HAL Id: hal-01713174
https://hal.archives-ouvertes.fr/hal-01713174
Submitted on 20 Feb 2018
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Collaborative Paradigm for Next Generation Wireless
Networks
Duc-Tuyen Ta, Nhan Nguyen-Thanh, van Tam Nguyen, Duy Nguyen
To cite this version:
Duc-Tuyen Ta, Nhan Nguyen-Thanh, van Tam Nguyen, Duy Nguyen. Collaborative Paradigm for
Next Generation Wireless Networks. EURASIP Journal on Wireless Communications and Networking,
SpringerOpen, In press. �hal-01713174�
Collaborative Paradigm for Next Generation Wireless
Networks
Duc-Tuyen Ta, Nhan Nguyen-Thanh, and Van-Tam Nguyen
LTCI, CNRS, T´el´ecom ParisTech, Universit´e Paris Saclay Paris, France
Duy H. N. Nguyen
Department of Electrical and Computer Engineering, San Diego State University San Diego, CA USA 92182
Abstract
The wireless revolution requires future wireless networks the capability of intelligently optimizing the spectrum by collaborating with, and learning from, the other systems that occupy the same spectrum bands. How to develop the wireless paradigm of col-laborative, therefore, is a crucial question. In this paper, we discuss how to model the collaborative power control in a wireless interference network, where users share the same frequency band. By collaborating with other users, each user exchanges in-formation to maximize not only its own performance but also others’ performances. A game theory framework is developed to determine the optimal power allocation. The proposed framework possesses several advantages over conventional methods, such as low-complexity and fast-converging algorithmic solution, distributed implementation and better user fairness. Simulation results state the proposed approach provides better fairness between users’ data rates, higher performance in the aggregate rate and lower convergence time.
Keywords: Collaborative power allocation, wireless interference network, game theory.
1. Introduction
The wireless revolution is creating a huge demand for ac-cess to the RF spectrum, while facing a serious problem of an increasing scarcity of RF spectrum [1, 2]. Static spectrum allo-cation approach, which divides the spectrum into exclusively
li-5
censed bands, is not adaptive to the dynamics of supply and de-mand. Unlicensed bands or shared spectrum provide more flex-ibility and efficiency in spectrum usage, but need more sophisti-cated interference avoidance techniques than simple sense-and-avoid techniques in future dynamic and flexible spectrum
allo-10
cations [3]. To harvest the full capacity out of the RF spectrum, future wireless networks will need to use greater intelligence to avoid interference while optimizing the spectrum by collaborat-ing with other systems that occupy the same spectrum bands. This is the main objective of DARPA spectrum challenge [2],
15
which proposes to develop the Collaborative Intelligent Radio Networks, where collaborative paradigm is underlined as a key area to be addressed.
In this paper, we discuss how to model the collaborative paradigm and present the advantage that this paradigm shift
20
Email addresses: {duc-tuyen.ta, nhan.nguyen-thanh, van-tam.nguyen}@telecom-paristech.fr (Duc-Tuyen Ta, Nhan Nguyen-Thanh, and Van-Tam Nguyen), duy.nguyen@sdsu.edu (Duy H. N. Nguyen)
can provide for the design of future wireless systems. Specif-ically, we tackle the collaborative power control problem be-tween network users to manage power allocation in a wireless interference network. A wireless interference network refers to a communication system in which multiple transmitter-receiver
25
pairs share a common frequency band and cause signal inter-ference in other receivers [4]. Such kind of network covers a broad range of wireless communication systems, such as wire-less ad-hoc networks [5] or Device-to-Device (D2D) communi-cations [6]. The power control in wireless interference network
30
is selected as an example for the collaborative paradigm due to the interaction between network users and the scarcity of radio spectrum. In this domain, recent works mainly focus on the two strategies: distributed strategy based on non-cooperative game framework [7, 8, 9, 10, 11, 12, 13] and centralized
strat-35
egy based on joint optimization [14, 15, 16, 17, 18, 19]. In the distributed power control, the power allocation among the users is considered as a non-cooperative game, where each user selfishly maximizes its own data rate [9, 20] or its packet transmission success rate [12] or its energy efficiency [13], or
40
minimize its transmit power while achieving a given target signal-to-interference-and-noise-ratio (SINR) [8]. In such games, pric-ing strategies [20] are added to encourage the users to adopt more socially optimal power control. As a result, the efficiency of the Nash equilibrium (NE) is substantially enhanced if
rea-45
sonable deviations from the target SINR are allowed. In the distributed approach, each user only needs the local informa-tion to make the independent and rainforma-tional decision. The feature makes it possible to use low-complexity distributed algorithms to determine the power allocation. However, the global
opti-50
mum may be less likely to be achieved and the system-level performance may be degraded.
In the centralized power control, the power allocation is co-ordinated by a joint optimization process, where all users aim to maximize a common utility function, such as the weighted
55
sum-rate [14, 15, 16, 17] or the total energy efficiency [18, 13, 19]. Mathematical frameworks, such as geometric program-ming [14, 15, 16, 17] or the factional prograprogram-ming [18, 13, 19], are employed to establish the optimal power allocation. The joint optimization approach allows all the users to coordinately
60
optimizetheir strategies and enables a dynamic allocation of the interference budget among users. However, this approach faces the problems of the increased complexity and overhead due to the demand for the channel information of all users and/or the requirement of a centralized unit. In addition, even the
65
global information are known, the optimization results show that users with poor channel conditions are allocated with much less power in order to optimize the performance of the whole network. It degrades the fairness between users in the network. Our goal in this paper is to propose a collaborative power
70
control framework, where each user optimizes its strategy in a collaborative manner through a modified objective which com-prises not only its own performance but also the others’ per-formance. The proposed paradigm possesses the advantages of the distributed approach, which is the low-complexity and
75
fast-converging algorithms with distributed implementation and overcomes the disadvantages of the centralized approach, which is the increased complexity, overhead and the requirement of a central unit. Specifically, we will formulate new games that will narrow the performance gap compared to the joint
optimiza-80
tion, while maintaining the distributed implementation. Unlike the previous works, which mostly focus on utility maximization with power constraint [9, 11, 12, 14, 17, 13, 19] or power min-imization with quality constraint (e.g., SINR) [8, 10, 20, 18], we consider the power control problem with utility
maximiza-85
tion under both power and SINR constraints. This allows to maximize the network performance while maintaining certain quality of transmission for each user. Typically, the optimiza-tion problems for multiuser power control are nonconvex. Ob-taining a global solution is highly complex. We then provide
90
a low-complexity method for efficiently solving this issues by approximating the utility function of the game for each region on SINR of the network to obtain a well-known game, such as a potential game [9] or a concave game [21], and analyze the power allocation through the NE of such game. The best
re-95
sponse dynamic then prompts the study on the existence and uniqueness of the NE in the game. We supplement the theory with numerical results, which show that the proposed paradigm provides better fairness between users, higher performance and lower convergence time. The paper makes the following
spe-100 cific contributions: 1N h 12 h 11 h 21 h h22 2N h 1 N h 2 N h NN h TX1 TX2 TXN RX1 RXN RX2 TXi: Transmitter i RXi: Receiver j Communication link
Figure 1: The system model of the power allocation problem between multiple users in a multiuser communication system.
