Optimal Power Control in Rayleigh-fadingHeterogeneous Networks

Chee Wei TanCity University of Hong Kong

Abstract—Heterogeneous wireless networks employ varyingdegrees of network coverage using power control in a multi-tierconfiguration, where low-power femtocells are used to enhanceperformance, e.g., optimize outage probability. We study theworst outage probability problem under Rayleigh fading. Asa by-product, we solve an open problem of convergence for apreviously proposed algorithm in the interference-limited case.We then address a total power minimization problem with outagespecification constraints and its feasibility condition. We proposea dynamic algorithm that adapts the outage probability specifi-cation in a heterogeneous network to minimize the total energyconsumption and simultaneously guarantees all the femtocellusers a min-max fairness in terms of the worst outage probability.

Index Terms— Optimization, nonnegative matrix theory, outageprobability, power control, femtocell networks.

I. INTRODUCTION

Conventional wireless cellular networks are designed toprovide network coverage over large areas and support manyusers. Most recently, 3GPP LTE-advanced investigated hetero-geneous network deployments to improve system performanceas well as effectively enhance network coverage, especiallyfor in-building coverage [1], [2]. Heterogeneous networksuse a mix of higher-tier macrocells to extend network reachand lower-tier femtocells to enhance performance, e.g., opti-mize outage probability, within the same frequency band [1],[2]. However, interference management is required to ensurethat the heterogeneous users control their transmit powerto mitigate performance loss due to mutual interference. Toenhance decentralized deployment, users also need to adapttheir transmit power with minimal signaling overhead [1], [2].

Outage probability is an important performance parame-ter for reliable wireless communication. A link outage isdeclared when the received Signal-to-Interference-and-NoiseRatio (SINR) falls below a given threshold that is often com-puted from the quality of service requirement. The statisticsof the SINR, and consequently that of the outage probability,are also dependent on other network parameters, includingimperfections due to the channel statistical fading, the additivebackground noise, user mobility and the dynamics of powercontrol updates. In this paper, we focus on the short-termchannel fading, which can be modeled by either a Rayleigh,a Ricean or a Nakagami distribution depending on the envi-ronment under consideration [3]. In particular, we focus onRayleigh fading that is relevant to in-building coverage modeland heavily built-up urban environments [1], [3].

In [4], the authors proposed tracking Rayleigh-fading fluc-tuation to reduce outage in a macrocell network. In [5], algo-

rithms that adapt SINR requirements for utility maximizationwere studied for a macro-femtocell network. The authors in [6]studied subcarrier and power control to minimize outage in anOFDMA cellular network. The authors in [7] studied the worstoutage probability problem and the total power minimizationproblem in interference-limited Rayleigh-fading networks. Theauthors in [8] proposed an iterative algorithm for the totalpower minimization problem, but left its feasibility issue open,which we address in this paper using the nonlinear Perron-Frobenius theory in [9], [10], [11] and nonnegative matrixtheory [12]. Interestingly, we also show that the worst outageprobability problem, similar to its certainty-equivalent margin(CEM) counterpart (also known as the max-min weightedSINR problem), is a nonlinear eigenvalue problem, whoseoptimal value and optimal solution are associated with aPerron eigenvalue and eigenvector, respectively.

It is imperative to design heterogeneous networks that arecapable of adapting their behavior to be both spectral andenergy efficient [1]. For this purpose, we propose a dynamicpower control algorithm that allows each femtocell user toadapt its outage probability specification to minimize the totalenergy consumption in the system and guarantees a min-max fairness in terms of worst outage probability to all thefemtocell users. This allows users to adapt their performancequality in a dynamic environment without the need of acentralized admission control mechanism (which is often notpractically feasible due to the ad-hoc architecture).

Overall, the contributions of the paper are as follows:1) We solve analytically the worst outage probability min-

imization problem subject to a total power constraint(for downlink scenario) or individual power constraints(for uplink scenario). We propose a fast algorithm tocompute the solution. As a by-product, we solve anopen problem of convergence for a previously proposedalgorithm in [7] for the interference-limited case.

2) We establish a tight relationship between the worstoutage probability problem and its certainty-equivalentmargin counterpart, and utilize the connection to finduseful bounds and the convergence rate of our algorithm.

3) We characterize analytically the feasibility conditionof the total power minimization problem with bothoutage specification and individual power constraints,and provide feasibility bounds based on the problemparameters, e.g., SINR thresholds, the outage thresholds.

4) We propose a dynamic algorithm for the graceful han-dling of infeasibility in a femto-macro cell network.

