, the $E{U}_m$ transmits the data directly to the BS. The attainable transmit rate $R_m^{dir}$ of uplink transmission between the $E{U}_m$ and the BS may be expressed as:

$$\begin{array}{c}\hfill R_m^{dir}={log}_2\left(1+\frac{h_{bm}{P}_m}{N_0}\right)\end{array}$$

(2)

where $P_m$ denotes the transmit power of the $E{U}_m$. Furthermore, $h_{bm}$ represents the channel gain of the uplink from the $E{U}_m$ to the BS and $N_0$ represents the power of Additive White Gaussian Noise (AWGN).

2.2.2. Cooperative Communication Model

When the $E{U}_m$ directly transmits its data to the BS, performance may suffer due to the poor link quality. Aiming for improving the connective experience of the $E{U}_m$, we design a RSWW-OSSS. Based on the proposed scheme, the original time period T allocated for the $E{U}_m$ may be divided into two time slots, namely $t_1={\tau }_{mn}$ and $t_2=1-{\tau }_{mn}$, as shown in Figure 2. Hence, based on the cooperative model, the communications may include two phases, as seen in Figure 2.

• PHASE 1: During the first time slot $t_1$, the $E{U}_m$ transmits data to $D{T}_n$ and BS. The achievable rate of the direct transmission from the $E{U}_m$ to the BS may be expressed as:

  $$\begin{array}{c}\hfill R_{E{U}_m\to BS}={\tau }_{mn}{log}_2\left(1+\frac{h_{bm}{P}_m}{N_0}\right)\end{array}$$

  (3)
  Furthermore, the transmit rate $R_{E{U}_m\to D{T}_n}$ achieved by the transmission from the $E{U}_m$ to the $D{T}_n$ may be expressed as:

  $$\begin{array}{c}\hfill R_{E{U}_m\to D{T}_n}={\tau }_{mn}{log}_2\left(1+\frac{h_{mn}{P}_m}{N_0}\right)\end{array}$$

  (4)

  where $h_{mn}$ represents the channel gain of the link from the $E{U}_m$ to the $D{T}_n$.
• PHASE 2: During the second time slot $t_2$, the $D{T}_n$ simultaneously transmits $E{U}_m$’s and itself data with the aid of NOMA technology. The $D{T}_n$ transmits its own data to $D{R}_n$ with the power of $P_{mn}^D$ where $P_{mn}^D={\alpha }_{mn}{P}_n$, while forwards the $E{U}_m$’s data with the power of $P_{mn}^B$ where $P_{mn}^B=\left(1-{\alpha }_{mn}\right)P_n$. Moreover, ${\alpha }_{mn}$ is power allocation coefficient. Equation (5) describes the data sent by the $D{T}_n$ when the NOMA technology is employed.

  $$\begin{array}{c}\hfill y=\sqrt[2]{P_{mn}^B}{S}_{mn}^1+\sqrt[2]{P_{mn}^D}{S}_{mn}^2\end{array}$$

  (5)

  where $S_{mn}^1$ is the $E{U}_m$’s data which has to be forwarded by the $D{T}_n$, while $S_{mn}^2$ denotes the $D{T}_n$’s data. The total power consumed by the $D{T}_n$ during the phase 2 is denoted by $P_n$, which is equal to $P_{MAX}$. The SIC technology is invoked at both the BS and the $D{R}_n$ for the sake of separating $E{U}_m$’s and $D{T}_n$’s data. During phase 2, the Signal to Interference Plus Noise Ratio (SINR) of the $E{U}_m$’s data received by the BS may be given by:

  $$\begin{array}{c}SINR=\frac{h_{bn}\left(1-{\alpha }_{mn}\right)P_n}{a{h}_{bn}{\alpha }_{mn}{P}_n+N_0}\hfill \end{array}$$

  (6)

  where $h_{bn}$ represents the channel gain of the link from the $D{T}_n$ to the BS. Moreover, a represents the relationship between the power $P_{mn}^D$ and $P_{mn}^B$. According to the NOMA and the SIC technologies, if $D{T}_n$ forwards $E{U}_m$’s data at a higher transmit power, namely $P_{mn}^B>P_{mn}^D$, a is equal to 1. Otherwise, the value of a is 0, as shown in Equation (7).

  $$a=\left\{\begin{array}{cc}\hfill 1& P_{mn}^B>P_{mn}^D\hfill \\ \hfill 0& P_{mn}^B\le P_{mn}^D\hfill \end{array}\right.$$

