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Joint uplink and downlink resource allocation for device to device communications

2021; Institution of Engineering and Technology; Volume: 15; Issue: 11 Linguagem: Inglês

10.1049/cmu2.12164

1751-8636

Sawsan Selmi, Ridha Bouallègue,

Advanced Wireless Communication Technologies

IET CommunicationsVolume 15, Issue 11 p. 1493-1506 ORIGINAL RESEARCH PAPEROpen Access Joint uplink and downlink resource allocation for device to device communications Sawsan Selmi, Corresponding Author Sawsan Selmi Sawssan.selmi@gmail.com orcid.org/0000-0002-6723-5723 Innovation of Communication and Cooperative Mobiles (Innov'Com) Laboratory (SUP'COM LR11/TIC03), National School of Engineers of Tunis, Tunis el Manar University, Tunis, Tunisia Correspondence Sawsan Selmi, 30 Street of Pyramids, El Ghazela City, 2083 Ariana, Tunisia. Email: Sawssan.selmi@gmail.comSearch for more papers by this authorRidha Bouallègue, Ridha Bouallègue orcid.org/0000-0001-7916-761X Innovation of Communication and Cooperative Mobiles (Innov'Com) Laboratory (SUP'COM LR11/TIC03), Higher School of Communication of Tunis, Carthage University, Ariana, TunisiaSearch for more papers by this author Sawsan Selmi, Corresponding Author Sawsan Selmi Sawssan.selmi@gmail.com orcid.org/0000-0002-6723-5723 Innovation of Communication and Cooperative Mobiles (Innov'Com) Laboratory (SUP'COM LR11/TIC03), National School of Engineers of Tunis, Tunis el Manar University, Tunis, Tunisia Correspondence Sawsan Selmi, 30 Street of Pyramids, El Ghazela City, 2083 Ariana, Tunisia. Email: Sawssan.selmi@gmail.comSearch for more papers by this authorRidha Bouallègue, Ridha Bouallègue orcid.org/0000-0001-7916-761X Innovation of Communication and Cooperative Mobiles (Innov'Com) Laboratory (SUP'COM LR11/TIC03), Higher School of Communication of Tunis, Carthage University, Ariana, TunisiaSearch for more papers by this author First published: 24 March 2021 https://doi.org/10.1049/cmu2.12164AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract With the ever-increasing number of intelligent devices, extensive research efforts have been spent towards developing efficient resource allocation in underlying device to device communications. Centralised and distributed approaches have been proposed to improve the network performances through uplink and downlink schemes. However, the full-duplex resource allocation remains challenging. Here, a joint uplink–downlink resource block assignment for device to device communications is proposed. The device's geometric distribution via the Poisson point process is formulated and the optimization problem as a mixed strategy non-cooperative game is modelled. The proposed resource allocation scheme allows many device to device communication pairs to share the same cellular resource block but does not restrict their number in advance; it is determined dynamically according to the channel's capacity and interference condition. Inversely, a device to device communication transmitter can simultaneously be allocated many resource blocks according to its bandwidth requirement and the number of its antennas. Furthermore, the optimal amount of data and power transmitted by each device is calculated. The efficiency of this algorithm is evaluated through computer simulations, and the results demonstrate good performances in the spectral and energy efficiency with relatively low complexity and convergence time. 1 INTRODUCTION The expansion of smart devices number and content sharing between users results in a tremendous increase in the wireless data traffic and local service demands. According to [1], there will be more than 30 billion connected devices by the end of 2021, and this trend will keep on growing in the next decade. However, with the limited availability of the cellular spectrum and the slow progress in the network infrastructure development, the conventional cellular system became incapable of satisfying these increasing demands. Facing this situation, device to device (D2D) communication was integrated as one of the key characteristics of 5G mobile networks [2]. It has drawn significant attention for its potential to improve the system performance and the user experience [3]. Its benefits mainly stem from the device's proximity, the single-hop nature of the communication link, and the radio resources reuse [4]. A general session setup of a D2D communication includes the following steps [3]; first, a D2D user initiates a communication