This paper was presented as part of the main technical program at IEEE INFOCOM 2011

978-1-4244-9921-2/11/$26.00 ©2011 IEEE 2552

The algorithm optimizes the overall energy consumptionbased on adaptive outage probability specification thatguarantees the worst outage probability to all the users.

This paper is organized as follows. We introduce the systemmodel in Section II. In Section III, we solve the problemof minimizing the worst outage probability analytically andpropose a fast algorithm. In Section IV, we study the feasi-bility condition of a total power minimization problem, andpropose a dynamic power control algorithm that adapts theoutage probability specification to minimize the total energy.In Section V, we illustrate the numerical performance of ouralgorithms. We conclude the paper in Section VI.

The following notation is used. Boldface uppercase lettersdenote matrices, boldface lowercase letters denote columnvectors, italics denote scalars, and u ≥ v denotes compo-nentwise inequality between vectors u and v. We also let(By)l denote the lth element of By. Let x/y denote thevector [x1/y1, . . . , xL/yL]

>. We write B ≥ F if Bij ≥ Fij

for all i, j. The Perron-Frobenius eigenvalue of a nonnegativematrix F is denoted as ρ(F), and the Perron (right) andleft eigenvector of F associated with ρ(F) are denoted byx(F) ≥ 0 and y(F) ≥ 0 (or, simply x and y, when thecontext is clear) respectively. Recall that the Perron-Frobeniuseigenvalue of F is the eigenvalue with the largest absolutevalue. Assume that F is an irreducible nonnegative matrix.Then ρ(F) is simple and positive, and x(F),y(F) > 0 [13].The super-script (·)> denotes transpose. We denote el as thelth unit coordinate vector and I as the identity matrix. For anyvector γ = (γ1, . . . , γL)> ∈ RL, let eγ = (eγ1 , . . . , eγL)>.

II. SYSTEM MODEL

Consider a multiuser communication system with L users(logical transmitter/receiver pairs) sharing a common fre-quency. Each user employs a singe-user decoder, i.e., treatinginterference as additive Gaussian noise, and has perfect chan-nel state information at the receiver. Our system with L userscan be modeled by a Gaussian interference channel havingthe baseband signal model yl = hllxl +

∑j 6=l hljxj + zl,

where yl ∈ C1×1 is the received signal of the lth user,

hlj ∈ C1×1 is the channel coefficient between the transmitterof the jth user and the receiver of the lth user, x ∈ CN×1 isthe transmitted (information carrying) signal vector, and zl’sare the i.i.d. additive complex Gaussian noise coefficient withvariance nl/2 on each of its real and imaginary components(thus nl being the noise power at the lth receiver). At eachtransmitter, the signal is constrained by an average powerconstraint, i.e., E[|xl|2] = pl, which we assume to be upperbounded by p for all l. The vector (p1, . . . , pL)

>is the transmit

power vector, which is the optimization variable of interest inthis paper.

Under Rayleigh fading, the power received from the jthtransmitter at lth receiver is given by GljRljpj where Gljrepresents the nonnegative path gain between the jth trans-mitter and the lth receiver (it may also encompass antennagain and coding gain) that is often modeled as proportionalto d−γlj , where dlj denotes distance, γ is the power fall-off

factor, and Rlj models Rayleigh fading and are independentand exponentially distributed with unit mean. The distributionof the received power from the jth transmitter at the l receiveris then exponential with mean value E[GljRljpj ] = Gljpj .

Next, we define the nonnegative matrix F with the entries:

Flj =

{0, if l = jGljGll

, if l 6= j(1)

and

v =(n1

G11,n2

G22, . . . ,

nLGLL

)>. (2)

Moreover, we assume that F is irreducible, i.e., each link hasat least an interferer.

We assume that the SINR for the lth receiver, e.g., a linearmatched-filter receiver, is given by [7]:

SINRl(p) =Rllpl∑

j 6=l

FljRljpj + vl. (3)

Now, (3) is a random variable that depends on Rayleighfading. The transmission from the lth transmitter to its receiveris successful if SINRl(p) ≥ βl (zero-outage), where βl is agiven threshold for reliable communication. An outage occursat the lth receiver when SINRl(p) < βl, and we denote thisoutage probability by P (SINRl(p) < βl).

Power constraint is a critical design parameter in a wirelessnetwork [3]. We focus on the power constraint P being eithera total power constraint or individual power constraints, i.e.,

P = {p | 1>

p ≤ P} or P = {p | pl ≤ p ∀ l}. (4)

III. WORST OUTAGE PROBABILITY MINIMIZATION

The problem of minimizing the worst outage probability canbe formulated as

minimize maxl

P (SINRl(p) < βl)

subject to p ∈ P,variables: p.

