Open access peer-reviewed chapter

Nash Equilibrium Study for Distributed Mode Selection and Power Control in D2D Communications

By Sameh Najeh and Ammar Bouallegue

Submitted: October 1st 2019Reviewed: April 2nd 2020Published: September 15th 2020

DOI: 10.5772/intechopen.92362

Downloaded: 135

Abstract

One of the main challenges of LTE-advanced (LTE-A) is to recover the local-area services and improve spectrum effciency. In order to reach those goals technical capabilities are required. D2D is a promising techniques for the 5G wireless com- munications system using several applications, as: network traffic offoading, public safety, social services and applications such as gaming and military applications. In this chapter, we investigate both mode selection and distributed power control in D2D system. Indeed, the mode selection is provided while respecting a predeter- mined SINR threshold relative to cellular and D2D users. The amount of minimum and maximum power are then derived to fulffill the predetermined requirements, by limiting the interference created by underlaid D2D users. In order to realize our proposed power control step, a new distributed control approach is proposed using game theory tools for several cellular and D2D users. This distributed approach is based on the mode selection strategy already proposed in the previous step. Finally, simulations were established in order to compare the proposed distributed algo- rithm in terms of coverage probability which is based on game theory, with other conventional centralized algorithms.

Keywords

  • mode selection
  • power control
  • distributed
  • Nash equilibrium

1. Introduction

The Internet of Things (IoT) is a developing and promising innovation, which were able to revolutionize the world [1]. IoT manages low-powered gadgets, using the internet by interacting with one another. IoT interconnect “Things” and also helps in machine-to-machine (M2M) communication, which is a way of data communication between varied gadgets without human intercession [2].

IoT applications can be classified into six main categories, such as [1]: smart cities, smart business, smart homes, healthcare, security and surveillance. Regarding these different applications, several requirements should be maintained, like [2]: (1) high scalability, (2) security and privacy, (3) high capacity, (4) security and privacy, (5) energy saving, (6) reduced latency, (7) quality of service (QoS), (8) built-in redundancy, (9) heterogeneity and (10) efficient network and spectrum.

The 5G enabled IoT contains a number of key communication techniques for IoT applications, in order to make the network with faster speeds and greater accessibility. A network that reaches all over the world and is accessible to all. Since 5G technology offers greater connectivity, more and more applications for this technology are likely to come to the field. Four main categories can be cited in the this context: (1) Wireless Network Function Virtualization, (2) Architecture of 5–IoT, (3) Heterogeneous Network (HetNet) and (4) Device to Device (D2D) Communication.

In this chapter, we mainly focus on the last type especially on D2D Communication [3, 4, 5, 6, 7, 8], which allows the exchange of data between user equipment without the use of the base station. The short distance communication between two devices (D2D) becomes a challenging way to transmit data, since it benefits the 5–IoT with low power consumption, load balancing and better QoS for edge users. Indeed, in IoT over 60%of applications require low power, a long battery and also wide connectivity coverage. Hence, for these reasons more light should needs to be shed on low-power wireless networks and their prospects in meeting these requirements. Integrating D2D in cellular networks poses challenges and design problems, in order to offer adequate Radio Resource Management (RRM) schemes [4, 5, 6, 9, 10, 11] and this taking into account all the constraints imposed by the different users. As has already been mentioned in the literature, RRM techniques can be classified into four groups as: (1) Mode Selection (MS): where the Mobile Station determines whether D2D candidates in the proximity of each other should communicate in direct mode using the D2D link or in cellular mode [3, 4, 5], (2) Power Control (PC): is an efficient solution to mitigate the interference for D2D underlaid cellular network, in order to improve the overall of the system [6, 9]. (3) Pairing: is a concept which exists only when D2D links are reusing cellular resources and consists on assigning one cellular uplink user (CUE) and one or more D2D uplink user (DUE) links for each resource block [18] and (4) Resource Allocation: is a process of selecting radio resources for each cellular and D2D link, this can be done jointly with MS and pairing [3, 4].