request. Then, the eNodeB checks if the receiver is in the same proximity. If some criteria are satisfied; as the distance, the interference situation, and the channel quality, the eNodeB triggers a direct D2D link between the transmitter and the receiver in overlay or underlay mode [2]. In the overlay mode, the communicating pairs use dedicated resource blocks (RBs) to route their data. Whereas underlay D2D mode, they dynamically reuse resources occupied by the primary cellular users [3]. The first mode easily splits the spectrum between cellular and D2D communications; however, it leads to resource waste if the reserved spectrum exceeds the D2D demands. The second enhances the spectrum reuse and the system capacity. Yet, it generates severe interference between co-channel communications if the resource allocation is processed randomly [4]. To tackle this issue, we focus in this paper on a set of D2D challenges and propose solutions to provide efficient and scalable underlay D2D communication. We try to answer a set of questions, basically how to synchronize and meet both the CUEs and DUEs throughput demands without relying on the eNodeB? How to manage the co-channel interferences in uplink and downlink channels? And what are the measurements to maximize the network spectral and energy efficiency? Extensive research efforts have been spent on solving these problems either in a centralized or distributed fashion. Centralized approaches such as [5-25] assume the channel state information (CSI) knowledge at the eNodeB. They rely on several tools as: stochastic geometry [5, 6], centralized graph-theoretic approaches [7, 8], and mixed-integer programming [9, 10] to enhance the network capacity. Paper [10] considers a centralized power control scheme to maximize the cellular and D2D channel quality. The authors analyse the non-convexity of the problem and propose an alternative sub-optimal solution. Ref. [11] investigates a novel scheme to maximize the sum-rate in one cellular and one D2D scenario. Whereas in [12, 13], they consider a partial CSI knowledge of the D2D links and suggest a solution to maximize the device's throughput. Ref. [14] proposes an interference control mechanism in uplink subcarriers. The D2D pairs broadcast their signal to interference plus noise ratio (SINR) through a control channel. If the received SINR at the eNodeB level is higher than the maximum threshold, it stops scheduling cellular users on the occupied resource blocks. Inversely, in refs. [15, 16], the cellular users measure their power levels and forward them to the eNodeB via a dedicated control channel. Consequently, it avoids allocating the same RBs to high-power D2D transmitters. Paper [17] proposes a downlink RA scheme to maximize the sum rate. The original concave problem is transformed into a convex formula and solved via approximations [18]. Paper [19] investigates a centralized spectrum and power control scheme. The optimization problem is also formulated as a concave formula and solved via non-linear programming [20]. Limited research investigated centralized full-duplex RA [21-23] to manage the co-tier/cross-tier interference. The authors of [21] propose a suboptimal approach, whereas, in paper [23], they use a heuristic suboptimal algorithm to solve the problem. Paper [24] presents a centralized two-step scheme to enhance the spectrum reuse [25]. However, a D2D pair can share only one cellular subcarrier either in uplink or downlink. Moreover, the spectral and energy efficiency were not analysed. The former centralized approaches [5-25] assume the knowledge of the CSI at the eNodeB. However, this information is not practical when the channels vary rapidly with time [10]. Besides, the CSI reporting requires additional resources, while a limited number is available for the network control [13]. It also increases the complexity and generates significant signalling overhead between the eNodeB and the connected devices [26, 27]. Hybrid methods proposed joint centralized-distributed RA [28-30] to address these issues. However, fully distributed approaches have gained more merit [31-58]. They generally perform a utility function that represents every device's preference [31]. Then develop distributed resource allocation algorithms via: pricing, auctions [32], cooperative [33], and non-cooperative games [34]. They combine the advantages of low information sharing, traffic offload, and low complexity [35, 37]. In this context, refs [36, 37] propose a two-stage energy-efficient