(5)

Assuming independent Rayleigh fading at all signals, theoutage probability of the lth user can be given analytically by[7]:1

P (SINRl(p) < βl) = 1− e−vlβlpl

∏j

(1 +

βlFljpjpl

)−1

. (6)

In the following, we denote Ol(p) = 1 −e−vlβl/pl

∏j

(1 + βlFljpj

pl

)−1

. Observe that the probabilityof successful transmission, i.e., the complement of (6), issimply the product of two factors, namely, e−vlβl/pl and∏j

(1 + βlFljpj

pl

)−1

, which are the probability of successfultransmission in a noise-limited Rayleigh-fading channel andan interference-limited Rayleigh-fading channel respectively.

1A closed form expression was first derived in [14], but we use anotherequivalent form derived in [7].

2553

Using (6), (5) simplifies to a deterministic problem:

minimize maxl

Ol(p) = 1− e−vlβlpl

∏j

(1 +

βlFljpjpl

)−1

subject to p ∈ P.(7)

Note that (7) is always feasible and its optimal solution isstrictly positive. Previous work in the literature, e.g., [7], onlyconsidered (7) for the interference-limited case, i.e., v = 0 andwithout any power constraint. In this special case, [7] showedthat (7) can be reformulated as a geometric program (GP), andbe solved efficiently using the interior point method [15].

In the following, we give a reformulation of (7) as a convexoptimization problem (not a convex GP formulation in general,but reduces to one in the interference-limited special case).By exploiting the nonlinear Perron-Frobenius theorem, wepropose a fast algorithm2 (no configuration whatsoever andorders of magnitude faster than standard convex optimizationalgorithms, e.g., interior-point method) to solve (7) optimally.As a by-product, it resolves an open problem on the con-vergence of a previously proposed heuristic algorithm in [7].Furthermore, we derive analytically the optimal value andsolution of (7) in terms of a Perron eigenvalue and eigenvectorof a specially constructed nonnegative matrix respectively.

We next introduce the auxiliary variable τ and write (7)in the epigraph form (augmenting the constraint set with anadditional L constraints):

minimize τ

subject to 1− e−vlβlpl

∏j

(1 + βlFljpj

pl

)−1

≤ τ ∀ l,p ∈ P,

variables: p, τ.

(8)

By letting α = − log(1−τ) and rewriting the L augmentedconstraints in (8), (8) is equivalent to the following problem:

minimize α

subject to vlβl/pl +∑j log

(1 + βlFljpj

pl

)≤ α ∀ l,

p ∈ P,variables: p, α.

(9)

We call the first L constraints of (9) the outage constraints,and denote the optimal solution to (9) by (p?, α?). Now, (9)is nonconvex in (p, α). However, by making a logarithmicchange of variable in p, i.e., pl = log pl for all l, (9) can beconverted into the following convex optimization problem in(p, α):3

minimize α

subject to vlβle−pl +

∑j log

(1 + βlFlje

pj−pl)≤ α ∀ l,

ep ∈ P,variables: p, α.

(10)

2Some key computational considerations are the extremely fast signal pro-cessing requirement at the transceiver chip and the decentralized environment.

3Note that (9) cannot be rewritten as a standard GP formulation, as has beendone in [7]. Nevertheless, after a logarithmic change of variables, a convexform can still be obtained as shown here.

Though solving the nonconvex problem (9) is equivalentto solving the convex problem (10), we next use a non-linear Perron-Frobenius theory-based approach to solve (9)optimally. Using nonnegative matrix theory, we then connect(9) to the Lagrange duality of (10) (cf. Lemma 2 later).

Lemma 1: At optimality of (9), the outage constraints in(9) are tight:

vlβl/p?l +

∑j

log(

1 +βlFljp

?j

p?l

)= α? ∀ l. (11)

Furthermore, if P = {p | 1>p ≤ P}, we have 1>

p? = P ,and if P = {p | pl ≤ p ∀ l}, we have p?i = p for some i.

Proof: First, we note that it has been pointed out in [7]that all the outage constraints are tight for the interference-limited case, i.e., v = 0. We prove the first part of Lemma1 for the general case here. Clearly, the function on thelefthand side of the lth outage constraint in (9) is monotoneincreasing in pj , j 6= l, and monotone decreasing in pl.Suppose the lth constraint is not tight at optimality, i.e.,vlβl/p

?l +

∑j log

(1 + βlFljp

?j

p?l

)< α?. Then, we choose a

feasible power pl < p?l such that the evaluated value of

vlβl/pl +∑j log

(1 + βlFljp

?j

pl

)is still less than α?. Now,

vjβj/p?j +

∑k 6=l log

(1 + βjFjkp

?k

p?j

)+ log

(1 + βjFjlpl

p?j

)for

all j 6= l. This implies that the value of α can be furtherdecreased, i.e., α < α?, which contradicts the assumption.Hence, the lth constraint must be tight at optimality for all l.