Several approaches have already been proposed in the literature in order to achieve MS and PC management, these approaches can be: (1) Centralized management: where the base station (BS) allocates directly to the designated DUE and require the knowledge of D2D links’ Channel State Information (CSI) at the BS level and (2) Distributed (or decentralized) management, in which the BS informs D2D users which Resource Blocks (RBs) they can use. In this chapter, we focus on the RRM algorithms in underlay D2D communication, for both centralized and distributed MS and PC, in order to improve the overall of the system using game theory tools. In [3, 4, 5, 6, 7, 9, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22], the authors have proposed centralized and distributed PC approaches with perfect channel state information (CSI) using powerful mathematical tools, such as game theory [7, 8, 9], stochastic approximation [20], mechanism design [21] etc.

In fact, Game theory (GT) is a branch of applied mathematics that provides models and tools for analyzing situations where multiple rational users interact to achieve their goals [23, 24]. Several examples based on Wireless Communications are investigated in the literature, as in PC, congestion control, load balancing, etc. In [6], a centralized and distributed PC algorithms are developed and evaluated for a D2D underlaid cellular system using stochastic geometry. The authors in [9] have focused on maximizing the total sum-rate in an heterogeneous network (HetNet) via game theoretic approaches. The authors in [11] have proposed a distributed PC, based on an appropriate interference management scheme in D2D underlaid Cellular Network by using GT approach. An iterative distributed power allocation algorithm for the two kinds of game: pure and mixed has investigated. Distributed vs. centralized MS and PC approaches have been suggested in [25] using GT tools.

This chapter investigates both MS and PC in D2D communications using centralized and distributed approaches based on GT tools. In the proposed MS approach, the CUE and DUE list, the minimum and maximum quantities of power are derived according to CUE and DUE signal-to-interference plus-noise-ratio (SINR) thresholds. The expression of the minimum amount of power known in the literature as the Pareto power [6, 9, 11], has always been used until now without mathematical proof. We propose in this chapter to show mathematically this Pareto power, which is considered as a key of the PC process. Then, a pure strategy non-cooperative game between the two kind of users is modeled as a distributed PC approach, based on the derived minimum and maximum power, where two utility functions are investigated for both type of users. This chapter reviews the work previously published by the authors in [12]. For this, all the proofs and demonstrations of the different results stated in this chapter will not be provided since they are already explained in [12].

The structure of this chapter is given as: In Section 2, the system model of a D2D communication and CUE and DUE SINR and coverage probability expressions are defined. In Section 3, the closed form expression of both minimum and maximum amounts of power with a mathematical demonstration are provided, based on the predetermined CUE and DUE SINR thresholds. In Section 4, our proposed centralized MS and PC approaches are investigated, which consists of generalizing a classical centralized MS and PC approaches. The Section 5, outlines an iterative NE distributed power approach which is proposed for both CUE and DUE and is based on the minimum and maximum amounts of power, derived from Section 3 and on the GT tools. This proposed distributed approach aims to achieve a better compromise between different users in terms of allocated powers is presented in Section 6. Several simulations are provided in order to assess the performance of the allocation approaches of the MS and PC thus proposed in Section 7. Section 8 is followed with conclusion and future scope.

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2. System model

In this section, a D2D uplink underlaid cellular network is considered illustrated in this section, which is shown in Figure 1. A system which is composed by a single-cell cellular network, where K1CUE and K2DUE communicate with as it is illustrated in Figure 1, where the BS is in the center of circular coverage area and each D2D user refers to a source-destination pair. The total number of users Kis defined as K=K1+K2. We assume that all users (CUE and DUE) are drawn in a circular disk Cwith radius Rand are randomly distributed in the whole 2and modeled as an independently homogeneous Poisson Point Process (PPP) Φwith density λ. For each kind of user (CUE or DUE) k, we denote ykas the received signal defined as follows.

  • CUE mode: if 1kK1, ykis the received signal at the CUE kfrom the BS.

  • DUE mode: if K1+1kK, ykis the received signal at the kthDUE receiver from the kthDUE transmitter.

Figure 1.

System model of D2D communication.

Let gk,idenotes the instantaneous channel gain from the kthtransmitter to the ithreceiver, where k,iK=1..K. Further, we denote K1=1..K1and K2=K1+1..K.

2.1 CUE and DUE SINR expressions

In order to ensure a QoS in terms of γcthand γdth, as SINR thresholds of both CUE and DUE (respectively), we assume the following statement for each user k, as performance yardsticks

γkP=gk,kpki=1,ikKgk,ipi+σ2γcth,kK1:cellularmode,1.aγkP=gk,kpki=1,ikKgk,ipi+σ2γdth,kK2:D2Dmode,11.bE1

where pkis the amount of the transmit powers for the kthuser (CUE or DUE), σ2is the receiver noise power, P=p1..pKis the vector of transmit powers and gk,iis defined as follows

gk,i=hk,i2dk,iα,E2

where, hk,iand dk,iare respectively the distance-independent fading and the distance from the transmitter kto receiver iand αis the path loss.