• We formulate a simple and effective framework to allo-cate power between users in a wireless interference net-work by considering the collaborative with others. • For the different network configurations (i.e., in terms of
105
SINR region and the number of users), we introduce the corresponding approximation methods to determine the optimum power for each user.
• We propose a simple and distributed implementation al-gorithm for finding the best power allocation.
110
• Comparisons between the various power control meth-ods for wireless interference networks are throughly pre-sented.
The paper is organized as follows: Section 2 contains the system model and the game formulation of collaborative power
115
control paradigm in wireless interference network, the approxi-mation method for each region on SINR of the network to over-come the nonconcave issues of the utility function and the cor-responding games are presented in Section 3 for networks with high SINR region, Section 4 for the networks with low SINR
120
region and Section 5 for the networks with a large number of users, Section 6 provides some numerical results and interpre-tations, and conclusions are summarized in Section 7.
2. System Model and Game Formulation 2.1. System Model
125
We consider a wireless interference network with M in-dependent users, each consists of a transmitter-receiver pair, sharing a common frequency (Figure 1). Transmission from each transmitter causes interference at the reception of other receivers. We model this scenario as a Gaussian interference
130
channel with flat fading, where each receiver perceives the trans-mitted signal with additive white Gaussian noise.
Let hji be the channel power gain from the transmitter j to the receiver i and σi2 be the noise power at receiver i. De-note by p= {p1, p2, . . . , pM} the vector comprising the allo-cated powerof all users and p−i the power vector of all users 2
except i. The SINR at the receiver i, denoted by γi, is given by γi(pi, p−i)= hiipi σ2 i + Í j,i hjipj . (1)
Suppose that each user can adjust its transmit power within a bounded region (0, pmaxi ) to meet a given target SINR con-straint. For user i, it means that
γi ≥γt ari , (2)
where γt ari is the given target SINR of user i.
Over the time-period of interest, we assume that the channel gains are fixed (i.e., fading effects take place at a much slower time-scale). Let ri(pi, p−i) be the rate of user i. Then,
ri(pi, p−i)= log2(1+ γi(pi, p−i)). (3) For each user i, we defined the performance metric fi(pi, p−i) that captures a trade-off between the obtained transmission rate and the power cost. This metric is given by
fi(pi, p−i)= ri(pi, p−i) − cipi, (4) where ci is the pricing factor of user i [20].
In the distributed strategy based on non-cooperative game approach [8, 9, 10, 11, 12, 13], the utility of each user is its own performance. In the centralized strategy based on joint op-timization approach [14, 15, 16, 17, 18, 13, 19], however, net-work users share a common utility function which is typically the total performance of the network. In this paper, we consider the collaboration between users by proposing the collaborative utility function. Instead of considering its own performance or the common utility function, each user optimizes its collabora-tive utility function which comprises not only its own perfor-mance but also the others’ perforperfor-mances. For simplicity, we suppose that the collaborative utility function of each user is
Uicol(pi, p−i)= fi(pi, p−i) | {z } performance metric + gi(pi, p−i) | {z } collaboration metric , (5)
where the collaboration metric gi(pi, p−i) is assumed to be the partial sum of the others’ performances, i.e.,
gi(pi, p−i)= Õ
j,i
αjfj pj, p− j, (6) and αj ≥ 0 is the collaboration factor for user j.
135
The collaboration between users is designated through a collaboration channel, which provides a direct means for net-work users to share potentially valuable information and to strate-gize with their peers. Example of the collaboration protocol is presented in the system specifications of [2]. The collaboration
140
factors are determined by each user based on its demand. In the collaborative power allocation, each user aims to max-imizeits collaborative utility function, i.e.,
max pi
Uicol(pi, p−i) ∀i = 1, . . . , M
s.t. pi ∈ [0, pimax] (7)
γi ≥γt ari .
2.2. Game Formulation
Due to the conflict and trade-off between the objectives of network users, the game-theoretic approach is employed to model the relationship between network users. We referred this game as the collaborative power control game
G , {M, {Pi(p−i)}i, {Uicol(pi, p−i)}i}, (8) where M= {1, 2, . . ., M} is the set of players and Pi(p−i) is
145
the player i’s strategy set such that γi ≥γit ar.
Generally, a Nash Equilibrium (NE) of a game is a feasi-ble strategy from which players cannot gain by independently adjusting their strategy. For G, (p∗i, p∗−i) is a NE if and only if
Uicol(p∗i, p∗−i) ≥ Uicol(pi, p∗−i)∀pi ∈ {Pi(p−i)}i. (9) To determine the NE, each user selects the best response (BR) (or one of the BRs) to the others’ strategies. Given the others’ powers p−i, the BR Bi(p−i) of player i is defined as
Bi(p−i) ∆
= arg max pi∈ Pi(p−i)
Uicol(pi, p−i). (10)
The NE of G is then determined by iteratively solving M coupled problems in (7). The iterative process is called the best-response dynamics (BRD) of G where any fixed point of BRD is a NE of the game [9, 19].