  (7)
  When the BS receives data from $D{T}_n$, SIC technology is exploited to separate the $E{U}_m$’s and $D{T}_n$’s data. Then, both the previous data from the $E{U}_m$ in phase 1 and the later data from the $D{T}_n$ in phase 2 are combined by the BS with the aid of Maximum Ratio Combining (MRC) [40,41]. The SINR of the merged data may be expressed as:

  $$\begin{array}{c}\hfill SIN{R}_{MRC}=\frac{h_{bn}\left(1-{\alpha }_{mn}\right)P_n}{a{h}_{bn}{\alpha }_{mn}{P}_n+N_0}+\frac{h_{bn}{P}_m}{N_0}\end{array}$$

  (8)
  After receiving the data from $D{T}_n$, $D{R}_n$ separates the $D{T}_n$’s and $E{U}_m$’s data with the aid of SIC technology. The SINR of the $D{T}_n$’s data may be expressed as:

  $$\begin{array}{c}SIN{R}_{D2D}=\frac{h_{nn}{\alpha }_{mn}{P}_n}{b{h}_{nn}\left(1-{\alpha }_{mn}\right)P_n+N_0}\hfill \end{array}$$

  (9)

  where $h_{nn}$ represents the channel gain of the D2D link. Coefficient b represents the relationship between the power $P_{mn}^D$ and $P_{mn}^B$, which may be given by:

  $$b=\left\{\begin{array}{cc}1\hfill & P_{mn}^B\le P_{mn}^D\hfill \\ 0\hfill & P_{mn}^B>P_{mn}^D\hfill \end{array}\right.$$

  (10)

3. Joint Cooperative Spectrum-Sharing Scheme

3.1. Utility of $E{U}_m$

During phase 1, $E{U}_m$ transmits its data to $D{T}_n$ and BS. After successfully receiving $E{U}_m$’s data, $D{T}_n$ conveys both the $E{U}_m$’s data and its own data to BS and $D{R}_n$, with the aid of the NOMA technology during phase 2. Provided that the BS successfully receive the data from $D{T}_n$, SIC is invoked at the BS for separating the $E{U}_m$’s data and $D{T}_n$’s data. Then the BS combines the direct transmission from $E{U}_m$ within phase 1 and the relayed transmission from the $D{T}_n$ during phase 2 with the aid of MRC. Hence, the utility of $E{U}_m$ may be expressed as:

$$\begin{array}{cc}\hfill U_{mn}^{EU}({\tau }_{mn},{\alpha }_{mn})& =R_{mn}^C=\left(1-{\tau }_{mn}\right){log}_2(1+SIN{R}_{MRC})\hfill \\ & =\left(1-{\tau }_{mn}\right){log}_2\left[1+\frac{h_{bn}\left(1-{\alpha }_{mn}\right)P_n}{a{h}_{bn}{\alpha }_{mn}{P}_n+N_0}+\frac{h_{bm}{P}_m}{N_0}\right]\hfill \end{array}$$

(11)

3.2. Utility of $D{T}_n$

During phase 2, $D{T}_n$ has to consume a part of the power, namely $P_{mn}^B=(1-{\alpha }_{mn})P_n$, to forward the $E{U}_m$’s data, in exchange of the opportunity to access the cellular spectrum for D2D communication. Hence, the utility of the $D{T}_n$ may be expressed as:

$$\begin{array}{cc}\hfill U_{mn}^{D2D}({\tau }_{mn},{\alpha }_{mn})=& R_{mn}^{D2D}-\frac{P_{mn}^B}{P_n}\hfill \\ \hfill =& \left(1-{\tau }_{mn}\right)\left[{log}_2(1+SIN{R}_{D2D})\right]-\frac{P_{mn}^B}{P_n}\hfill \\ \hfill =& \left(1-{\tau }_{mn}\right)\left\{{log}_2\left[1+\frac{h_{nn}{\alpha }_{mn}{P}_n}{b{h}_{nn}(1-{\alpha }_{mn})P_n+N_0}\right]\right\}-(1-{\alpha }_{mn})\hfill \end{array}$$

(12)

where $R_{mn}^{D2D}$ is the achievable transmit rate of the D2D link by accessing the the cellular spectrum and $\frac{P_{mn}^B}{P_n}$ is the loss of $D{T}_n$.