Let us define for each user k(CUE and DUE), the SINR threshold γkth, as

γkth=γcthifkK1:CellularmodeγdthifkK2:D2Dmode.E3

2.1.1 CUE and DUE coverage probabilities

In order to simplify the notations used in the chapter, we will consider vector rather than analytical expressions. According to each kind of user, we define the coverage probabilities expressions denoted as Pc,covPΓcthand Pd,covPΓdthfor both CUE and DUE (respectively) as

Pc,covPΓcthProbΓcPΓcth,E4
Pd,covPΓdthProbΓdPΓdth,E5

where,

ΓcP=γ1P..γK1P,Γcth=γcth..γcth.E6
ΓdP=γK1+1P..γKP,Γdth=γdth..γdth.E7

3. The minimum and maximum amount of power: PminPmax

This section investigates the study of the existence of the two minimum and maximum powers Pminand Pmax, necessary to verify the constraints imposed by the previous system (1). First, based on this system (1), the minimum power Pminis derived, already known in the literature under the name of Pareto Power. Second, by limiting the quantity of power by a quantity, which we denote Pmaxfrom Pmin, we make sure more that the system (1) remains satisfied as long as we are in the power range PminPmax.

To do this, we start by providing another vector form of the (1) system to build this Pareto power.

3.1 Vector form of system

Let us take into consideration the following definition and proposition (1),

Definition 1.If A=ai,j1i,jKand B=bi,j1i,jKare two matrices, then we define

ABai,jbi,j,1i,jK.E8

Proposition 1.The previous Eq. (1) can be written compactly in the following vector form as

IKFPb,E9

where IKdenotes the identity matrix of order K, F=fk,ik,iKand b=bkkK, k,iK, are defined as below [11, 12].

fk,i=gk,igk,kγkthifki0otherwiseandbk=σ2γkthgk,k.E10

3.2 Power lower band Pmin

All previous works [6, 8, 9, 11] have dealt with the resolution of the problem presented in (9) by using a minimum power Pminwithout proof, which is defined as

Pmin=IKF1b,E11

The authors in [6, 8, 11] have shown that: if ρF<1then the matrix IKFis invertible. Next, we propose to suggest another sufficient condition on the matrix F, which guarantees the existence of the inverse of the matrix IKF, in order to derive the quantity of minimum power Pmin, already given in Eq. (11).

To do this, we propose theorems, definitions and propositions in order to outline all the necessary steps which allow to build this sufficient condition. Obviously, to make reading easier, all the demonstrations relating to these theorems and propositions are already detailed in [11, 12].

Theorem 1.We assume thatρF<1, the following statement is true

IKFPbPPmin=IKF1b.E12

Hence, ifρF<1then the minimum powerPmindefined inEq. (11)exists and we can consider the following notation

Pmin=Pmin1,..PminK1Pc,minPminK1+1,PminKPd,min.E13

We can note in another way

Pmin=Pc,minPd,min.E14

Proposition 2.Let consider the following iterative power process, relative to each iteration i, as [26].

Pi+1=FPi+b.E15

Hence,

limi+Fi=0i=0+Fib=IKF1b.E16

3.3 Power upper band Pmax

To let Pmaxgreater than Pminfor all users, it is proposed in this paragraph to build a quantity of power Pmaxfrom Pmin, in order to guarantee the conditions required by the users already depicted in Eq. (11). The two maximum power quantities dedicated to the different CUE and DUE users, denoted as Pc,maxand Pd,max(respectively), are defined as follows

Pc,max=Pc,min+ΔPcPd,max=Pd,min+ΔPd,E17

where, ΔPcand ΔPdare the power margins allocated to different users CUE and DUE (respectively), to ensure that both Pc,maxand Pd,maxremain greater than Pc,minand Pd,min(respectively). Almost all previous work [6, 8, 9, 11] carried out in this context proposes an amount of powers not to be exceeded for both the CUEs and the DUEs. Since these maximum powers quantities may not verify the criteria already mentioned in the system (1), it is proposed in this chapter to guarantee this condition, by assuming that the power Pmaxremains always greater than Pmin. The difference between the two power quantities Pmaxand Pminis denoted by ΔPcfor the CUEs and by ΔPdfor the DUEs.