150
2.3. Approximation
The convergence of the BRD is an important question to investigate the existence and/or the uniqueness of the NE point of the game (i.e., the optimal power control strategy). In our game, the question is more challenging since
155
• the objective function in (7) is nonconcave, and
• both the utility function and the strategy set of each player are dependent on other players’ actions due to the SINR constraints.
To this end, we propose low-complexity methods for efficiently
160
solving these problems by approximating the utility function of the game to obtain a well-known game and analyze the power allocation through the NE of such game.In particular, we do approximation for each region on SINR of the network to over-come the nonconcave issues of the utility function thanks to the
165
following features: i) for high SINR region, the rate log2(1+γi) can be approximated by log2(γi), i.e., log2(1+ γi) ≈ log2(γi), ii) otherwise, since the utility function is continuous and dif-ferentiable, it hence can be approximated by a linear function through the first-order Taylor approximation. The difference
170
between these approximated games and how each game is ap-plicable for certain scenarios are shown in Figure 2. The analy-sis on these games will be presented in subsequent sections.
Collaborative Power
Control Game
Large Network
Small Network
High SINR
Low SINR
Log approximation
Modified utility function
Potential Game
First-order Taylor approximation
Modified utility function
Concave Game
(i.e., network with a small number of users) ( i.e., network with a large number of users)
All SINR
Step 1: log approximation
Step 2: First-order Taylor approximation
Modified utility function
Concave Game 2 2 log (1i)log ( )i ( , ) (p , )pizi ( , )
i i p zi i T p i i i i i i i i g p g g p p z p p p 2 1 3
2 2 log (1i)log ( )i
1 { ,{ } ,{ˆ , } } col i pi i Ui pipi i
ˆ( , ) ˆ(p , ) ˆ( , ) pizi i i p zi i T p i i i i i i i i g p g g p p z p p p
,
ˆ
,
col col i i i i i i U p p U p p
,
,
col col i i i i i i U p p U p p
,
,
col col i i i i i i U p p U p p
3{ ,{ ipi} ,{i U pi i,pi} }i
2 { ,{ } ,{ , } } col i pi i Ui pipi i Figure 2: The approximated games for each region on SINR and network size.
3. The Potential Game Approximation
In this section, we do the approximation the utility function
175
of G for the wireless interference network with high SINR re-gion, i.e., the network users are far apart. For such scenario, the performance metric and the collaboration metric of each user can be approximated as ˆ fi(pi, p−i)= log2(γi) − cipi, i = 1, . . ., M (11) ˆ gi(pi, p−i)= Õ j,i αjfˆj pj, p− j. (12) Consequently, the modified utilities of player i is given by
ˆ
Uicol(pi, p−i)= ˆfi(pi, p−i)+ ˆgi(pi, p−i). (13) We formulate a new game with the modified utilities, which is referred as the potentialized game, and denote it by
G1 , {M, {Pi(p−i)}i, { ˆUicol(pi, p−i)}i}. (14) In game G1, given the power vector p−i, we define the best response ˆBi(p−i) of player i as
ˆ Bi(p−i) ∆ = arg max pi∈ { Pi(p−i)}i ˆ Uicol(pi, p−i). (15)
3.1. Properties of the Potentialized Game
180
Hereafter, we obtain some basic properties of the game G1. First, we prove that G1is an exact potential game.
Proposition 1. The game G1 is an exact potential game. The corresponding potential function Φ(p) is given by
Φ(p)=Õ i log2(hiipi) − cipi− Õ j,i αjlog2 Õ k,j hk jpk+σ2j ! ! . (16)
Proof 1. For user i ∈ M, we have:
∂2Uˆcol i (pi, p−i) ∂pi∂pj = Õ t,i, j αt ln (2) © « hithjt σ2 t + hjtpj + Í k,j,t hktpk !2 ª ® ® ® ® ® ® ¬ , (17) and ∂2Φ(p) ∂pi∂pj = ∂ ∂pj Í t,i 1 pi − ci− Í j,t αj ln(2) hi j σ2 j+ Í k ,t hk jpk ! ! = Í t,i, j αt ln(2) © « hi thj t σ2 t+hj tpj+ Í k , j, t hk tpk !2 ª ® ® ® ¬ . (18)
According to the characterization of (exact) potential game in [22], i.e., ∂2Φ(p) ∂pi∂pj = ∂2Uˆcol i (pi, p−i) ∂pi∂pj , ∀i ∈ M
we therefore conclude that G1 is an exact potential game with potential function Φ(p).
185
Proposition 2. If Í j,i
αj ≤ 1,∀ j ∈ M, the game G1is strictly concave potential game.
Proof 2. First, the potential function (16) is continuously dif-ferentiable:∀i ∈ M, it is easy to check that
ln(2)∂ 2Φ(p) (∂pi)2 =−1 p2i + Õ j,i αj© « pi+ Õ k,j,i hk j hi j pk+ σ2 j hi j ª ® ¬ −2 . (19) 4
Thus, if
Õ
j,i
αj ≤ 1,∀i, j ∈ M (20)
is satisfied then the second-order derivative of the potential function with variable piis smaller than0. Hence, the potential function Φ(p) is strictly concave. Also, the SINR constraint is a linear function with pi since
γi−γit ar = hiipi Í j,i hjipj+ σi2 −γit ar ≥ 0,∀i ∈ M. (21)
Therefore, from [22] (Definition 2.3), if (20) then G1 is a strictly concave potential game.
3.2. Analysis of the Equilibria
190
The existence and uniqueness of the NE points of the game G1are now studied by exploiting the potential function (16). Proposition 3. If condition (20) holds then the game G1admits a unique NE point.
Proof 3. If condition (20) holds then G1 is a strictly concave
195
potential game. It is proved in [22] that every strictly concave potential game admits a unique equilibrium. Proposition 3 thus follows.
