Distributed Power Allocation with Rate Constraints in Gaussian Parallel Interference Channels

0.49 MB·English

by  Jong-Shi Pang

Distributed Power Allo
ation with Rate
Constraints in Gaussian Parallel Interferen
e
Channels
1,∗ 2,♮,† 3,‡ 4
Jong-Shi Pang , Gesualdo S
utari , Fran
is
o Fa

hinei , and Chaoxiong Wang .
1 1
Email: jspanguiu
.edu, s
utariinfo
om.uniroma .it, fa

hineidis.uniroma .it, wang
rpi.edu.
1
Dept. of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign,
8
104 S. Mathews Ave., Urbana IL 61801, U.S.A..
0
0 2
2 Dept. INFOCOM, University of Rome (cid:16)La Sapienza", Via Eudossiana 18, 00184, Rome, Italy.
n 3
a Dept. of Computer and Systems S
ien
e (cid:16)Antonio Ruberti", University of Rome (cid:16)La Sapienza",
J Via Ariosto 25, 00185Rome, Italy.

8 4
1 Dept. of Mathemati
al S
ien
es, Rensselaer Polyte
hni
Institute, Troy, New York, 12180-3590,U.S.A..

♮
]
T Corresponding author
I
s. Submitted to IEEE Transa
tions on Information Theory, February 17, 2007.
c
[ Revised January 11, 2008.

2
v
2
6 Abstra
t
1
2 Thispaper
onsidersthe minimizationoftransmitpowerinGaussianparallelinterferen
e
hannels,
0
subje
t to a rate
onstraint for ea
h user. To derive de
entralized solutions that do not require
7
0 any
ooperationamongthe users, weformulate this power
ontrol problemasa (generalized)Nash
/
s equilibrium game. We obtain su(cid:30)
ient
onditions that guarantee the existen
e and nonemptiness
c
: of the solution set to our problem. Then, to
ompute the solutions of the game, we propose two
v
i
X distributed algorithmsbased on the single user water(cid:28)lling solution: The sequential and the simul-
r taneous iterativewater(cid:28)llingalgorithms,whereintheusersupdatetheirownstrategiessequentially
a
and simultaneously, respe
tively. We derive a uni(cid:28)ed set of su(cid:30)
ient
onditions that guaranteethe
uniqueness of the solution and global
onvergen
e of both algorithms. Our results are appli
able
to all pra
ti
al distributed multipoint-to-multipoint interferen
e systems, either wired or wireless,
where a quality of servi
e in terms of information rate must be guaranteed for ea
h link.
Index Terms: Gaussian parallel interferen
e
hannel, mutual information, game theory, general-
ized Nash equilibrium, spe
trum sharing, iterative water(cid:28)lling algorithm.
∗
Theworkofthisauthorisbasedonresear
hsupportedbytheNationalS
ien
eFoundationundergrantDMI-0516023.
†
Thework of this authoris based onresear
h partially supportedbytheSURFACEproje
t fundedbytheEuropean
CommunityunderContra
t IST-4-027187-STP-SURFACE, and bytheItalian Ministry of Universityand Resear
h.
‡
Theworkofthisauthorisbasedonresear
hsupportedbytheProgramPRIN-MIUR2005(cid:16)InnovativeProblemsand
Methods in Nonlinear Optimization".
1
1 Introdu
tion and Motivation
The interferen
e
hannel is a mathemati
al model relevant to many
ommuni
ation systems where
multiple un
ordinated links share a
ommon
ommuni
ation medium, su
h as wireless ad-ho
networks
orDigitalSubs
riber Lines(DSL).Inthispaper, wefo
usontheGaussianparallelinterferen
e
hannel.
A pragmati
approa
h that leads to an a
hievable region or inner bound of the
apa
ity region
is to restri
t the system to operate as a set of independent units, i.e., not allowing multiuser en
od-
ing/de
oding or the use of interferen
e
an
elation te
hniques. This a
hievable region is very relevant
in pra
ti
al systems with limitations on the de
oder
omplexity and simpli
ity of the system. With
thisassumption, multiuser interferen
e istreated asnoise and the transmission strategy for ea
h useris
simplyhispower allo
ation. Thesystemdesignredu
esthento (cid:28)ndingtheoptimum power distribution
for ea
h user over the parallel
hannels, a