Likewise, we consider the following notation

Pmax=Pmax1,..PmaxK1Pc,maxPmaxK1+1,PmaxKPd,max.E18

Hence,

Pmax=Pc,maxPd,max.E19

4. On the proposition of a centralized MS and PC approaches

This section investigates centralized MS and PC approaches, which aims to select CUE and DUE from a predetermined list and to minimize the consumed amount of power, in order to satisfy the QoS depicted in Eq. (1). A centralized approach is proposed in this section, which is a generalized version of the algorithm CPCA (denoted GCPCA) to more than one CUE.

The condition assumed during the MS process is ρF<1. Thus only the users who check this last condition are retained in the final list. Then, the minimum power Pminis allocated to the different types of users (CUE and DUE) based on this selection criterion, in order to optimize the amount of power.

4.1 Proposed generalized centralized power control algorithm (GCPCA)

Unlike the CPCA algorithm which is based on a MS relating to a system containing only one CUE, the GCPCA (see algorithm 1) generalizes this latter for several CUEs, based on the same condition K11. In fact, this assumption is more realistic and illustrates a more real case.

As shown in step 1 from algorithm 1, we first test if the matrix Fl(relative to the iteration l), verifies the condition ρFl<1. If this is true, the Pareto power Pminalready defined in the Eq. (11) is assigned to admitted users, as the steps 5 and 6 indicate. Otherwise, we select the k̂-th user transmitter (CUE or DUE) who can increase the maximum of interference power compared to other receivers, as shown in step 2. Mathematically, this results in finding user k̂, such as: k̂=argmaxkfkl2.

Thus, this user k̂will now be eliminated from the list of users admitted into the system, as is confirmed at step 3.

his most annoying user elimination strategy is repeated until the constraint is verified. Finally, the matrix obtained satisfies the sufficient condition ρFl<1, for the existence of the pareto power Pmin(see step 5).

All these steps are more detailed in algorithm 1.

Algorithm 1: Proposed GCPCA

initialisation:Flfor l=0, for all active CUE and DUE. [Step 1]

Step 1: if ρFl<1, go to step 5and 6. Otherwise, go to Step 2.

Step 2.k̂=argmaxkfkl2

Step 3. remove the k̂-th column and row vectors of the matrix Fl.

Step 4. update: Fl+1=Fl, l=l+1. Go to Step 1.

Step 5. evaluate the power Pminusing the equation (11).

Step 6.P=Pmin

This GCPCA algorithm converges after an iteration number, since the condition ρFl<1must each time be checked by the selected users during each iteration l. In fact, the proposition 2 developed in the previous section provides a convergence certificate of this algorithm.

The maximum power Pmaxdeduced from the previous Eq. (17), will be useful in the next section in order to limit the powers allocated for each type of user.

5. On the proposition of a distributed PC approach based on GT

The power control problem proposed in this paper is considered as a distributed strategies non-cooperative game, where the utility functions as well as the strategies adopted by each user are defined and justified.

5.1 Proposed utility functions

Several utility functions have suggested in [3, 9, 27, 28], using a pricing coefficient to enhance both efficiency and fairness among users. The proposed CUE and DUE utility functions considered in this section are defined as follows [6, 11],

  1. CUE utility function:The utility function ukPrelative to a CUE kis defined as

ukP=γkPγcth2,kK1.E20

  1. DUE utility function:The utility function ukPrelative to a DUE kis defined as

ukPRewkPPenkP,kK2,E21

where

  • The reward function RewkP, relative to the kthDUE user, evaluates the payoff of the kthDUE based on both γdthand on a nonnegative weighting factor pricing coefficient ak, as follows

RewkP=1eakγkPγdth,kK2,E22

  • The penalty function, PenkP, relative to the kthDUE user, is defined as

PenkP=bkCpkPkIkPk,kK2E23

where,

CpkPk=j=1,jkKpkgk,jandIkPk=j=1,jkKpjgk,j+σ2,E24

and bkis a constant and nonnegative weighting factor, which reflects the relative impact of the kthDUE user in terms of power. We denote Pkas the vector of transmit powers of all users other than k, defined as follows

Pk=p1pk1pk+1..pK,kK.E25

From which it follows

P=pkPk.E26

Afterwards, we denote the utility function vector as: uP=u1Pu2P..uKP, where ukPcan be evaluated from (20) or (21), depending on whether the user kis CUE or DUE (respectively).