scheme. The first deals with the uplink resource allocation, and the second controls the device's power. Finally, the problem is solved using graph theory. Paper [38] proposes a distributed power control scheme to maximize the uplink D2D SINR in a massive MIMO system. Paper [25] applies a joint mode selection, RA, and power control scheme to minimize the interference at the D2D level. Ref. [39] proposes a distributed game-theoretic scheme to solve the uplink RA problem in a multi-cell scenario. In [40], the uplink RB assignment between cellular and DUEs is performed based on a Q-learning solution. Paper [41] evaluates the resource allocation in a cognitive D2D network, whereas many other papers consider approximations such as stochastic geometry to manage the uplink spectral resources [42-44]. In our previous work [45-47], we also focused on distributed uplink RA to avoid the eNodeB downlink interference. In [45], we proposed an interference-aware algorithm based on a mixed-strategy game. This approach prioritizes the RBs occupied by quite far cellular users. In [46] and [47], we added a power control stage to investigate the optimal transmission power and RB maximizing the network EE. Other distributed algorithms propose downlink RA as [48-52]. In [48], the D2D pairs share the downlink RBs with cellular UEs located far from the eNodeB. In [49], the RA problem is modelled and solved via a distributed Stackelberg game. Paper [50] investigates a downlink RA scheme based on eNodeB power mitigation. Paper [51] considers only D2D power control to maximize the system EE, whereas refs. [52-54] yield the joint EE and SE maximization for D2D pairs. Refs. [55, 56] propose joint uplink/downlink bipartite matching to optimize the EE in a relay-assisted D2D scenario [44]. They investigate two-stage algorithms to reduce the complexity [25, 60]. The first stage manages the power consumption, and the second allocates the resources between the active D2D pairs. However, they ignored the QoS requirements of CUEs [57-59]. In [56], the authors proposed a maximum weight bipartite matching [25] to select the RB requiring the minimum power transmission. They suggest a distributed power-based scheme [58] to deal with the interference problem. However, they also focused on D2D performance maximization and ignored the primary cellular communications [59, 60]. In light of the related works, we investigate a joint uplink/downlink RA to maximize the total spectral and energy efficiency of the network. Our main contributions are summarised as follows: The proposed algorithm maximizes the spectrum reuse in both uplink and downlink subcarriers. It simultaneously addresses two main critical issues in cellular networks: the interference and the power consumption. It improves the cellular capacity and further enhances the network's energy efficiency. Given that centralized RA algorithms require additional data broadcast and generate significant signalling overhead, we perform a distributed algorithm based on a mixed strategy game. Many distributed RA approaches adopt pure strategy games in which a UE can select only one RB. Consequently, the devices must run the algorithm each time slot to be allocated a new RB. This process is taught to converge, especially when the devices number increases. The proposed RA algorithm allows many D2D pairs to share the same cellular RB but did not restrict their number in advance. It is set dynamically according to the channel's capacity and interference condition. Besides, a D2D can simultaneously be allocated many RBs according to its bandwidth requirement and antennas. Each timeslot, many devices can access the cell sequentially and play the game. The proposed approach calculates the optimal amount of data and power transmitted by each device. To the best of our knowledge, this is the first study that provides such results. Since the available bandwidth and the devices' transmission power are essential to satisfy the QoS requirements, the proposed algorithm determines the optimal RB, data and transmission power for each D2D pair based on the UEs density, locations and co-channel devices power. The proposed algorithm reduces the calculation and the complexity. Moreover, it guarantees a fast convergence to the mixed strategy Nash equilibrium (MSNE) due to the limited amount of information broadcast. Simulations results prove the efficiency of the proposed approach even with many devices number. Throughout the rest of the paper, we present in