ording to a spe
i(cid:28)ed performan
e metri
.
Within this
ontext, existing works [1℄-[13℄
onsidered the maximization of the information rates
of all the links, subje
t to transmit power and (possibly) mask
onstraints on ea
h link. In [1℄-[3℄,
a
entralized approa
h based on duality theory [14, 15℄ was proposed to
ompute, under te
hni
al

onditions,thelargesta
hievablerateregionofthesystem(i.e., thePareto-optimal setofthea
hievable
rates). In [4℄, su(cid:30)
ient
onditions for the optimal spe
trum sharing strategy maximizing the sum-rate
to be frequen
y division multiple a

ess (FDMA) were derived. However, the algorithms proposed
in [1℄-[4℄ are
omputationally expensive and
annot be implemented in a distributed way, require the
full-knowledge of the system parameters, and are not guaranteed to
onverge to the global optimal
solution.
Therefore, in[5℄-[13℄,usingagame-theory framework, theauthorsfo
usedondistributedalgorithms
with no
entralized
ontrol. In parti
ular, the rate maximization problem was formulated asa strategi
non-
ooperative game, where every link is a player that
ompetes against the others by
hoosing the
transmission strategy that maximizes its own information rate [5℄. Based on the
elebrated notion
of Nash Equilibrium (NE) in game theory [16℄, an equilibrium for the whole system is rea
hed when
every player's rea
tion is (cid:16)unilaterally optimal(cid:17), i.e., when, given the rival players'
urrent strategies,
any
hange in a player's own strategy would result in a rate loss. In [6℄-[13℄, alternative su(cid:30)
ient

onditions were derived that guarantee the uniqueness of the NE of the rate maximization game and
−
the
onvergen
eofalternativedistributedwater(cid:28)llingbasedalgorithms,eithersyn
hronous sequential
−
[6℄-[11℄ and simultaneous [12℄ or asyn
hronous [13℄.
Thegametheoreti
al formulationproposedinthe
itedpapers,isausefulapproa
h todevisetotally
distributed algorithms. However, due to possible asymmetries of the system and the inherent sel(cid:28)sh
nature of the optimization, the Nash equilibria of the rate maximization game in [5℄-[13℄ may lead
to ine(cid:30)
ient and unfair rate distributions among the links even when the game admits a unique NE.
This unfairness is due to the fa
t that, without any additional
onstraint, the optimal power allo
ation
2
orresponding to a NE of the rate maximization game is often the one that assigns high rates to the
users with the highest (equivalent)
hannels; whi
h strongly penalizes all the other users. As many
realisti

ommuni
ation systems require pres
ribed Quality of Servi
e (QoS) guarantees in terms of
a
hievable rate for ea
h user, the system design based on the game theoreti
formulation of the rate
maximization might not be adequate.
Toover
omethisproblem,inthispaperweintrodu
eanewdistributedsystemdesign,thattakesex-
pli
itlyintoa

ounttherate
onstraints. Morespe
i(cid:28)
ally,weproposeanovelstrategi
non-
ooperative
game, where every link is a player that
ompetes against the others by
hoosing the power allo
ation
over the parallel
hannels that attains the desired information rate, with the minimum transmit power.
We will refer to this new game as power minimization game. An equilibrium is a
hieved when every
user realizes that, given the
urrent power allo
ation of the others, any
hange in its own strategy
would result in an in
rease in transmit power. This equilibrium is referred to as Generalized Nash
1
Equilibrium (GNE) and the
orresponding game is
alled Generalized Nash Equilibrium Problem.
The game theoreti
al formulation proposed in this paper di(cid:27)ers signi(cid:28)
antly from the rate maxi-
mizationgamesstudied in[5℄-[13℄. Infa
t, di(cid:27)erently from thesereferen
es, where theusersareallowed
to
hoose their own strategies independently from ea
h other, in the power minimization game, the
rate
onstraints indu
e a
oupling among the players' admissible strategies, i.e., ea
h player's strategy
set depends on the
urrent strategies of all the other players. This
oupling makes the study of the
proposed game mu
h harder than that of the rate maximization game and no previous result in [5℄-[13℄