5.2 Pure strategies game

We denote our game G=KPuPof complete information between Kplayers. The strategies of such game are considered to be the vector power PΩ, where Ωis given by

Ω=P=p1p2..pKpkΔkkK,E27

where,

Δk=pminkpmaxk.E28

The two powers pminkand pmaxkare derived from Eqs. (13) and (18) (respectively).

The NE is a strategy profile in which the strategy used by each user is at least as good a reply as any other strategy available to him to the strategies played by the other users. In this sense, to derive the NE of our proposed game, we propose in the following paragraph to study the best response relative to each user k, by improving the utility function of each user (CUE and DUE).

5.3 Nash equilibrium

Definition 2. Best-response:The best-response function BRkPkof a user k(CUE or DUE) to the profile of strategies Pk, is a set of strategies pkfor that user kshould satisfy the following condition [24]

BRkPk=pkΔkukpkPkukpkPkpkΔk.E29

Hence, each element of the best-response function BRkPkis a best-response of the user k, relative to other strategies Pk.

Definition 3. Nash Equilibrium (NE)A pure strategies NE (PSNE) G=KPuP, is a set of strategies P=p1p2pK, such that no player can unilaterally enhance its own utility [24], i.e.,

ukpkPkukpkPk,pkΔk.E30

Hence, a PSNE Pis a stable outcome of a game G, if no user has any incentive to change its strategy.

5.4 Example: 2-users game

5.4.1 Best-responses expressions of CUE and DUE

We assume that (K1=K2=1) and (a1=a2=aand b1=b2=b), the expressions of the two Best-responses relatives to CUE and DUE are studied and evaluated in this section. As already explained in the preceding sections, the amount of power for each type of user kshould be greater than a minimum power (pc,min=pmin1for CUE and pd,min=pmin2for DUE) and also less than a maximum power (pc,max=pmax1for CUE and pc,max=pmax2for DUE). Hence, if we denote P=p1p2as the allocated power vector, where p1is the power relative to the CUE which should belong to Δ1and p2is the power relative to the DUE which should belong to Δ2. We remind that Δ1and Δ2are already defined in Eq. (28).

In this case, the feasible region of the power is defined as a region where the amount of power P=p1p2Ωshould verify the following condition

pc,minp1pc,maxp1Δ1E31
pd,minp2pd,maxp2Δ2E32

Proposition 3.The Best-response relative to the first user (CUE), denoted as BR1p2, is given by

BR1p2=p1Δ1p1=γthcg1,2g1,1p2+σ2g1,1p2Δ2.E33

Proof.Based on the expression of the CUE utility function u1p1p2, which is defined in Eq. (20) and where γ1Pcan be derived from the Eq. (1.a), we can easily deduce the following result

u1p1p2p1=02g1,1g1,2p2+σ2g1,1p1g1,2p2+σ2γthc=0.E34

So, the expression of BR1p2found in (33) is derived by deducing the expression of p1according to p2from the last equation. This completes the proof. □

Proposition 4.The Best-response relative to the second user (DUE), denoted as BR2p1, is given by

BR2p1=p2Δ2p2=g2,1p1+σ2g2,21alogbg1,2ag2,2+γthd+p1Δ1.E35

Proof.Based on the utility function expression u2p1p2defined in Eq. (21) and on the expression of γ2Pdefined in Eq. (1), we can easily get the following expression

u2p1p2p2=0ag2,2g2,1p1+σ2eag2,2p2g2,1p1+σ2γthdbg1,2g2,1p1+σ2=0.E36

After simplification, the BR2p1depicted in Eq. (35) is readily derived by deducing the expression of p2according to p1from the last equation. □

5.4.2 Simulations and result interpretations

The two best-response BR1p2and BR2p1of both CUE and DUE (respectively), which are derived from (33) and (35) (respectively) and the NE are depicted in Figure 2. The simulation parameters used in this figure are presented in Table 1. As a note from this figure, we can notice that the NE exists and is unique, because the two curves of BR1p2and BR1p2intersect at a single point in the feasible region Ω.

Figure 2.