Next, given the power vector p−i, the BR of the user i ∈ M in (15) is determined as follows: 200 Lemma 1. If pmaxi ≥ γ t ar i hii Õ j,i hjipmaxj + σi2 ! (22)
then ˆBi(p−i) takes the form ˆ
Bi(p−i)= min{pimax, max{p∗i, pt ari }} (23) wherein pt ari (p−i) ∆ = γ t ar i hii Õ j,i hjipj + σi2 ! (24) and p∗i = argmax∆ pi∈R+ Φ (p). (25)
Proof 4. Please refer to Appendix A. 3.3. Distributed Implementation
To determine the NE strategy of the game, we employ the Best Response Dynamic (BRD) algorithm by iteratively finding the BR of each player given others’ power profile until reaching
205
the NE point [9].
To implement Algorithm 1, information is exchanged be-tween users through a collaboration paradigm. Specifically, the network users using a collaboration channel to distribute the information to be shared and the resulting decisions. For
210
each user i, the direct channel (i.e., hii) can be estimated at
1: Initialize: k= 1 and ∀i : pi[0] ∈ R+in the feasible set
2: repeat
3: for i= 1 to M do
4: Compute p∗i[k] from (25)
5: Update the power as (23)
6: end for
7: Update k= k + 1
8: until convergence
Algorithm 1: Best Response Dynamic (BRD)
its receiver and sent back to its transmitter through a feedback channel. Since the power (pi) is locally available at the trans-mitter, the interference plus noise (i.e., Í
j,i
hjipj + σi2) can be observed. Then, the computation of (24) is available. The
po-215
tential function Φ(p) can be written simply in term of pi as in (26). For user i, we observe that the computation of (25) does not depend on the other’s direct channel and power (i.e., hj j and pj,∀ j , i) and only requires: 1) knowledge of its di-rect channels, 2) knowledge of its cross-channel (i.e., hi j, j , i)
220
and 3) knowledge of other users’ interference plus noise. Since the cross-channel can be estimated through the other users’ in-terference plus noise and its locally available power (pi), (25) can be obtained by exchanging the information about the in-terference plus noise of each user. Therefore, the distributed
225
implementation can be adopted to solve (23). We conclude that Algorithm 1 not only guarantees the convergence to a unique NE but can also be implemented in a distributed manner.
4. The Concave Game Approximation
The potentialized game approach in Section 3 only serves as a good approximation for the true rate when the users operate in the high SINR region. Hereafter, we do the approximation the utility function of G for the wireless network with low SINR re-gion. For such scenario, the utility function (5) is approximated by linearizing the convex part as in follows:
˜
gi(pi, p−i, zi) ≈ g(zi, p−i)+5pig(pi, p−i)
T|
pi=zi(pi− zi), (27)
where zi ∈ {Pi(p−i)}i is the initial point of user i for the
ap-230
proximation process .
The modified utility function is given by ˜
Uicol(pi, p−i, zi)= f (pi, p−i)+ ˜gi(pi, p−i, zi). (28) We then formulate a non-cooperative game with modified utility function as follows
G2, {M, {Pi(p−i)}i, { ˜Uicol(pi, p−i)}i}. (29) In game G2, the players’ feasible action sets are non-empty, closed, and convex. In addition, the modified utility function (28) is concave. Hence, from [21], we refer to this game as the con-cave game. The BR ˜Bi(p−i) of user i is defined as
˜ Bi(p−i) ∆ = arg max pi∈ Pi(p−i) ˜ Uicol(pi, p−i). (30) 5
Φ(p)= log2(hiipi) − cipi− Õ j,i αj ! log2© « Ö j,i hi jpi+ Õ k,i, j hk jpk+ σ2j ª ® ¬ −Õ j,i © « Õ k,j αklog2 hikpi+ Õ t,i,k htkpt+ σk2 ! ª ® ¬ +Õ j,i log2 hj jpj − cjpj (26)
4.1. Properties of the Concave Game
We obtain some properties of the game G2as follows: Lemma 2. If condition (22) holds then ˜Bi(p−i) takes the form
˜
Bi(p−i)= min{pimax, max{p+i, pt ari }} (31) wherein pt ar
i (p−i) is defined as in (24) and p+i = arg max∆
pk∈R+
˜
Ui(pi, p−i) (32) Proof 5. Please refer to Appendix A.
Lemma 3. For any given p−i, the solution to(32) is found to be p+i = 1 ci+ Í j,i αjϕi(zi) −Õ j,i hji hii pj− σ2 i hii , (33) whereϕi(zi)= ln 21 hj jhi jpj hijzi+ Í k ,i, j hk jpk+σ2j ! hijzi+ Í t ,i ht jpt+σ2j . 235
Proof 6. Please refer to Appendix B. 4.2. Analysis of the Equilibria
The existence of the NE points are now studied under the assumption that condition (22) holds.
Proposition 4. The game G2admits a nonempty set of NE points,
240
which can be obtained by iteratively updating the transmit pow-ers according to(31) from any starting point.
Proof 7. In a game, the existence of a NE is guaranteed under the following assumptions [23]:
• The users’ feasible action sets Pi(p−i) are non empty,
245
closed, convex, and contained in some compact set Ci for all p−i ∈ Pi(p−i)= Îj,iPj.
• The set Pi(p−i) vary continuously with pi.
• Each user’s payoff function ˜Ui(pi, p−i) is quasi-concave in pi ∀p−i ∈ P−i.
250
In our setting, if condition(22) holds, then:
• The sets Pi(p−i) are non-empty, closed convex, and bounded for every p−i.
• Each of the sets Pi(p−i) varies continuously with p−i since the SINR constraint in P−k is itself continuous in
255
Pi(p−i).
• ˜Uicol(pi, p−i) is a concave function and consequently is a quasi-concave function.
Therefore, G2admits to a nonempty set of NE points.
4.3. Distributed Implementation
260
The BRD Algorithm is used to determine the NE of the game. Similar to G1, by exchanging the information about the interference plus noise of each user, (24) and (32) can be ob-tained. BRD Algorithm guarantees the convergence to a NE and the distributed implementation. However, in such a game,
265
the NE may not be unique and it depends on the starting points (i.e., zi) of the algorithm.