an be used. Re
ently, the
al
ulation of generalized Nash equilibria has been the subje
t of a renewed
attention also in the mathemati
al programming
ommunity, see for example [17℄-[21℄. Nevertheless,
in spite of several interesting advan
es [21℄, none of the game results in the literature are appli
able to
the power minimization game.
The main
ontributions of the paper are the following. We provide su(cid:30)
ient
onditions for the
nonemptiness and boundedness of the solution set of the generalized Nash problem. Interestingly,
these su(cid:30)
ient
onditions suggest a simple admission
ontrol pro
edure to guarantee the feasibility
of a given rate pro(cid:28)le of the users. Indeed, our existen
e proof uses an advan
ed degree-theoreti
result for a nonlinear
omplementarity problem in order to handle the unboundedness of the users'
rate
onstraints. We also derive
onditions for the uniqueness of the GNE. Interestingly, our su(cid:30)
ient

onditions be
ome also ne
essary in the
ase of one sub
hannel. To
ompute the generalized Nash
solutions, weproposetwo alternative totally distributedalgorithms basedonthesingleuserwater(cid:28)lling
solution: The sequential IWFA and thesimultaneous IWFA.The sequential IWFAisaninstan
e of the
Gauss-Seidel s
heme: The users update their own strategy sequentially, one after the other, a

ording
1
A

ording to re
ent use, we term generalized Nash equilibrium problem a Nash game where the feasible sets of the
players depend on the other players' strategy. Su
h kind of games have been
alled in various di(cid:27)erent ways in the
literature, for exampleso
ial equilibrium problems or justNash equilibrium problems.
3
to the single user water(cid:28)lling solution and treating the interferen
e generated by the others as additive
noise. The simultaneous IWFA is based on the Ja
obi s
heme: The users
hoose their own power
allo
ation simultaneously, still using the single user water(cid:28)lling solution. Interestingly, even though the
rate
onstraintsindu
ea
ouplingamong thefeasible strategiesofalltheusers,bothalgorithms arestill
totally distributed. In fa
t, ea
h user, to
ompute the water(cid:28)lling solution, only needs to measure the
power of the noise plus the interferen
e generated by the other users over ea
h sub
hannel. It turns out
that the
onditions for the uniqueness of the GNE are su(cid:30)
ient for the
onvergen
e of both algorithms.
Our
onvergen
e analysis is based on a nonlinear transformation that turns the generalized game in
the power variables into a standard game in the rate variables. Overall, this paper o(cid:27)ers two major

ontributions to the literature of game-theoreti
approa
hes to multiuser
ommuni
ation systems: (i)
a new non
ooperative game model is introdu
ed for the (cid:28)rst time that dire
tly addresses the issue of
QoS in su
h systems, and (ii) a new line of analysis is introdu
ed in the literature of distributed power
allo
ation that is expe
ted to be broadly appli
able for other game models.
The paper is organized as follows. Se
tion 2 gives the system model and formulates the power
minimization problem as a strategi
non-
ooperative game. Se
tion 3 provides the su(cid:30)
ient
onditions
for the existen
e and uniqueness of a GNE of the power minimization game. Se
tion 4
ontains the
des
ription of the distributed algorithms along with their
onvergen
e
onditions. Finally, Se
tion 5
draws the
on
lusions. Proofs of the results are given in the Appendi
es A(cid:21)F.
2 System Model and Problem Formulation
In this se
tion we
larify the assumptions and the
onstraints underlying the system model and we
formulate the optimization problem expli
itly.
2.1 System model
Q N Q
We
onsider a -user Gaussian -parallel interferen
e
hannel. In thismodel, there are transmitter-
re
eiver pairs, where ea
h transmitter wants to
ommuni
ate with its
orresponding re
eiver over a
N
set of parallel sub
hannels. These sub
hannels
an model either frequen
y-sele
tive or (cid:29)at-fading
time-sele
tive
hannels [9℄. Sin
e our goal is to (cid:28)nd distributed algorithms that do not require neither
a
entralized
ontrol nor a
oordination among the links, we fo
us on transmission te
hniques where no
interferen
e
an
elationisperformedandmultiuserinterferen
e istreatedasadditive
olorednoisefrom
ea
h re
eiver. Moreover, we assume perfe
t
hannel state information at both transmitter and re
eiver
2
side of ea
h link; ea
h re
eiver is also assumed to measure with no errors the power of the noise plus
N q
the overall interferen
e generated by the other users over the sub
hannels. For ea
h transmitter ,
2
Notethat ea
h user is only requiredto known its own
hannel, butnot the
hannels of the otherusers.
4
N
the total average transmit power over the sub
hannels is(in units of energy per transmitted symbol)
N
1
P = p (k),
q q
N (1)
k=1
X
p (k) q k
q
where denotes the power allo
ated by user over the sub
hannel .
−
Under these assumptions, invoking the
apa
ity expression for the single user Gaussian
hannel
− q
a
hievable using random Gaussian
odes from all the users the maximum information rate on link
3
for a spe
i(cid:28)
power allo
ation is [23℄
N
R (p ,p )= log(1+sinr (k)),
q q −q q
(2)
k=1
X
sinr (k) q k
q
with denoting the Signal-to-Interferen
e plus Noise Ratio (SINR) of link on the -th sub-