3.3. Analysis of the Proposed Two-Stage Stackelberg Game

According to the considered system model, the interaction between the $E{U}_m$ and the $D{T}_n$ may be modeled as a two-stage Stackelberg game where one is the leader and the other is the follower. $E{U}_m$ owns the authorised spectrum and shares the the authorised spectrum with $D{T}_n$ in the propose RSWW-OSSS. Moreover, in economic matters, the leader of Stackelberg game is the one who decides the strategy firstly and the follower decides the strategy based on the leader’s strategy. Hence, $E{U}_m$ is the leader, and the $D{T}_n$ is the follower [42,43]. More explicitly, during Stage $I$, the $E{U}_m$ first calculates the optimal time allocation coefficient ${\tau }_{mn}^{\ast }$ for the sake of maximizing its utility. During stage $II$, as the follower of Stage $I$, $D{T}_n$ calculates the optimal power allocation coefficient ${\alpha }_{mn}^{\ast }$ for the sake of maximizing its utility. According to the proposed two-stage Stackelberg game, the strategies in different stages have sequential dependence. Hence, the backward induction is introduced to analyse the game and to derive the solution, namely Stackelberg equilibrium. Based on the backward induction [34], we commence from the $D{T}_n$’s strategies in Stage $II$, given the $E{U}_m$’s time allocation strategy. Then, based on the analysis of $D{T}_n$’s optimal power allocation coefficient, namely ${\alpha }_{mn}^{\ast }$, the optimal time allocation of $E{U}_m$ in Stage $I$, namely ${\tau }_{mn}^{\ast }$ is derived.

3.3.1. $D{T}_n$’s Strategy in Stage $II$

Given the time allocation coefficient ${\tau }_{mn}^{\ast }$ offered by the $E{U}_m$ in Stage $I$, $D{T}_n$ has to calculate the optimal power allocation coefficient ${\alpha }_{mn}^{\ast }$ for the sake of maximizing its utility. Hence, the optimal power allocation coefficient ${\alpha }_{mn}^{\ast }$ may be expressed as:

$$\begin{array}{c}{\alpha }_{mn}^{\ast }=arg\underset{{\alpha }_{mn}}{max}{U}_{mn}^{D2D}({\alpha }_{mn},{\tau }_{mn}^{\ast })\hfill \end{array}$$

(13)

$$\begin{array}{c}s.t.R_{mn}^C\ge \xi R_m^{dir}\hfill \end{array}$$

(14)

$$\begin{array}{c}0<{\tau }_{mn}^{\ast }<1\hfill \end{array}$$

(15)

$$\begin{array}{c}0<{\alpha }_{mn}<1\hfill \end{array}$$

(16)

$$\begin{array}{c}\forall m\in \{1,\dots ,M\},\forall n\in \{1,\dots ,N\}\hfill \end{array}$$

(17)

The first constraint guarantees that $E{U}_m$ is willing to cooperate. The $E{U}_m$ has a transmit rate requirement of $R_{req}$, which the $D{T}_n$ should help achieve. Hence, transmission rate $R_{mn}^C$ achieved by the cooperative communication has to be higher than the $E{U}_m$’s target rate, namely $R_{mn}^C\ge \xi R_m^{dir}$. $\xi$ is the rate requirement coefficient.

Based on Equation (12), the second derivative of $U_{DT}({\alpha }_{mn},{\tau }_{mn})$ is smaller than zero, namely $\frac{d^2{U}_{D2D}}{d{{\alpha }_{mn}}^2}<0$. Hence, $U_{D2D}({\alpha }_{mn},{\tau }_{mn})$ is a convex function of ${\alpha }_{mn}$. Therefore, the optimal power allocation coefficient of ${\alpha }_{mn}^{\ast }$ may be calculated by solving the KKT conditions [44], as shown in Equation (18).

$$\begin{array}{cc}\hfill {\alpha }_{mn}^{\ast }=& arg\underset{{\alpha }_{mn}}{max}{U}_{mn}^{D2D}({\alpha }_{mn},{\tau }_{mn}^{\ast })\hfill \\ \hfill =& \frac{h_{bn}{P}_n-N_0\left[{\left(1+\frac{h_{bm}{P}_m}{N_0}\right)}^{\frac{\xi }{1-{\tau }_{mn}^{\ast }}}-1-\frac{h_{bm}{P}_m}{N_0}\right]}{h_{bn}{P}_n+a{h}_{bn}{P}_n\left[{\left(1+\frac{h_{bm}{P}_m}{N_0}\right)}^{\frac{\xi }{1-{\tau }_{mn}^{\ast }}}-1-\frac{h_{bm}{P}_m}{N_0}\right]}\hfill \end{array}$$

(18)