Best-responseBR1p2,BR2p1and NE.

ParametersValues
R700(m)
D2D link range dk,k50(m)
D2D link density (λ)5×105
Path loss exponent (α)4
γthcfrom 6to 18(dB)
γthdfrom 10to 14(dB)
Δpc50mw
Δpd0.002mw
σ2(for 1MHzbandwidth)143.97dBm
25
b1
λ5105
ε1mw
Iterations number103

Table 1.

Simulation parameters.

In the next section, we propose to extend this study for a system which contains K1CUE and K2DUE and furthermore to determine the NE SINR and power closed forms for both CUE and DUE, if it exists.

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6. Proposed distributed power control algorithm based on GT

A NE Distributed PC Algorithm (NEDPCA) relative to both CUE and DUE is investigated in this section, in which our proposed game and CUE and DUE utility functions already defined in the previous section are considered. First, to do this, the SINR NE expressions for each user (CUE and DUE) are presented. Afterwards, the amount of power allocated to each user (CUE and DUE) relative to the derived NE are also studied. Thirdly, a power allocation algorithm will be suggested, based on the obtained results to derive the NE power quantities and the power limitation already discussed in Section 3.

6.1 SINR and power NE for CUE and DUE

The authors in [11] have shown that for a CUE and DUE k, the unique SINR NE γc,kPand γd,kP(respectively) have the following expressions:

γc,kP=γcth,kK137.aγd,kP=1aklogakgk,kbkikgj,k+γdth+kK2,37.bE37

where f+=maxf0.

Assumption 1.Based on the Eq. (37.b), we assume the following statement for each DUE k

1aklogakgk,kbkikgj,k0E38

In fact, if: 1aklogakgk,kbkikgj,k<0, we can have one of the two following cases

case1:if1aklogakgk,kbkikgj,k+γdth0γd,kP<γdth.case2:if1aklogakgk,kbkikgj,k+γdth<0γd,kP=0.E39

The unique NE power Pof both CUE and DUE can be derived as follows

P=p1..pK1pK,wherepk=IkPkγkgk,kpminkpmaxk,kK,E40

where, γc,kPand γd,kPare defined in Eq. (37.a) if kK1and (37.b) if kK2(respectively).

We propose in the next step a distributed PC algorithm for the mentioned pure strategy game, which is based on the allocated power Ppreviously defined in Eq. (40).

6.2 Proposed distributed power control algorithm based on GT

The algorithm depicted in 2 outlines the different steps of the proposed algorithm NEDPCA offered to each CUE and DUE, which is based on the previous pure strategy game. In fact, this algorithm NEDPCA offers a NE power for the two kinds of users, based on both the previous constraints (1) and on the power expression depicted in Eq. (40). First, the MS process is derived from the GCPCA, in order to guarantee the constraints imposed by the CUEs and the DUEs in terms of SINR thresholds (see Eq. (1)). This is shown in step 1 of the Algorithm 2. Second, the SINR NE γc,kPand γd,kPrelative each user k(CUE and DUE) are obtained, which are evaluated in Eqs. (37.a) and (37.b) (respectively) (as it is shown from steps 2 and 3). Third, the PC process is investigated based on the iterative approach which is executed by using Eq. (40), as shown in steps 4 and 5. Step 6 allows to finalize the power distribution step, with an error of nearly δfor each user k. If pk(t+1pkt<δuser k, then the amount of power allocated to each user k, is given as: Pkt=Pk, otherwise we repeat this process and go to step 2.

Algorithm 2:Proposed NEDPCA

initialisation:t=0,F,b,δ>0and pk=pk0,kuser.

Step 1: Evaluate from the Algorithm 1 GCPCA:

1) the CUE and DUE set of users: K1, K2

2) Pminand Pmaxusing equations (17) and (19).

2. for each CUE k, evaluate γc,kPusing equation (37).a.

Step 3. for each DUE k, evaluate γd,kPusing equation (37).b.

Step 4. Derive for each CUE and DUE k, the amount of power pktusing equation (40), where pminkand pmaxkare derived from the 1ststep. Evaluate Pt=p1t..pKt

Step 5: update Pt+1=FPt+b.

Step 6. if pk(t+1pkt<δ,k, derive the solution Pkt=Pk, otherwise t=t+1and go to the 2ststep.