hannel: |H (k)|2p (k)
sinr (k) , qq q ,
q σ2(k)+ |H (k)|2p (k) (3)
q r6=q qr r
|H (k)|2 P q r σ2(k)
qr
where is the power gain of the
hannel between destination and sour
e ; q is the
k q p , (p (k))N
varian
e of Gaussian zero mean noise on sub
hannel of re
eiver ; and q q k=1 is the power
q N p , (p )
allo
ation strategy of user a
ross the sub
hannel, whereas −q r r6=q
ontains the strategies
of all the other users.
2.2 Game theoreti
formulation
We formulate the system design within the framework of game theory [25, 26℄, using as desirability

riterion the
on
ept of GNE, see for example [16, 27℄. Spe
i(cid:28)
ally, we
onsider a strategi
non-

ooperative game, in whi
h the players are the links and the payo(cid:27) fun
tions are the transmit powers
oftheusers: Ea
hplayer
ompetesagainsttheothersby
hoosingthepowerallo
ation(i.e.,itsstrategy)
that minimizes its own transmit power, given a
onstraint on the minimum a
hievable information rate
onthe link. A GNEof thegameisrea
hed whenea
h user, given thestrategy pro(cid:28)le oftheothers, does
not get any power de
rease by unilaterally
hanging its own strategy, still keeping the rate
onstraint
satis(cid:28)ed. Stated in mathemati
al terms, the game has the following stru
ture:
G = Ω,{P (p )} ,{P } ,
q −q q∈Ω q q∈Ω (4)
n o
Ω , {1,2,...,Q} P (p ) ⊆ RN
q −q +
where denotes the set of the a
tive links, is the set of admissible
p ∈ P (p ) q N , {1,...,N}
q q −q
power allo
ation strategies of user over the sub
hannels , de(cid:28)ned as
P (p ), x ∈ RN : R (x ,p ) ≥ R⋆ .
q −q q + q q −q q
(5)
(cid:8) (cid:9)
3
Observe that a GNE is obtained if ea
h user transmits using Gaussian signaling, with a proper power allo
ation.
However, generalized Nash equilibria a
hievable using non-Gaussian
odes may exist. In this paper, we fo
us only on
transmissions using Gaussian
odebooks.
5
R (p ,p ) R⋆
q q −q q
with given in (2), and denotes the minimum transmission rate required by ea
h user,
whi
h we assume positive without loss of generality. In the sequel we will make referen
e to the ve
tor
R⋆ , (R⋆)Q q
q q=1 as to the rate pro(cid:28)le. The payo(cid:27) fun
tion of the -th player is its own transmit power
P P (p )
q q −q
, given in (1). Observe that, be
ause of the rate
onstraints, the set of feasible strategies
q p
−q
of ea
h player depends on the power allo
ations of all the other users.
q
The optimal strategy for the -th player, given the power allo
ation of the others, is then the
solution to the following minimization problem
N
minimize p (k)
q
pq ,
Xk=1 (6)
subject to p ∈ P (p )
q q −q
P (p ) q p ,
q −q q
where is given in (5). Note that, for ea
h , the minimum in (6) is taken over for a (cid:28)xed
p . p ,
−q −q
but arbitrary Interestingly, given the solution of (6)
an be obtained in (cid:16)
losed(cid:17) form via
the solution of a singly-
onstrained optimization problem; see [28℄ for an algorithm to implement this
solution in pra
ti
e.
p , p⋆ = {p⋆(k)}N
Lemma 1 Forany(cid:28)xed andnonnegative −q theoptimalsolution q q k=1 oftheoptimization
problem (6) exists and is unique. Furthermore,
p⋆ = WF (p ,...,p ,p ,...,p )= WF (p ) ,
q q 1 q−1 q+1 Q q −q (7)
WF (·)
q
where the water(cid:28)lling operator is de(cid:28)ned as
+
σ2(k)+ |H (k)|2p (k)
q qr r
[WF (p )] , λ − r6=q , k ∈ N,
q −q k  q X|H (k)|2  (8)
qq
 