3.3.2. $E{U}_m$’s Strategy in Stage I

Given the $D{T}_n$’s optimal power allocation coefficient ${\alpha }_{mn}^{\ast }({\tau }_{mn}^{\ast })$ which is calculated in Stage $II$, the $E{U}_m$ derives the optimal time allocation coefficient for maximizing its utility. Based on the $D{T}_n$’s optimal power allocation coefficient ${\alpha }_{mn}^{\ast }({\tau }_{mn})$ which is calculated in Stage $II$, the $E{U}_m$’s optimal time allocation coefficient of ${\tau }_{mn}^{\ast }$ may be expressed as:

$$\begin{array}{c}{\tau }_{mn}^{\ast }=arg\underset{{\tau }_{mn}}{max}{U}_{EU}^{mn}({\tau }_{mn},{\alpha }^{\ast }({\tau }_{mn}^{\ast }))\hfill \end{array}$$

(19)

$$\begin{array}{c}s.t.U_{mn}^{D2D}\ge 0\hfill \end{array}$$

(20)

$$\begin{array}{c}0<{\tau }_{mn}<1\hfill \end{array}$$

(21)

$$\begin{array}{c}0<{\alpha }_{mn}^{\ast }<1\hfill \end{array}$$

(22)

$$\begin{array}{c}\forall m\in \{1,\dots ,M\},\forall n\in \{1,\dots ,N\}\hfill \end{array}$$

(23)

where $U_{mn}^{D2D}$ is the utility of $D{T}_n$ as shown in Equation (12). The first constraint guarantees that $D{T}_n$ is willing to cooperate. The utility of $D{T}_n$ is at least zero. $U_{mn}^{D2D}=0$ is the minimum requirements when $D{T}_n$ cooperatives with $E{U}_m$.

Proposition 1.

Given the $D{T}_n$’s optimal power allocation coefficient of ${\alpha }_{mn}^{\ast }({\tau }_{mn}^{\ast })$, $E{U}_m$’s optimal time allocation coefficient, namely ${\tau }_{mn}^{\ast }$ is the unique solution of the following equation:

$$\begin{array}{c}\hfill \left(1-{\tau }_{mn}^{\ast }\right){log}_2\left\{1+\frac{h_{nn}{\alpha }^{\ast }({\tau }_{mn}^{\ast })P_n}{b{h}_{nn}\left[1-{\alpha }^{\ast }({\tau }_{mn}^{\ast })\right]+N_0}\right\}-[1-{\alpha }^{\ast }({\tau }_{mn}^{\ast })]=0\end{array}$$

(24)

See proof in Appendix A.

Let us now prove the existence and the uniqueness of the Nash Equilibrium for the proposed two-stage Stackelberg game.

Definition 1.

A strategy profile $({\tau }_{mn}^{\ast },{\alpha }_{mn}^{\ast })$ is a Nash Equilibrium of the proposed two-stage Stackelberg game, if:

$$\begin{array}{c}\hfill U_{mn}^{EU}({\tau }_{mn}^{\ast },{\alpha }_{mn}^{\ast })\ge U_{mn}^{EU}({\tau }_{mn}^{\ast },{\alpha }_{mn}^{\ast }),\end{array}$$

(25)

$$\begin{array}{c}\hfill U_{mn}^{D2D}({\tau }_{mn}^{\ast },{\alpha }_{mn}^{\ast })\ge U_{mn}^{D2D}({\tau }_{mn}^{\ast },{\alpha }_{mn}^{\ast }).\end{array}$$

(26)

As above discussion, Equation (12) is convex. Hence, the optimal power allocation coefficient ${\alpha }^{\ast }$ exists and is unique. Then we have $U_{DT}(\tau ,{\alpha }^{\ast })\ge U_{DT}(\tau ,\alpha )$. Furthermore, based on the Proposition 1, we have $U_{EU}({\tau }^{\ast },{\alpha }^{\ast })\ge U_{EU}(\tau ,{\alpha }^{\ast })$. Therefore, according to the Definition 1, strategy profile (${\tau }^{\ast },{\alpha }^{\ast }$) is the Nash Equilibrium of the proposed two-stage Stackelberg game. The proposed two-stage Stackelberg game algorithm to determine the Stackelberg equilibrium is shown in

Table 2. The proposed two-stage Stackelberg game algorithm.