Section 2 our system model. In Section 3, we formulate the joint spectral efficiency (SE) and energy efficiency (EE) maximization problem through a mixed strategy non-cooperative game. Section 4 explains the proposed algorithm and solves the RA problem. Section 5 analyses the system performances via computer simulations and compares the obtaining results to the existing approaches. Finally, Section 6 concludes the paper and proposes some perspectives for future work. 2 SYSTEM MODEL 2.1 roposed system model Figure 1 represents the system model of the proposed approach. We consider the joint uplink/downlink scenario of a single cellular network; an eNodeB placed at the cell centre, and two device types, namely cellular UEs (CUEs) and D2D UEs (DUEs). Each CUE is allocated an orthogonal link and communicates with the base station via uplink/downlink channels, whereas DUEs communicate directly in pairs and reuse the same cellular RBs. Accordingly, the D2D receivers’ achievable data rate in the RB c is: R i c = W × l o g 2 ( 1 + S i c ) (1)Where S i c and W are the SINR of the ith D2D receiver at the RB c and the channel bandwidth. FIGURE 1Open in figure viewerPowerPoint System model of the proposed approach Let N be the total UEs number in the cell. N = {C, D}. C denotes the cellular users’ number and D represents the D2D pair's number. The spatial distributions of C and D follow an independent homogeneous Poisson point process (PPP) with intensities λ c and λ d respectively. 2.2 Uplink and downlink SINR distribution In a dense network, there is no dominant line of sight propagation. So Rayleigh fading is the most applicable. We assume the independent and identically distributed (iid) Rayleigh fading for all the network channels. Let g i B c be the channel gain between the eNodeB and the ith CUE, g j B d the fading coefficient between the eNodeB and the jth DUE, g i j c d the one between the ith CUE and the jth D2D pair, g j k d d the fading coefficient between the jth DUE and the kth DUE, and g j j d d the channel gain between the transmitter and the receiver of the j th D2D pair. Considering the large scale fading effects; the interference channel gain between the j th D2D pair and the ith cellular channel in the RB c can be modelled by: g i j c d = h i , j 2 d i j − α (2)where d i j is the distance between the D2D pair and the CUE, h i , j is a complex Gaussian channel coefficient satisfying h i , j ≈ C N ( 0 , 1 ) , α represents the channel path loss exponent and N0 is the average white Gaussian noise AWGN. We assume the symmetry between the downlink and the uplink channels. Accordingly, the communication channel can be modelled as: P r = P t g 2 d α (3)Where P t and P r are the transmitted and received power. For a CUE: the interferences occur only from the co-channel D2D pairs. For a D2D receiver: The uplink interferences came from the co-channel D2D communications and the primary CUE, and the downlink interference came from the co-channel D2D communications and the eNodeB. In the rest, we characterize the sensed interference at the i th CUE as I i , c c and that received by the j th D2D in the uplink and the downlink subcarriers as I j , d U , c and I j , d D , c respectively. A D2D pair can reuse downlink or uplink cellular channels. Therefore, we obtain the following interference model for the CUE and the D2D receiver. In the downlink mode: I i , c D , c = I i , c c = ∑ j = 1 D P d h i j d c 2 d i j − α (4) I j , d D , c = ∑ k = 1 , k ≠ j D P d k h j k d d 2 d j k − α + P b h j B d 2 d b j − α (5) In the uplink mode: I i , c U , c = I i , c c = ∑ j = 1 D P d h i j d c 2 d i j − α (6) I j , d U , c = ∑ k = 1 , k ≠ j D P d k h j k d d 2 d j k − α + P c h i j c d 2 d i j − α + N 0 (7) d i j represents the distance from the ith CUE to the jth D2D pair. d j k is the distance from the kth D2D pair to jth D2D pair. d b j represents the distance from the eNodeB to the jth D2D pair, d b i is the distance from the eNodeB to the ith D2D pair and d i i is the distance between the transmitter and the receiver of the same D2D pair. Accordingly, the downlink SINR of a cellular and a D2D receiver can be expressed by: δ c , i D L = P b h i B c 2 d b i − α ∑ j = 1 N P d h i j d c 2 d i j − α + N 0 (8) δ d , j D L = P d h i i d d 2 d j j − α ∑ k = 1 , k ≠ j D P d k h j k d d 2 d j k − α + P b h j B d 2 d b j − α (9) And the SINR in the uplink channel for a cellular and D2D receiver can be expressed by: δ c , i U L = P c h i B c 2 d b i − α ∑ j = 1 N P d h i j d c 2 