4.4. Assigning Approximation Points
The utility function of player i, as well as the convergence of the game, depend on the initial point zi(i ∈ M) for the approx-imation process. The question on how to choose an efficient initial point must be considered. For a given p−i, the solution of (32) is an increasing function with the variable zi ≥ 0. The maximum value of p+(zi) is p+ pimax = 1 ci+ Í j,i αjϕi pmaxi −Õ j,i hji hii pj− σ2 i hii . (34)
The individual rate of each player is an increasing function with its own transmission power, but a decreasing function with
oth-270
ers’ powers. For the network in the low-SINR region, the in-dividual rate of each player hence will be worse if the initial points are the maximum power. Instead of selecting the maxi-mum power as the initial point for the approximation process, we claim that the zero points (i.e., zi = 0 ∀i ∈ M) will be better
275
for the network performance in term of the aggregate rate.
5. The Concave-Potential Game Approximation
For a large network, the distributed implementations by it-eratively solving M coupled problems (i.e., the problem (7)) in G1 and G2 are more complex. In such a case, we adopt a hy-brid approach, which is referred as the concave-potential game, to approximate the utility function of G. In particular, we lin-earizes the convex part in (13) as follows:
¯
gi(pi, p−i, zi) ≈ ˆg(zi, p−i)+5pig(pˆ i, p−i)
T|
pi=zi(pi− zi). (35)
The modified utility function is then given by ¯
Ui(pi, p−i, zi)= ˆfi(pi)+ ¯gi(pi, p−i, zi), (36) where zi ∈ {Pi(p−i)}i is the initial point of the player i for the approximation process.
We formulate the concave-potential game as:
G3 , {M, {Pi(p−i)}i, { ¯Ui(pi, p−i)}i}. (37) For game G3, the BR ˜Bi(p−i) of user i is defined as
¯ Bi(p−i) ∆ = arg max pi∈ Pi(p−i) ¯ Ui(pi, p−i). (38)
5.1. Properties of the Concave-potential Game
We obtain some properties of the game G3as follows: Lemma 4. If condition (22) holds then ¯Bi(p−i) takes the form
¯
B−i(p−i)= min{pimax, max{p?i, pt ari }}, (39) wherein pt ari (p−i) is defined as in (24) and
p?i = arg max∆ pk∈R+
¯
Ui(pi, p−i). (40)
Proof 8. Please refer to Appendix A.
280
Lemma 5. For a given p−i, the solution to(40) is found to be
p?i = 1 ci+ Í j,i αj hi j ln(2)hi jzi+ Í k ,i, j hk jpj+σ2j ! . (41)
Proof 9. Please refer to Appendix C. 5.2. Analysis of the Equilibria
The existence of the NE points of are now studied under the assumption that condition (22) holds.
Proposition 5. The game G3admits a nonempty set of NE points,
285
which can be obtained by iteratively updating the transmit pow-ers according to(39) from any starting point.
Proof 10. In our setting, if condition (22) holds, then:
• The sets Pi(p−i) are non-empty, closed convex, and bounded for every p−i.
290
• Each of the sets Pi(p−i) varies continuously with p−i since the SINR constraint in P−k is itself continuous in Pi(p−i).
• ˜Ui(pi, p−i) is a concave function and thus is a quasi-concave function.
295
Therefore, the game admits to a nonempty set of NE points. The proof of the uniqueness of the NE builds upon the standard function framework [24], which states that a non-cooperative game admits a unique NE (reachable by iteratively computing the players’ best-responses) provided that the game admits at
300
least one equilibrium and the best-response function is a stan-dard function. The stanstan-dard function is defined as follows:
Definition 1. A function l(p) is standard if for all p ≥ 0, the following properties are satisfied:
1. Positivity: l(p) > 0,
305
2. Monotonicity: If p ≥ p0then l(p) ≥ l(p0), 3. Scalability: For all > 1, l(p) > l( p).
Proposition 6. The game G3admits a unique NE point, which can be obtained by iteratively updating the transmit powers ac-cording to(39) from any starting point.
310
Proof 11. From Proposition 5, the game G3admits to a nonempty set of NE point. We then prove that the best-response func-tion(39) is a standard function. The proof is presented in Ap-pendix D.
5.3. Distributed Implementation
315
The BRD algorithm is then adopted to determine the NE point of G3. Similar to the game G1, by exchanging the in-formation about the interference between users, (24) and (40) can be obtained. Algorithm 1 guarantees the convergence to a unique NE and the distributed implementation.
320
5.4. Assigning Approximation Points
The utility function of each player in game G3depends on its initial point. The question on how to choose an efficient initial point must be considered. We observed that the value of p?(zi) in (41) does not depends on the direct channel hii. In
325
other words, p?(zi) only depends on the interference plus noise of other users and the initial point. Similar to game G2, we claim that the minimum points (i.e., all initial points are zero) will be better for the aggregate rate of the network.
6. Numerical results
330
To validate the performance of collaborative power control in the wireless interference networks, we consider a network with M = 3 users at f = 2.4 GHz. Two scenarios are consid-ered: i) the network in the high-SINR region (i.e., the users are far apart), and ii) the network in the low-SINR region
(other-335
wise). For comparison purposes, we use the rate of each user and the aggregate rate of the network obtained by considering the collaborative power control, the distributed power control with BRD algorithm [9, 20], and the centralized power control with branch-and-bound algorithm [17]. These algorithms will
340
stop if the total tolerance between two consecutive power allo-cations (or between the upper-bound and the lower-bound of the weighted sum-rate) is smaller than = 10−6. Other parameters are in Table 1.
6.1. The Potential Game Approximation
345
First, we consider the wireless interference network in the high-SINR region. The potentialized game approach is adopted to determine the optimal collaborative power allocation. The collaboration factors are αi = 1/3 (i = 1, 2, 3). Figure 3a shows
1 5 10 15 20 25 30 35 0.5
1 1.5 2
Collaborative Power Allocation Distributed Power Allocation Centralized Power Allocation
(a) Individual rate
1 5 10 15 20 25 30 35 3.2 3.25 3.3 3.35 3.4 3.45 3.5 3.55 3.6 (b) Sum-rate 3.3 0.1 3.35 0.2 3.4 0.3 0.4 0.1 0.2 0.3 0.5 0.4 0.5
(c) sum-rate varying collaboration factors.