Require
Let $E{U}_m$ and $D{T}_n$ exchange their profile information.
${\mathit{DT}}_{\mathit{n}}$’s Strategy in Stage $\mathit{I}\mathit{I}$:
▹ Given the time allocation coefficient ${\tau }_{mn}^{\ast }$ offered by the $E{U}_m$ in Stage $I$.
▹$D{T}_n$ calculates the optimal power allocation coefficient ${\alpha }_{mn}^{\ast }$ according Equation (18)
${\mathit{EU}}_{\mathit{m}}$’s strategy in Stage $\mathit{I}$:
▹ Given the optimal power allocation coefficient ${\alpha }_{mn}^{\ast }({\tau }_{mn}^{\ast })$ which is shown in Equation (18)
▹$E{U}_m$ calculates the optimal power allocation coefficient ${\tau }_{mn}^{\ast }$ which is the solution of Equation (24)
(${\mathbf{\tau }}_{\mathit{mn}}^{\mathbf{\ast }}\mathbf{,}{\mathbf{\alpha }}_{\mathit{mn}}^{\mathbf{\ast }}$) is the Nash Equilibrium of the proposed two-stage Stackelberg game.

.

3.4. Matching Game for EUs and D2D Pairs

As seen in Figure 1, there are multiple EUs and multiple D2D transmission pairs in the considered system model. In order to maximizing the utility given by Equation (11), $E{U}_m$ intends to share its spectrum resource with a best D2D transmission pair $(D{T}_n,D{R}_n)$. On the other hand, a D2D transmission pair $(D{T}_n,D{R}_n)$ intends to provide cooperative assistance for an appropriate EU in order to win a transmission opportunity within the licensed band for its own traffic, while simultaneously maximizing its utility formulated by Equation (12). Hence, based on the matching theory, both the EUs and the DTs in our system are considered as a pair of disjoint sets. Each EU intends to be matched with an appropriate DT who can provide a higher transmit rate. On the other hand, a DT is matched to an appropriate EU for the sake of pursuing higher utility. Hence, the spectrum-sharing problem can be formulated as a matching game, which is capable of producing a cooperative matching between an EU and a D2D transmission pair. Based on matching theory, we define a cooperative matching as follow:

Definition 2.

A one-to-one matching between M and N can be represented by a one-to-one correspondence $\mathsf{\Omega }(\ast )$, where $\mathsf{\Omega }(m)=n$ if and only if $\mathsf{\Omega }(n)=m$ and $\mathsf{\Omega }(m)\in N,\mathsf{\Omega }(n)\in M$, for $\forall m\in M,\forall n\in N$.

According to Definition 2, in the two-sided matching market, each EU selects one D2D pair namely $\mathsf{\Omega }(m)=n$, while each D2D pair selects one EU, namely $\mathsf{\Omega }(n)=m$ for $\forall m\in M,\forall n\in N$. Based on the Definition 2, a matching algorithm is developed for the sake of producing cooperative matchings between multiple EUs and multiple D2D pairs in the considered system, as shown in

Table 3. The proposed matching algorithm.

Initialization:
Let $L_m^{EU}$ and $L_n^{D2D}$ be the preference list of $E{U}_m$ and $D{T}_n$ respectively.
Repeat:
for all $\forall m\in \{1,\dots ,M\}E{U}_m$ do
if $E{U}_m$ is not matched
▹$E{U}_m$ broadcasts proposal $\varphi (m,n)$ to favourable $D{T}_n$ according to $L_m^{EU}$.
Set $i=0$, $V_m=0$.
while $V_m=0$ and $i\le n$ for $0<n\le N$
▹$i=i+1,n=m_i$.
if $E{U}_m$ in the preference list $L_n^{D2D}$ of $D{T}_n$
if $D{T}_n$ is not matched
▹$D{T}_n$ accepts the proposal $\varphi (m,n)$, set $V_m=1$.
if $D{T}_n$ is matched with $E{U}_{\widehat{m}}$
if ${\alpha }_{\widehat{m}n}>{\alpha }_{mn}$
▹$D{T}_n$ rejects the former $E{U}_{\widehat{m}}$ and accepts the proposal $\varphi (m,n)$ of $E{U}_m$, set $V_m=1$.
else
▹ set $V_m=0$.
for all $\forall n\in \{1,\dots ,N\}$ $D{T}_n$ do
if $D{T}_n$ achieves more than one proposal
▹$D{T}_n$ selects the one who requires lower transmit power.
for all $\forall m\in \{1,\dots ,M\}E{U}_m$ do
if $E{U}_m$ is already matched with $D{T}_{\widehat{n}}$
if $D{T}_{\widehat{n}}$ divorces matched pair $\varphi (m,\widehat{n})$
▹$E{U}_m$ is not matched.
Until: Each EU are either assigned to one of the D2D pairs or rejected by all of D2D pairs in $L_m^{EU}$.