(x)+ , max(0,x) λ R (p⋆,p ) = R⋆
q q q −q q
with and the water-level
hosen to satisfy the rate
onstraint ,
R (p ,p )
q q −q
with given in (2).
G ( )
The solutions of the game in 4 , if they exist, are the Generalized Nash Equilibria, formally
de(cid:28)ned as follows.
p⋆ = (p⋆)Q G
De(cid:28)nition 2 A feasible strategy pro(cid:28)le q q=1 is a GNE of the game if
N N
p⋆(k) ≤ p (k), ∀p ∈ P (p⋆ ), ∀q ∈ Ω.
q q q q −q
(9)
k=1 k=1
X X
A

ording to Lemma 1, all the Generalized Nash Equilibria of the game must satisfy the
ondition
expressed by the following Corollary.
6
p⋆ = (p⋆)Q G
Corollary 3 A feasible strategy pro(cid:28)le q q=1 is a GNE of the game if and only if it satis(cid:28)es
the following system of nonlinear equations
p⋆ = WF p⋆,...,p⋆ ,p⋆ ,...,p⋆ , ∀q ∈ Ω,
q q 1 q−1 q+1 Q (10)
(cid:16) (cid:17)
WF (·)
q
with de(cid:28)ned in (8).
Given the nonlinear system of equations (10), the fundamental questions we want an answer to are:
i) Does a solution exist, for any given users' rate pro(cid:28)le? ii) If a solution exists, is it unique? iii) How

an su
h a solution be rea
hed in a totally distributed way?
An answer to the above questions is given in the forth
oming se
tions.
3 Existen
e and Uniqueness of a Generalized Nash Equilibrium
In this se
tion we (cid:28)rst provide su(cid:30)
ient
onditions for the existen
e of a nonempty and bounded
solution set of the Nash equilibrium problem (4). Then, we fo
us on the uniqueness of the equilibrium.
3.1 Existen
e of a generalized Nash equilibrium
R⋆ = (R⋆)Q k ∈ N Z (R⋆) ∈ RQ×Q
Given the rate pro(cid:28)le q q=1, de(cid:28)ne, for ea
h , the matrix k as
|H11(k)|2 −(eR⋆1 −1)|H12(k)|2 ··· −(eR⋆1 −1) |H1Q(k)|2
 
−(eR⋆2 −1) |H21(k)|2 |H22(k)|2 ··· −(eR⋆2 −1) |H2Q(k)|2
Z (R⋆) ,   .
k  
  (11)
 .. .. .. .. 
 . . . . 
 
 −(eR⋆Q −1) |HQ1(k)|2 −(eR⋆Q −1) |HQ2(k)|2 ··· |HQQ(k)|2 
 
 
We also need the de(cid:28)nition of P-matrix, as given next.
A ∈ RN×N
De(cid:28)nition 4 A matrix is
alled Z-matrix if its o(cid:27)-diagonal entries are all non- positive.
A ∈ RN×N A
A matrix is
alled P-matrix if every prin
ipal minor of is positive.
Many equivalent
hara
terizations for a P-matrix
an be given. The interested reader is referred to
[31, 32℄ for more details. Here we note only that any positive de(cid:28)nite matrix is a P-matrix, but the
reverse does not hold.
G
Su(cid:30)
ient
onditions for the nonemptiness of a bounded solution set for the game are given in
the following theorem.
G R⋆ = (R⋆)Q > 0
Theorem 5 The game with rate pro(cid:28)le q q=1 admits a nonempty and bounded solution
Z (R⋆) k ∈ N Z (R⋆) p⋆ =
k k
set if is a P-matrix, for all , with de(cid:28)ned in (11). Moreover, any GNE
7
(p∗)Q
q q=1 is su
h that
p∗(k) p (k) σ2(k)(eR⋆1 −1)
1 1 1
 ...  ≤  ...  , (Zk(R⋆))−1 ... , k ∈ N. (12)
     
 p∗(k)   p (k)   σ2(k)(eR⋆Q −1) 
 Q   Q   Q 
     
Proof. See Appendix A.
A more general (but less easy to
he
k) result on the existen
e of a bounded solution set for the
G
game is given by Theorem 23 in Appendix A.
We now provide alternative su(cid:30)
ient
onditions for Theorem 5 in terms of a single matrix. To this
end, we (cid:28)rst introdu
e the following matrix
1 −(eR⋆1 −1)βmax ··· −(eR⋆1 −1)βmax
12 1Q
 