Figure 3: The performances of the collaborative power control based on the potentialized game approach, the distributed power control, and the centralized power
control for the wireless interference network with high SINR region where (a, b) αi = 1/3 ∀i = 1, 2, 3, and (c) for varying collaboration factors.
1 10 20 30 40 50 60 70 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Collaborative Power Allocation Distributed Power Allocation
Centralized Power Allocation
(a) Individual rate
1 10 20 30 40 50 60 70 1.4 1.6 1.8 2 2.2 2.4 2.6 (b) Sum-rate 1.5 0.5 1.7 1.9 0.4 2.1 2.2 0.5 0.3 0.4 0.3 0.2 0.2 0.1 0.1
(c) sum-rate varying collaboration factors.
Figure 4: The performance of the collaborative power control based on the potentialized game approach, the distributed power control, and the centralized power
control for the wireless interference network with low SINR region where (a, b) αi= 1/3 ∀i = 1, 2, 3, and (c) for varying collaboration factors.
Table 1: Wireless network simulation parameters
Parameter Value
Cell size (rectangular) 1 km
Spectral noise density (200C) - 174 dBm/Hz Maximum transmit power pmax = 20 dBm SINR constrain γit ar= 5 dB (i = 1, 2, 3)
Cost factor ci = 0.01 (i = 1, 2, 3)
the rates, which are obtained by the collaborative power con-trol paradigm, are fairness than the distributed and the central-ized approaches. Specially, the fairness among users is quan-titatively evaluated by using the Jain’s fairness index [25]. For the power control problem, the fairness index in term of rate for the power control profile p is defined as:
f (p)= ÍM i=1ri(p) 2 MÍM i=1ri2(p) (42)
The index measures the equality of power allocation p as fol-lows: if all users get the same rate, i.e., riare all equal, then the fairness index is 1 and the system is 100% fair; if the disparity increases, then fairness decrease. For the collaborative power allocation, the fairness index is 0.989, which is much higher
350
than the distributed power allocation (0.893) and the central-ized power allocation (0.793).
Next, Figure 3b indicates that the sum-rate of the network by following the collaborative power control paradigm outper-forms the distributed one and is close to the centralized one.
355
The reason is that each player aims to improve not only its own rate (i.e., through the (approximated) performance metric) but also the others’ rate (i.e., through the (approximated) collabo-ration metric). Hence, the sum-rate of the network is higher than the one in distributed power control. Moreover, since the
360
game is an exact potential game with strictly concave poten-tial function, fewer iterations are required to converge to the NE point. We, therefore, conclude that the collaborative power control paradigm provides better fairness between users, higher performance and lower convergence time.
365
Figure 3c shows the influence of the collaboration factors to the network performance in term of the sum-rate. We fix the collaboration factor of user 1 (α1) and change it’s of other users while ensuring (20). We observe that the sum-rate of the collaborative power control is higher than the distributed one.
370
Moreover, it reaches its maximum if the collaboration factors are (0.5, 0.1, 0.5). We claim that if the network is in low high-SINR region, it should be better to set αismall. In addition, we conclude that the sum-rate of the system strongly depends on the collaboration factors.
375
0 25 50 75 100 125 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8
Figure 5: The sum-rate of the network which is obtained by employing the collaborative power allocation based on the concave game framework for some initial points zi.
6.2. The Concave Game Approximation
We next consider the wireless interference network in the low-SINR region. The concave game approach is adopted to determine the optimal collaborative power allocation. The col-laboration factors are αi = 1/3, (i = 1, 2, 3). In order to simplify
380
the problem, we suppose that the initial points for the approx-imation process in (28) are zero, i.e., zi = 0, ∀i. Figure 4a shows the rates, which are obtained by the collaborative, the distributed and the centralized power control approaches. The fairness index are 0.72, 0.84 and 0.67, respectively. It means
385
the collaborative one is more fair in term of rate than the central-ized one but less than the distributed one. For the initial points z1= 1.5935 dBm, z2= 1.8166 dBm, and z3= 1.2335 dBm, the fairness index of collaborative power control is f (p) = 0.92, which much higher than the other power control approaches.
390
Moreover, for both cases, the collaborative power control pro-vides a faster convergence rate than others approaches.
Next, Figure 4b indicates that the sum-rate of the network by following the collaborative power control outperforms the distributed one and is close to the centralized power control
395
one. Figure 4c then shows the impact of the collaboration fac-tor to the performance of the network by fixing the collabora-tion factor of user 1 (α1) and change the collaboration factors of other users. We observed that the sum-rate in the collaboration scheme is always higher than the distributed one and gets max
400
when α1 = 1/3. Therefore, similar to the collaborative power control based on the potential game approach, the sum-rate of the system strongly depends on the collaboration factors.
Next, we evaluate the influence of the initial points zi to the system performance. Figure 5 shows the sum-rate of the
405
system as well as the number of iterations to reach the NE for varying initial points. We observed that the best initial point in both terms of sum-rate and the convergence rate is pi = 0 ∀i. Moreover, the sum-rate in the collaborative power allocation is greater than or equal to the one in the distributed power
alloca-410
tion scheme. Interestingly, if the initial points are the maximum power or the NE of the distributed power allocation, the sum-rate in the collaborative power allocation scheme closes to the
one in the distribution power allocation case. We conclude that the selection of zero points for the approximation process will
415
be most beneficial when investigating the collaborative power allocation based on the concave game framework.