.

Before any offer is made to the D2D transmission pairs, each EU first calculates the optimal time allocation coefficients $\{{\tau }_{m1}^{\ast },{\tau }_{m2}^{\ast },\dots ,{\tau }_{mn}^{\ast }\}$ for $0<m\le M$ and $0<n\le N$, by playing the proposed two-stage Stackelberg Game with different D2D transmission pairs. Then, each EU constructs a preference list of potential D2D transmission pairs, which can satisfy the EU’s rate requirement, namely $R_{mn}^C\ge \xi R_m^{dir}$. More explicitly, the $E{U}_m$ constructs a preference list $L^{E{U}_m}$, which orders the potential D2D pairs according to the optimal time allocation coefficients. Thus, the preference list for $E{U}_m$ is given by: $L^{E{U}_m}={\tau }_{m1}<{\tau }_{m2}<\dots <{\tau }_{mn}$ for $0<n\le N$. Based on the optimal power allocation coefficients $\{{\alpha }_{1n}^{\ast },{\alpha }_{2n}^{\ast },\dots ,{\alpha }_{mn}^{\ast }\}$ for $0<m\le M$ and $0<n\le N$, generated by the proposed two-stage Stackelberg Game in Section 3.3, $D{T}_n$ constructs a preference list of potential EUs which may share the licensed spectrum with it. More explicitly, the $D{T}_n$ constructs a preference list $L_n^{D2D}$, according to the transmit power consumed by forwarding data for different potential EUs. Hence, the preference list for $D{T}_n$ is given by: $L^{D2D_n}={\alpha }_{1n}<{\alpha }_{2n}<\dots <{\alpha }_{mn}$ for $0<m\le M$. According to the preference list, the $E{U}_m$ first offers the optimal time allocation coefficient ${\tau }_{mn}^{\ast }$ to the first D2D of its preference list, namely $D{T}_n$, which requires the shortest time slot for D2D transmission, as shown in Table 3. If $D{T}_n$ receives a proposal from $E{U}_m$ which is in its preference list, $D{T}_n$ and $E{U}_m$ form a cooperative matching, provided that $D{T}_n$ has not been matched. However, if the $D{T}_n$ is already matched with $E{U}_{\widehat{m}}$, the $D{T}_n$ may accept the proposal from $E{U}_m$, when lower transmit power is required for forwarding $E{U}_m$’s data. Based on the proposed RSWW-OSSS, each DT only has a single cooperative partner. Hence, $D{T}_n$ has to divorce its current cooperative pair $\mathsf{\Phi }(E{U}_{\widehat{m}},D{T}_n)$ and proceeds to form the new pair $\mathsf{\Phi }(E{U}_m,D{T}_n)$. If the $D{T}_n$ rejects the proposal from $E{U}_m$, the $E{U}_m$ sends the optimal time allocation coefficient ${\tau }_{mi}^{\ast }$ to the next favourable cooperative partner, namely $D{T}_i$ in the preference list. Then $E{U}_m$ repeats the above discovery procedure either until it finds an appropriate partner or is rejected by all potential D2D transmission pairs.

The complexity of the proposed matching algorithm is $O(MN)$ when each EU was rejected by $N-1$ D2D pairs of its preference list and cooperates with the final D2D pair of its preference list. However, when each EU receives the cooperation from the first D2D of its preference list, the complexity of the proposed matching algorithm is $O(M)$. For example, $M=20$ and $N=30$. The worst case performance is $O(600)$ and the best case performance is $O(20)$.

A common and realistic assumption in a cooperative communication network is that both the EUs and D2D pairs are selfish. It implies that both EUs and D2D pairs intend to maximize their own utilities when they are seeking for an appropriate cooperative partner. Hence, bearing in mind the selfish behaviour of the EUs and D2D pairs, it is necessary to analyse the stability of the proposed matching algorithm. Let us first introduce the definition of ’stable matching’. Let $m{\succ }_n\widehat{m}$ denote $D{T}_n$ prefers $E{U}_m$ to $E{U}_{\widehat{m}}$.

Definition 3.

A stable matching is not blocked by the $E{U}_m$ and $D2D_n$ pair $\mathsf{\Phi }(m,n)$, such that $n{\succ }_m\mathsf{\Omega }(m),m{\succ }_n\mathsf{\Omega }(n)$ and $\mathsf{\Omega }(m)\in N,\mathsf{\Omega }(n)\in M$, for $\forall m\in M,\forall n\in N$.