 −(eR⋆2 −1)β2m1ax 1 ··· −(eR⋆2 −1)β2mQax 
Zmax(R⋆) ,   ,
 
  (13)
 
 ... ... ... ... 
 
 −(eR⋆Q −1)βmax −(eR⋆Q −1)βmax ··· 1 
 Q1 Q2 
 
where |H (k)|2
βmax , max qr , ∀r 6= q, q ∈ Ω.
qr k∈N |H (k)|2 (14)
rr
eR⋆−1 Q q eR⋆q −1, q = 1,...,Q
We also denote by the -ve
tor with -th
omponent for . Then, we have
the following
orollary.
Zmax(R⋆) {Z (R⋆)}
k
Corollary 6 If in (13) is a P-matrix, then all the matri
es de(cid:28)ned in (11) are
p⋆ = (p⋆)Q G
P-matri
es. Moreover, any GNE 1 q=1 of the game satis(cid:28)es
maxσ2(k)
r
p∗(k) ≤ p (k) = r∈Ω (Zmax(R⋆))−1 eR⋆ −1 , ∀q ∈Ω, ∀k ∈ N.
q q  |Hqq(k)|2  q (15)
h (cid:16) (cid:17)i
 
Proof. See Appendix B.
To give additional insight into the physi
al interpretation of the existen
e
onditions of a GNE,
|H (k)|2
qr
we make expli
it the dependen
e of ea
h
hannel (power) gain on its own sour
e-destination
d |H (k)|2 = |H (k)|2dγ , γ
qr qr qr qr
distan
e by introdu
ing the normalized
hannel gain where is the path
loss exponent. We have the following
orollary.
{Z (R⋆)}
k
Corollary 7 Su(cid:30)
ient
onditions for the matri
es de(cid:28)ned in (11) to be P-matri
es are:
|H (k)|2dγ 1
qr qq
< , ∀r ∈ Ω, ∀k ∈ N.
|Hqq(k)|2dqγr eR⋆q −1 (16)
r6=q
X
8
Proof. The proof
omes dire
tly from the su(cid:30)
ien
y of the diagonally dominan
e property [32, De(cid:28)-
Z (R⋆)
k
nition 2.2.19℄ for the matri
es in (11) to be P-matri
es [31, Theorem 6.2.3℄
Remark 8 A physi
al interpretation of the
onditions in Theorem 5 (or Corollary 7) is the following.
G
Given the set of
hannels and the rate
onstraints, a GNE of is guaranteed to exist if multiuser
interferen
e is (cid:16)su(cid:30)
iently small(cid:17) (e.g., the links are su(cid:30)
iently far apart). In fa
t, from (16), whi
h
quanti(cid:28)es the
on
ept of small interferen
e, one infers that, for any (cid:28)xed set of (normalized)
hannels
and rate
onstraints, there exists a minimum distan
e beyond whi
h an equilibrium exists,
orrespond-
ing to the maximum level of interferen
e that may be tolerated from ea
h user. The amount of su
h
a tolerable multiuser interferen
e depends on the rate
onstraints: the larger the required rate from
ea
h user, the lower the level of interferen
e guaranteeing the existen
e of a solution. The reason why
G
an equilibrium of the game might not exist for any given set of
hannels and rate
onstraints, is
that the multiuser system we
onsider is interferen
e limited, and thus not every QoS requirement is
G
guaranteed to be feasible. In fa
t, in the game , ea
h user a
ts to in
rease the transmit power to
satisfy its own rate
onstraint; whi
h leads to an in
rease of the interferen
e against the users. It turns
out that, in
reasing the transmit power of all the users does not guarantee that an equilibrium
ould
exist for any given rate pro(cid:28)le.
Observe that
onditions in Theorem 5 also provide a simple admission
ontrol pro
edure to
he
k
if a set of rate
onstraints is feasible: under these
onditions indeed, one
an always (cid:28)nd a (cid:28)nite power
budget for all the users su
h that there exists a GNE where all the rate
onstraints are satis(cid:28)ed.
3.2 Uniqueness of the Generalized Nash Equilibrium
G
Before providing
onditions for the uniqueness of the GNE of the game , we introdu
e the following
R⋆ = (R⋆)Q > 0, B(R⋆) ∈ RQ×Q
intermediate de(cid:28)nitions. For any given rate pro(cid:28)le q q=1 let be de(cid:28)ned
as e−R⋆q, q = r,
B(R⋆) ≡ if
qr  −eR⋆q βmax, (17)
 qr
(cid:2) (cid:3) otherwise,
where βmax , max |Hqr(k)|2σr2(k)+b r′6=r|Hrr′(k)|2pr′(k) ,
qr k∈N |Hrr(k)|2 P σq2(k) ! (18)
p (k) b χ ρ,
r′
with de(cid:28)ned in (12). We also introdu
e and de(cid:28)ned respe
tively as
χ , 1−max eR⋆q −1 βmax ,
q∈Ω  qr  (19)
(cid:16) (cid:17)Xr6=q
 