6.3. The Concave-Potential Game Approximation
Finally, for the network with a large number of users, we employed the collaborative power control based on the
concave-420
potential game approach. A wireless interference network with M= 5 users is considered, where the initial points for the ap-proximation process are (i) zero, and (ii) the maximum power, respectively. Figure 6 shows the power allocation strategy as well as the sum-rate of the network. The collaboration
fac-425
tors are αi= 1/5 ∀i ∈ M. We observe that, by considering the collaborative power control based on the concave-potential game approach, the obtained sum-rate is higher than the one obtained by the distributed power control and is close to the one obtained by the centralized power control. Moreover, its
430
convergence rate is much faster than otherwise. The reason is that we can directly find the best response through an analyti-cal solution (41). Also, we observed that, for the collaborative power control based on the concave-potential game approach, the selection of the minimum points for the approximation
pro-435
cess provides a better sum-rate. In contrast, the selection of the maximum power for the approximation process provides a better convergence rate.
Finally, we compare the convergence rate of the collabo-rative power control based on the concave-potential game
ap-440
proach with the conventional power control approaches. Ta-ble 2 shows the number of iterations which is used by the al-gorithms (BRD and branch-and-bound) to determine the power allocation. The maximum power is chosen as the initial point for the approximation process. We observe that the
collabora-445
tive power control outperforms the other approaches in term of the convergence rate. We conclude that, for the large network, it will be better to adopt the concave-game approximation ap-proach. Otherwise, we adopt the potential game approach for the network in the high-SINR region and the concave game
ap-450
proach for the network in the low-SINR region.
7. Conclusion
In this paper, we presenting through the example an ap-proach to model the collaborative paradigm between wireless systems that occupy the same spectrum bands. The issue in
455
collaborating the network users to manage power allocation in a wireless interference network is addressed. By collaborating with other users, each user maximizes not only its own perfor-mance but also others’ perforperfor-mance. A game theory framework is developed to determine the optimal power allocation.
Com-460
pared with existing work, the proposed collaborative paradigm provides better fairness between users, higher performance and lower convergence time.
Table 2: The average number of iterations which is used by BRD/branch-and-bound algorithm to determine the power allocation strategy in: (left) the distributed strategy based non-cooperative game approach, (middle) the collaborative strategy based on the concave-potential game approach, and (right) the centralized strategy based joint optimization.
M = 3 M = 5 M = 7 M = 9 Low Interference 29 9 35 46 25 61 113 45 133 225 77 203 High Interference 65 14 70 79 43 103 301 85 286 597 151 488 0 20 40 60 80 100 0 20 40 60 80 100
Collaborative power allocation Distributed power allocation
(a) Power allocation strategies
0 20 40 60 80 100 2.4 2.6 2.8 3 3.2 3.4 (b) Sum-rate
Figure 6: The collaborative power allocation strategy based on the concave-potential game framework and the distributed power allocation strategy for a wireless
interference network with M= 5 users.
Appendix A. Proof of Lemma 1
The first part of the Lemma easily follows from the SINR constraints (γi ≥γit ar) as pi ≥ γt ar i hii Õ j,i hjipj+ σi2 ! . (A.1)
Since pi ≤ pmaxi for all i ∈ M, then γt ar i hii Õ j,i hjipmaxj + σi2 ! ≥ γ t ar i hii Õ j,i hjipj+ σi2 ! . (A.2) Hence, if∀i ∈ M, (19) holds, then there always exists a power
465
pi ∈0, pmaxi such that γi ≥γit aris fulfilled.
Next, since the game is a potential game with a strictly con-cave potential function, it admits a unique maximizer pi ∈ R+. Accounting for the SINR constraint (2) and imposing (22) even-tually yields (20).
470
Appendix B. Proof of the Lemma 3 The utility function of player i is given by
˜
Ui(pi, p−i, zi)= f (pi, p−i)+ ˜gi(pi, p−i, zi). (B.1) Since ˜Ui(pi, zi) is a concave function with pi, there are unique maximizer point p∗i which is determined by
p+i = arg max∆ pk∈+ ˜ Ui(pi, p−i) (B.2) Thus, we have: ∂ ˆUi(pi, zi) ∂pi pi=p+i = 0 (B.3) ⇒ p+i = 1 ci−Í j,i αjϕi(zi) −Õ j,i hji hii pj− σ2 i hii (B.4)
Appendix C. Proof of the Lemma 5 The utility function of player i is given by
¯
Ui(pi, p−i, zi)= ˆf(pi)+ ˆg(zi)+ 5pig(zˆ i)
T(p i− zi)
(C.1) Since ¯Ui(pi, p−i, zi) is a concave function with pi, there are unique maximizer point p?i which is determined by
p?i = arg max∆ pk∈+ ¯ Ui(pi, p−i, zi) (C.2) ⇒p?i = 1 ci+ Í j,i αj hi j ln 2hi jzi+ Í k ,i, j hk jpk+σ2j ! . (C.3)
Appendix D. Proof of the Proposition 6
We prove that both the best response function meet the three requirements of a standard function. We first consider the
func-475
tion pt ari (pi) as following. 10
• Positively: pt ari (p−i)> 0 • Monotonicity: pt ar
i (p−i) is increasing in all {pj}j,i. • Scalability: take any ω > 1 then it holds
pt ari (ωp−i)= ω γt ar i hii Õ j,i hjipj+ σ2 i ω ! < ωpt ar i (p−i). (D.1) Thus, pt ari (pi) is a standard function. Next, we prove that p?i (pi) is a standard function as followings:
480
• Positively: p?i (p−i)> 0
• Monotonicity: p?i (p−i) is increasing in all {pj}j,i. If p+−i ≥ p++−i then
p?i p+−i = 1 ci+ Í j,i αj hi j ln 2hi jzi+ Í k ,i, j hk jp+k+σ 2 j ! > p? i p++−i . (D.2) • Scalability: take any ε > 1 then it holds
εp? i = 1 ci ε + Í j,i αj hi j ln 2εhi jzi+ Í k ,i, j εhk jpk+εσ2j ! > p∗ i(εp−i) (D.3) Since p?i (pi) is a standard functions and pimax does not depend on p−i, we conclude that the best response function
¯
B−i(p−i) is also a standard function with variable pi.
485
References
[1] N. Alliance, 5G white paper, Next Generation Mobile Networks, White paper.