See proof in Appendix B.

Definition 3 illustrates that the specific EU and D2D pair, which constitute a cooperative pair according to our RSWW-OSSS can not simultaneously change their strategies for pursing higher utilities, if they select another D2D pair or EU as their cooperative partner.

4. Simulation Results

Simulation Configuration

In order to evaluate the achievable performance of the proposed RSWW-OSSS, we consider a cellular network with radius 500 m where a BS is located in the center of the cell. Furthermore, EUs are randomly located on the edge of the cellular network. D2D transmission pairs are randomly distributed across the entire network. The maximum power of both the each EU and each D2D pairs are assumed to be 0.1 W [13]. Rate requirement coefficient $\xi$ is set to be $2.5$. The simulation parameters are summarized in

Table 4. Simulation Configure Parameters.

Parameters	Value
$Cellradius$	500 m
$Noisepower(N_0)$	−114 dBm
$D2Dlinkdistance$	50 m
$EachEUTxpower$	100 mW
$EachD2DpairTxpower$	100 mW
$Pathlossconstant(L)$	${10}^{-2}$
$Pathlossexponent(\eta )$	4
$Raterequirementcoefficient(\xi )$	2.5
$NumberofEUs(M)$	20

[13,39].

We consider the pricing and spectrum leasing scheme (PSLS) which was proposed in [13] as the cooperative benchmarker of our scheme. The D2D transmission pair in the PSLS shares the spectrum resources with the cellular users by using the overlay approach. More explicitly, based on the PSLS, the DT provides cooperative assistance for the EU in order to obtain a D2D transmission opportunity and to improve the QoS of the uplink transmission from the EU to the BS. After successfully forwarding EU’s data, the DT in the PSLS conveys its own data during the successive sub-timeslot [13]. It is noted that power allocation was not considered in the PSLS [13].

Figure 3 compares the attainable rates of both the uplink transmissions from EUs to BS and the D2D transmissions versus the different number of D2D transmission pairs. Given the number of D2D transmission pairs, the proposed RSWW-OSSS is capable of achieving higher rates for both the EUs and the D2D transmission pairs than the PSLS which was proposed in [13], as seen in Figure 3. More explicit, when the number of D2D pairs is 20, the achievable rate of EUs in our RSWW–OSSS is about 12% more than that in PSLS and the achievable rate of D2D pairs in our RSWW–OSSS is about 28% more than that in PSLS. When the network hosts more D2D transmission pairs, the attainable rates of EUs are significantly improved by our RSWW-OSSS, due to the increased number of potential D2D transmission pairs which can provide cooperative assistance for the EUs, as seen in Figure 3. When the number of D2D transmission pair is increased from 4 to 20, more D2D transmission pairs may share the cellular spectrum for D2D communications. Therefore, the achievable rate of D2D communications is increased when the D2D network becomes larger, as shown in Figure 3. However, when the number of D2D transmission pairs is more than 30, the achievable rates of D2D communications almost reach the saturation point, because all the EUs are matched with appropriate D2D transmission pairs.

Figure 4 illustrates the transmit power consumed by the DTs for forwarding EUs’ data versus the different number of D2D transmission pairs. As shown in Figure 4, compared to the PSLS, our RSWW-OSSS is capable of decreasing the power for forwarding EU’s data with the aid of the proposed two-stage Stackelberg game and the proposed matching algorithm. When the D2D network becomes larger, cooperative matchings may be generated due to the increased number of D2D transmission pairs which can be the potential cooperative partner of EUs. Hence, the transmit power for forwarding EUs’ data is increased when the network hosts more D2D transmission pairs. However, when the number of D2D transmission pairs is more than 30, all the EUs may be matched with an appropriate D2D transmission pair. Hence, the number of cooperative matchings may not increase when more than 30 D2D transmission pairs intend to share cellular spectrum with the EUs. It implies that the transmit power for forwarding EUs’ data may be saturated, when the number of D2D transmission pairs is more than 30, as seen in Figure 4. When given the number of D2D pairs is 37, the saturated transmit power for forwarding EUs’ data in our RSWW–OSSS is about 2.5 W while the saturated transmit power for forwarding EUs’ data in the PSLS is about 3.4 W. The saturated transmit power for forwarding EUs’ data in our RSWW–OSSS is $26\%$ less than that in PSLS.