βmax
qr
with given in (14), and
eR⋆max −1
ρ, ,
eR⋆min −1 (20)
9
with
R⋆ , maxR⋆, R⋆ , minR⋆.
max q min q
q∈Ω and q∈Ω (21)
G
Su(cid:30)
ient
onditionsfortheuniquenessoftheGNEofthegame aregiveninthefollowingtheorem.
G R⋆ = (R⋆)Q > 0,
Theorem 9 Given the game with a rate pro(cid:28)le q q=1 assume that the
onditions of
B(R⋆) G
Theorem 5 are satis(cid:28)ed. If, in addition, in (17) is a P-matrix, then the game admits a unique
GNE.
Proof. See Appendix D.
Morestringentbutmoreintuitive
onditionsfortheuniquenessoftheGNEaregiveninthefollowing

orollary.
G R⋆ = (R⋆)Q > 0,
Corollary 10 Given the game with rate pro(cid:28)le q q=1 assume that
0 < χ < 1,
(22)
G χ
so that a GNE for the game is guaranteed to exist, with de(cid:28)ned in (19). Then, the GNE is unique
if the following
onditions hold true
σ2(k) σ2 (k) ρ 1
βmax max r + max max r′ −1 < , ∀q ∈ Ω,
r6=q qr (cid:26)(cid:18)k∈N σq2(k)(cid:19) (cid:20)r′∈Ω (cid:18)k∈N σq2(k)(cid:19)(cid:21) (cid:18)χ (cid:19)(cid:27) e2R⋆q (23)
X
ρ
with de(cid:28)ned in (20).
σ2(k) = σ2(n), ∀r,q ∈ Ω ∀k,n ∈ N,
r q
In parti
ular, when and
onditions (23) be
ome
|H (k)|2 dγ γ
qr rr
max < , ∀q ∈ Ω,
r6=qk∈N (cid:26)|H¯rr(k)|2(cid:27) dqγr eR⋆q −1 (24)
X
with max e−R⋆q −e−2R⋆q
γ , q∈Ω < 1.
eR⋆max −1 (cid:8) (cid:9) (25)
+max e−R⋆q −e−2R⋆q
eR⋆min −1 q∈Ω
(cid:8) (cid:9)
Proof. See Appendix E.
3.3 On the
onditions for existen
e and uniqueness of the GNE
It is natural to ask whether the su(cid:30)
ient
onditions as given by Theorem 5 (or the more general ones
given by Theorem 23 in Appendix A) are tight. In the next proposition, we show that these
onditions
N = 1
be
ome indeed ne
essary in the spe
ial
ase of sub
hannel.
R⋆ = (R⋆)Q ,
Proposition 11 Given the rate pro(cid:28)le q q=1 the following statements are equivalent for the
G N = 1
4
game when :
4 N =1 pq(k)=pq |Hrq(k)|2 =|Hrq|2
Inthe
aseof ,thepowerallo
ation ofea
huser,the
hannelgains andthenoise
σq2(k)=σq2 k Zk(R⋆)=Z(R⋆) |Hrq(k)|2
varian
es are independentonindex . Matrix isde(cid:28)nedas in (11), whereea
h is
|Hrq|2.
repla
ed by
10