d i j − α + N 0 (10) δ d , j U L = P d h j j d d 2 d j j − α ∑ k = 1 , k ≠ j D P d k h j k d d 2 d j k − α + P c h i j c d 2 d i j − α + N 0 (11) 2.3 Joint spectral and energy efficiency γ i , c 0 , 1 , γ j , c = 1 , i f t h e j t h l i n k a l l o c a t e s t h e R B c 0 , o t h e r w i s e Consequently, the joint uplink/downlink spectral efficiency SE of the ith CUE can be expressed by: S E c , i t o t = log 2 1 + δ c , i D L + log 2 1 + δ c , i U L (12) A cellular UE is allocated a RB, and a D2D link could be allocated many RBs. Accordingly, the jth D2D achievable data rate is the sum of the allocated RBs data rates and its joint uplink/downlink system capacity can be expressed as: S E d , j t o t = ∑ k = 1 C γ j , c l o g 2 1 + δ d , j D L + l o g 2 1 + δ d , j U L (13) The overall network system capacity is given by: S E s y s t o t = ∑ i = 1 C ∑ j = 1 D ( S E c , i t o t + S E d , j t o t ) (14) The network SE is maximized when all the transmitted signals arrive at the receiver with the maximum SINR. Accordingly, the system SE and EE optimization can be expressed respectively as: m a x δ c , i D L , δ c , i U L , δ d , j D L , δ d , j U L ∑ i = 1 C ∑ j = 1 D ( S E c , i t o t + S E d , j t o t ) (15) max ∑ j = 1 D ∑ i = 1 C ( C c , i t o t + C d , j t o t ) ∑ k = 1 C 1 η P j d k + 1 η P k (16) Subject to: δ c , i D L , δ c , i U L ≥ δ m i n C ; δ d , j D L , δ d , j U L ≥ δ m i n D 0 ≤ P c , i ≤ P c m a x ; 0 ≤ P b ≤ P b m a x ; 0 ≤ P d , j ≤ P d m a x P j d k is the power of the jth D2D transmitter in the kth cellular channel, P i is the transmission power of the jth CUE, δ m i n C and δ m i n D characterize the minimum CUE and D2D SINR requirements. P d m a x and P c m a x are the maximum transmission power of D2D and CUE, respectively. The SINR is constrained by a minimum threshold to guarantee the QoS. Besides, the transmission powers are kept under a maximum value to preserve the energy consumption and reduce the interferences. The objective function defined in Equation (16) is non-convex. It is a mixed-integer programming NP-hard complex. 3 MIXED STRATEGY GAME FORMULATION Game theory is proven as an efficient tool to model the strategic interaction among rational decision-makers. It addresses the games, in which each player's gains or losses depend on those of the other participants [45]. The optimal outcome of the game is Nash equilibrium. It describes a situation where no player can improve unilaterally its payoff when the others’ remain unchanged [46]. Pure strategy games have been extensively used to analyse interactive and conflict decisions in cellular networks. However, the convergence is not always guaranteed, mostly when the player's number increases. Yet, a mixed strategy game can appropriately fit the requirements of the studied problem. 3.1 ame theory model We model the proposed problem via a mixed strategy non-cooperative game. Each UE is a rational player. It acts independently from others and aims to maximize its payoff. The strategy sets of the ith CUE and C∖ { i } cellular UEs are denoted s i c and s − i c respectively. The jth D2D pair's strategy s j d depends on all the strategies taken by the other UEs s i c , s − i c , and s − j d . The utility function U j of the jth player is a mathematical representation of its preferences. It assigns a value (payoff) to each alternative. If the player prefers the strategy S1 than S2; its payoff applied to S1 is higher than this applied to S2. We define the utility function of the ith CUE as: U c i c s i c , s − i c , s j d , s − j d = S E c , i t o t = l o g 2 ( 1 + δ c , i D L ) + l o g 2 1 + δ c , i U L (17) And, the utility function of the jth D2D pair as: U c j d s j d , s − j d , s i c , s − i c = S E d , j t o t = ∑ k = 1 C γ i , c l o g 2 1 + δ d , j D L + l o g 2 1 + δ d , j U L (18) Similarly, the ith device power mitigation is expressed through the utility function u p o w i as: u p o w i s j d , s − j d , s i c , s − i c = − ∑ k = 1 C 1 η P j d k + 1 η P k (19) 3.2 Mixed strategy payoff The mixed strategy δ i of the ith player, is the probability distribution over the pure strategies space S i . δ i σ i ∈ R m i ∑ j = 1 m i σ j i = 1 (20) where σ j i is the probability assigned to the pure strategy S j i . δ = ∏ i ∈ N δ i is the strategy space of the game that is, when a UE plays a mixed strategy σ, the probability that the pure strategies combinations ( S j 1 1 , S j 2 2 , … , S j n n ) occurs is: σ ( s ) = ∏ i ∈ N σ j i . In pure strategy games, a player makes only one choice without involving chance or probability. In a mixed strategy game, all the players’ payoffs become random variables, and the mixed strategy payoff of the ith player u i ( σ ) is defined as: u i ( σ ) = ∑ s ∈ S u i ( S ) σ ( S ) (21)Where u i ( s ) is the i th player payoff in the pure strategy space S{N}. Given that a UEs’ payoff depends not only on its own strategy, but also on the strategies taken by others UEs in S∖{i}, the mixed strategies σ is a combination of σ i and σ − i . 