In order to evaluate the performance of the proposed RSWW-OSSS, we consider a stable matching with the fixed time allocation (SM-FTA) as the cooperative benchmarker of our RSWW-OSSS. Based on the SM-FTA, an EU shares a fixed portion of the whole time slot to its cooperative partner. Additionally, a stable matching with the fixed power allocation (SM-FPA) is introduced as another cooperative benchmarker of our RSWW-OSSS. Based on the SM-FPA, the transmit power which is allocated by a DT for forwarding EU’s data is fixed. Furthermore, we also introduce a random-matching-based overlay spectrum-sharing (RM-OSS) scheme, where both the EUs and the DTs calculate their optimal strategies by playing the proposed two-stage Stackelberg Game. However, according to the RM-OSS scheme, an EU is randomly matched with a D2D transmission pair which is capable of satisfying its transmit rate requirement.

Figure 5 shows our comparison between the total rates of EUs achieved by our RSWW-OSSS, the SM-FTA, the SM-FPA as well as the RM-OSS versus different numbers of D2D transmission pairs. Furthermore, Figure 6 compares the total rates of D2D transmissions achieved by our RSWW-OSSS, the SM-FTA, the SM-FPA as well as the RM-OSS versus different numbers of D2D transmission pairs. Compared to the benchmarker schemes, the proposed RSWW-OSSS is capable of significantly improving the attainable rates for both the D2D communications and the uplink transmissions from EUs to BS, as shown in Figure 5 and Figure 6. More explicit, when the number of D2D pairs is 20, the achievable rate of EUs in the proposed RSWW–OSSS is 62 b/s/Hz which is about $100\%$ more than that in RM-OSS, $48\%$ more than that in SM-FPA and $22\%$ more than that in SM-FTA, while, the achievable rate of D2D pairs in our RSWW–OSSS is 91 /b/s/Hz which is about $111\%$ more than that in RM-OSS, $52\%$ more than that in SM-FPA and $34\%$ more than that in SM-FTA. Given the number of D2D pairs, the RM-OSS scheme achieves the lowest transmit rate, because that both the EUs and DTs randomly select their cooperative partner without considering the other potential cooperative partners, as seen in Figure 5 and Figure 6. According to the total transmit rate of the uplink transmissions and that of the D2D communications shown in Figure 5 and Figure 6, the SM-FTA outperforms the SM-FPA. Based on the SM-FPA, less DT can afford the total power for successfully forwarding EU’s data and for conveying its own data, due to the fixed power allocation and maximum power constraint. Hence, compared to the SM-FTA, the SM-FPA reduces the number of cooperative matching. Observe in Figure 5 and Figure 6 that the total transmit rate of uplink transmissions and that of D2D communications are increased when the network hosts more D2D transmission pairs, due to the increased number of EUs’ potential cooperative partners.

5. Conclusions

This paper proposed a RSWW-OSSS for supporting overlay D2D communication and cooperative uplink transmission in a cellular network hosting multiple EUs and D2D transmission pairs. Based on the proposed RSWW-OSSS, the EUs intend to improve the QoS with the aid of cooperative assistance provided by the D2D transmission pairs. Moreover, the D2D transmission pairs are encouraged to forward data for the EUs, in exchange for accessing the cellular spectrum to convey their own data. In order to further improve the spectrum efficiency, NOMA technology is used by the D2D transmission pairs to simultaneously transmit the EU’s and its own data within the same spectrum and at different transmission power. In this paper, the behaviours of EUs and D2D pairs are modelled as a two-stage Stackelberg game. Furthermore, the backward induction is exploited to derive the optimal strategies for the sake of maximizing the utilities of both the EUs and the D2D transmission pairs. Moreover, a one-to-one stable matching algorithm is developed for the sake of generating the stable cooperative matchings between the EUs and the D2D transmission pairs. Finally, we analyse the stability of the matchings generated by the proposed matching algorithm. The simulation results show that the proposed RSWW-OSSS is capable of significantly providing $E{U}_m$ the transmission rate $41\%$ higher than the non-cooperative direct transmission rate $R_m^{dir}$. Moreover, the D2D transmissions can achieve 105 b/s/Hz by exclusively occupying EUs’ licensed spectrum resource. According to the simulation results, the proposed RSWW-OSSS performs greater than exiting methods as well.

There are two time slots while NOMA is used during the second time slot for the sake of improving the spectrum efficiency. In future work, NOMA will be exploited in both the first time slot and the second time slot. EUs decide their transmission power during the first time slot while D2D pairs decide their transmission power during the second time slot versus two equal time slots, namely $t_1=t_2=0.5$ T.