All the NEDPCA steps relative to the distributed MS and PC for both CUE and DUE are detailed in Algorithm 2. Indeed, the first step of NECPCA makes it possible to realize the MS approach and all the other steps allow to deduce the PC approach.

Like the GCPCA algorithm, the NEDPCA algorithm converges after an iteration number, since it is based on the same condition ρFl<1which must be checked during each iteration l by all the selected users. In fact, by applying the step 1 of the algorithm GCPCA (see Algorithm 2), the last condition should be guaranteed. It is also due to the proposition 2, that the convergence of NEDPCA is proved.

7. Analysis of simulations

In order to evaluate the performance of the algorithms already mentioned and proposed in the following sections, we consider in this section to study the simulations of these algorithms: GCPCA and NEDPCA. A Monte Carlo simulation is applied according to the Table 1, already given in the previous section.

The CUE and DUE Total powers are evaluated in Figures 3 and 4 versus γcthand γdth(respectively).

We remain that the CPCA algorithm considers only one CUE and possibly several DUE. The GCPCA allocates to the different users the minimum power derived from Eq. (11), while respecting the condition ρF<1. Thereby, any CUE and/or DUE that does not verify this condition will be eliminated from the user list.

First, we can notice from Figure 3 that all the curves relative to GCPCA and NEDPCA algorithms are decreasing when γcthincreases. This is because when the threshold γcthincreases, it becomes difficult to find CUEs that check the constraints depicted in (1). On the other side from Figure 4, only the DUE performances are improved when GCPCA is considered and γdthis improved. This is due to the fact that, the NE allows to maximize the utility function relative to each user k.

Figure 3.

CUE Total power vs.γcth.

Figure 4.

DUE Total power vs.γdth.

As it is shown from these two figures, the DUE coverage probability is significantly improved compared to that of DUE, because the DUEs benefit much more from TG compared to CUE, by using the NEPCA.

Second, the proposed centralized approach GCPCA which is a generalization of the CPCA approach offers less total power compared to the NEPCA, for both types of users CUE and DUE. This is due to the fact that when we want to reach a NE, by increasing the utility functions of each type of user, we should increase the total power consumed. Moreover, the difference between the two types of curves represents the power gain that must be added in order to reach this NE. From Figures 3 and 4, it is clear that for DUEs, this difference in terms of power is smaller compared to that of the CUEs. This is explained by the fact that the DUEs require less power to transmit. Third, by limiting the power consumed in PminPmaxwhen GCPCA and NEDPCA are used, more flexibility and possibility to integrate the two types of users are offered. The greater the amount of maximum power Pmax, the higher the probability of coverage using the proposed GCPCA algorithm compared to CPCA one. It is for these reasons that it would be judicious to choose adequate margins of power ΔPcand ΔPd, relative to the types of users (CUE and DUE).

8. Conclusions

This chapter allows to invoke the problem of selection mode and power control for a D2D underlaid cellular networks in 5G. The basic idea of this chapter is to generalize the classic allocation algorithms by applying Game Theory, for many CUEs and DUEs in system.

First, we assume that the amount of power allocated to each kind of user should be between two amounts of power: a minimum power defined as a Pareto solution and a maximum power. Thus, a mathematical demonstration was provided in this chapter, in order to prove the expressions of these two powers, based on constraints imposed by the users in terms of SINR thresholds to be respected.

Second, our proposed system is modeled as a non-cooperative pure game between the different types of users, where the utility functions should be maximized. From the built-in utility functions, NE SINR and PC solution existence and uniqueness are investigated and studied.

Third, simulations have been established in this context, in order to assess the performance of the algorithms thus proposed in terms of total powers relative to both users CUE and DUE. Through these simulations which compare these results without and with GT, we noticed that by applying the TG, the total power consumed increases in order to reach the NE for the two types of users: CUE and DUE. This is due to the fact that to increase the utility functions relating to the two types of users, a power margin must be added. However, this difference in terms of power becomes less important for the DUEs, since they require less power relative to the DUEs.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Sameh Najeh and Ammar Bouallegue (September 15th 2020). Nash Equilibrium Study for Distributed Mode Selection and Power Control in D2D Communications, Systems-of-Systems Perspectives and Applications - Design, Modeling, Simulation and Analysis (MS&A), Gaming and Decision Support, Tien M. Nguyen, IntechOpen, DOI: 10.5772/intechopen.92362. Available from:

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