3.3 ixed strategy Nash equilibrium Definition 1: A mixed strategy profile σ* is called Nash equilibrium (MSNE) of the game Γ if: u i σ ∗ ≥ u i ( σ − i ∗ σ i ) i ∈ N σ i ∈ δ i (22) The MSNE is reached when no player in the game can maximize its payoff by unilaterally changing his strategy when the other players keep theirs' unchanged. Theorem (Nash 1950): Every finite n-player game in strategic form has a mixed strategy Nash equilibrium. According to Nash theorem, there is at least a mixed strategy equilibrium for any game in a strategic form. The MSNE s* is mathematically represented by the combination of the optimal strategies for all the players in the game: s ∗ = s j d ∗ , s − j d ∗ , s i c ∗ , s − i c ∗ (23)With U i , E E s j d ∗ , s − j d ∗ , s i c ∗ , s − i c ∗ ≥ U i , E E s j d , s − j d ∗ , s i c ∗ , s − i c ∗ (24) ∀ i , j ∈ N , ∀ s i ∈ S i The MSNE represents the optimal resource blocks and power allocation of all the devices in the cell. 4 PROPOSED MSJUD APPROACH We consider a cellular network with C cellular and D D2D pairs sharing the same bandwidth. During the pair discovery step, the eNodeB broadcasts the players’ coordinates. Accordingly, a DUE can calculate the interference sensed by each cellular link and co-channel D2D pair, and can easily derive the SINR combinations from the distances and the random power distributions. This information serves to develop the strategic form of the game. 4.1 Mathematical formulation The strategic form of the proposed n-person non-cooperative game is represented by: Γ = ( N , { s i } i ∈ N , { u i } i ∈ N ) . Where N denotes the player's number, s i is the pure strategies space and u i is the utility function of the ith player. The number of possible pure strategies in the game is P = C × D and the space of the pure strategies combinations is: P s c = ∏ i ∈ D s i ( 2 C ) D (25) All the possible pure strategies combination can be grouped in the M P s c matrix as follows: s 1 1 , s 1 2 , … s 1 D − 1 , s 1 D ⋯ line 1 ⋮ ⋮ s p 1 , s p 2 , … , s p D − 1 , s p D … line P s c = C D For instance, suppose a cell with four D2D pairs and three CUEs. D = 4 and s 1 = s2 = s3 = s 4 = 3. So, the number of the pure strategies is C × D = 12 and the number of the pure strategies combinations is 2 C D = 81 . The pure strategy combination matrix M P s c is P s c × D , so the ith player can calculate its payoff according to each combination. M U s c characterises the payoff matrix of all the players in the game, it is a ( P s c × D ) matrix, and u j i represents the payoff of the ith player at the jth strategy considering the other players’ strategies. By the same reasoning, we investigate the MSNERA and MSNEPow. The first is a (C × D) matrix. It represents the optimal RB allocation of all the devices in the game, and the second provides their optimal transmission power. P = s 1 1 s 1 2 s 1 3 s 1 4 s 1 1 s 1 2 s 1 3 s 2 4 s 1 1 s 1 2 s 1 3 s 3 4 s 1 1 s 1 2 s 2 3 s 1 4 s 1 2 s 1 2 s 2 3 s 2 4 s 1 2 s 1 2 s 2 3 s 3 4 s 1 2 s 2 2 s 1 3 s 1 4 s 1 2 s 2 2 s 2 3 s 2 4 s 1 3 s 2 2 s 3 3 s 3 4 s 1 3 s 2 3 s 3 3 s 1 4 s 1 3 s 2 3 s 3 3 s 2 4 s 1 3 s 2 3 s 3 3 s 3 4 M U s c = u 1 1 u 1 2 ⋯ u 1 D u 2 1 u 2 2 ⋯ u 2 D ⋮ ⋱ ⋱ ⋮ u P s c − 1 1 u P s c − 1 2 ⋯ u P s c − 1 D u P s c 1 u P s c 2 ⋯ u P s c D M P s c = s 1 1 s 1 2 ⋯ s 1 D s 2 1 s 2 2 ⋯ s 2 D ⋮ ⋱ ⋱ ⋮ s P s c − 1 1 u P s c − 1 2 ⋯ s P s c − 1 D s P s c 1 u P s c 2 ⋯ u P s c D The ith column of the final MSNE matrix represents the optimal RB allocation σ i of the ith D2D pair. It represents the optimal probability assigned to the jth pure strategy played by the ith D2D. The sum of the optimal mixed strategies associated to a players’ pure strategy combination: ∑ k = 1 P σ k i = 1 . Let u m a x i ( σ i , σ − i ) be the optimal payoff of the ith D2D pair when fixing other players strategies at σ − i . Theorem: A necessary and sufficient condition for σ to be a Nash equilibrium of the game Γ is to be an optimal solution of the following minimization problem: m i n ∑ i ∈ D u m a x i σ i , σ − i − u