URL https://www.ngmn.org/5g-white-paper [2] Darpa Spectrum Collaboration Challenge (SC2) (2016-2019). 490
URL https://spectrumcollaborationchallenge.com/ [3] I. F. Akyildiz, W.-Y. Lee, M. C. Vuran, S. Mohanty, Next
genera-tion/dynamic spectrum access/cognitive radio wireless networks: A sur-vey, Computer networks 50 (13) (2006) 2127–2159. doi:10.1016/j. comnet.2006.05.001.
495
[4] K. Gomadam, V. R. Cadambe, S. A. Jafar, A distributed numerical ap-proach to interference alignment and applications to wireless interference networks, IEEE Transactions on Information Theory 57 (6) (2011) 3309– 3322. doi:10.1109/TIT.2011.2142270.
[5] H. Hartenstein, L. P. Laberteaux, A tutorial survey on vehicular ad hoc 500
networks, IEEE Communications Magazine 46 (6) (2008) 164–171. doi: 10.1109/MCOM.2008.4539481.
[6] K. Doppler, M. Rinne, C. Wijting, C. B. Ribeiro, K. Hugl, Device-to-device communication as an underlay to lte-advanced networks, IEEE Communications Magazine 47 (12) (2009) 42–49. doi:10.1109/MCOM. 505
2009.5350367.
[7] S. Baks¸i, D. C. Popescu, Distributed power allocation for spectrum shar-ing in mutually interfershar-ing wireless systems, Physical Communication 22 (2017) 42–48. doi:10.1016/j.phycom.2016.12.003.
[8] G. J. Foschini, Z. Miljanic, A simple distributed autonomous power con-510
trol algorithm and its convergence, IEEE Transactions on Vehicular Tech-nology 42 (4) (1993) 641–646. doi:10.1109/25.260747.
[9] G. Scutari, S. Barbarossa, D. P. Palomar, Potential games: A framework for vector power control problems with coupled constraints, in: 2006 IEEE International Conference on Acoustics Speech and Signal Process-515
ing Proceedings, Vol. 4, 2006, pp. IV–IV. doi:10.1109/ICASSP.2006. 1660950.
[10] U. O. Candogan, I. Menache, A. Ozdaglar, P. A. Parrilo, Near-optimal power control in wireless networks: A potential game approach, in: 2010 Proceedings IEEE INFOCOM, 2010, pp. 1–9. doi:10.1109/INFCOM. 520
2010.5462017.
[11] A. Ghosh, L. Cottatellucci, E. Altman, Normalized nash equilibrium for power allocation in cognitive radio networks, IEEE Transactions on Cognitive Communications and Networking 1 (1) (2015) 86–99. doi: 10.1109/TCCN.2015.2496578.
525
[12] D. Goodman, N. Mandayam, Power control for wireless data, IEEE Per-sonal Communications 7 (2) (2000) 48–54. doi:10.1109/98.839331. [13] A. Zappone, E. Jorswieck, et al., Energy efficiency in wireless networks
via fractional programming theory, Foundations and Trends in Com-R
munications and Information Theory 11 (3-4) (2015) 185–396. doi: 530
10.1561/0100000088.
[14] A. Gjendemsjø, D. Gesbert, G. E. Øien, S. G. Kiani, Binary power control for sum rate maximization over multiple interfering links, IEEE
Trans-actions on Wireless Communications 7 (8) (2008) 3164–3173. doi:
10.1109/TWC.2008.070227. 535
[15] L. Venturino, N. Prasad, X. Wang, Coordinated scheduling and power allocation in downlink multicell ofdma networks, IEEE Transactions on Vehicular Technology 58 (6) (2009) 2835–2848. doi:10.1109/TVT. 2009.2013233.
[16] L. Zheng, C. W. Tan, Maximizing sum rates in cognitive radio networks: 540
Convex relaxation and global optimization algorithms, IEEE Journal on Selected Areas in Communications 32 (3) (2014) 667–680. doi:10. 1109/JSAC.2014.140324.
[17] P. C. Weeraddana, M. Codreanu, M. Latvaaho, A. Ephremides, C. Fis-chione, et al., Weighted sum-rate maximization in wireless networks: A 545
review, Foundations and Trends in Networking 6 (1–2) (2012) 1–163.R
doi:10.1561/1300000036.
[18] A. Zappone, Z. Chong, E. A. Jorswieck, S. Buzzi, Energy-aware compet-itive power control in relay-assisted interference wireless networks, IEEE Transactions on Wireless Communications 12 (4) (2013) 1860–1871. 550
[19] A. Zappone, L. Sanguinetti, G. Bacci, E. Jorswieck, M. Debbah,
Energy-efficient power control: A look at 5G wireless technologies, IEEE
Trans-actions on Signal Processing 64 (7) (2016) 1668–1683. doi:10.1109/ TSP.2015.2500200.
[20] C. U. Saraydar, N. B. Mandayam, D. J. Goodman, Efficient power control
555
via pricing in wireless data networks, IEEE Transactions on Communica-tions 50 (2) (2002) 291–303. doi:10.1109/26.983324.
[21] J. B. Rosen, Existence and uniqueness of equilibrium points for con-cave n-person games, Econometrica: Journal of the Econometric Society (1965) 520–534doi:10.2307/1911749.
560
[22] Q. Zhu, A lagrangian approach to constrained potential games: Theory and examples, in: 2008 47th IEEE Conference on Decision and Control, 2008, pp. 2420–2425. doi:10.1109/CDC.2008.4738596.
[23] G. Debreu, A social equilibrium existence theorem, Proceedings of the National Academy of Sciences 38 (10) (1952) 886–893. doi:10.1073/ 565
pnas.38.10.886.
[24] R. D. Yates, A framework for uplink power control in cellular radio sys-tems, IEEE Journal on Selected Areas in Communications 13 (7) (1995) 1341–1347. doi:10.1109/49.414651.
[25] R. Jain, D.-M. Chiu, W. R. Hawe, A quantitative measure of fairness and 570
discrimination for resource allocation in shared computer system, Vol. 38, Eastern Research Laboratory, Digital Equipment Corporation Hudson, MA, 1984.