We next prove the second part for P = {p | pl ≤ p ∀ l}.Suppose p?l < p at optimality for all l. Let a positivescalar a = minl p/p?l > 1, and choose a feasible powerp = ap?, which evaluates the outage constraints as vlβl/pl +∑j log

(1 + βlFljpj

pl

)= vlβl/ap

?l +

∑j log

(1 + βlFljp

?j

p?l

)<

vlβl/p?l +

∑j log

(1 + βlFljp

?j

p?l

)= α? for all l. This implies

that α can be further decreased, i.e., α < α?, which contradictsthe assumption. Hence, p?i = p for some i. A similar proofcan be given when P = {p | 1>p ≤ P} and is omitted.

Remark 1: We give an analytical solution to (11) and thusthe optimal solution of (7) in Section III-C (see Table I later).

By exploiting a connection between the nonlinearPerron-Frobenius theory in [9], [10] and thealgebraic structure of (9), we propose the followingalgorithm (with geometric convergence rate and noconfiguration whatsoever) that computes the optimalsolution of (9). We let k index discrete time slots.

Algorithm 1 (Worst Outage Probability Minimization):

1) Update power p(k + 1):

pl(k + 1) = − log (1−Ol(p(k))) pl(k) ∀ l. (12)

2) Normalize p(k + 1):

p(k + 1)← p(k + 1) · P1>p(k + 1)

if P = {p | 1>

p ≤ P}.(13)

2554

p(k + 1)← p(k + 1) · pmaxj pj(k + 1)

if P = {p | pl ≤ p ∀ l}.(14)

Theorem 1: Starting from any initial point p(0), p(k) inAlgorithm 1 converges geometrically fast to the optimal solu-tion of (7).

Proof: Let us write the left-hand side of the lth outageconstraint in (9) as fl(p)

pl≤ α, where

fl(p) = vlβl +∑j

pl log(

1 +βlFljpjpl

). (15)

In the following, we show that fl(p) is a positive concaveself-mapping on the standard cone K = R

L+. The definition of

a concave self-mapping is given in [10] as follows.Definition 1 (Concave Self-mapping [10]): A mapping T :

K → K is concave if

T (ax+(1−a)y) ≥ aTx+(1−a)Ty ∀ x,y ∈ K, a ∈ [0, 1],

and monotone if 0 ≤ x ≤ y implies 0 ≤ Tx ≤ Ty.Let ‖·‖ be a norm on RL that is monotone, i.e., ‖x‖ ≤ ‖y‖. Aconcave self-mapping of K is monotone on K and continuouson the interior of K with respect to ‖ · ‖ [10].

We first show that T = fl(p) is a cone mapping with respectto the interior of K. Taking the derivative of fl(p) with respectto pl, the jth entry of the first derivative Ofl(p) is given by:

(Ofl(p))j =∑k

(log(

1 +βlFlkpkpl

)− βlFlkpkpl + βlFlkpk

), if j = l

βlFljplpl + βlFljpj

, if j 6= l.

Since z/(1 + z) ≤ log(1 + z) for z ≥ 0, (Ofl(p))l ≥ 0. Thus,(Ofl(p))j ≥ 0 for all j, i.e., fl(p) increases monotonically inp. Now, we state the following result [16].

Theorem 2 (Proposition 3.2 in [16]): Let K be the set ofcone mappings with respect to the interior of the positivestandard cone. Suppose T : K → K is differentiable andthe following inequalities hold for the component mappings:Tl : K → R+ for all l:

∑j |

∂Tl∂pj

(p) |≤ Tlp on K. ThenT ∈ K.

Now, we have∑j |

∂Tl∂pj

(p) |=∑j

pj(Ofl(p))j

=∑k

(pl log

(1 +

βlFlkpkpl

)− βlFlkpkplpl + βlFlkpk

)+∑j 6=l

βlFljplpjpl + βlFljpj

=∑k

pl log(

1 +βlFlkpkpl

)≤ fl(p).

(16)Hence, by Theorem 2, fl(p) is a strictly positive and monotonecone mapping on K.

We next show that T = fl(p) is a concave self-mapping.Taking the second derivative, we obtain the Hessian O2fl(p)

with entries given by:

(O2fl(p))jk =

− (βlFlj)2pj(pl + βlFljpj)2

, if j = k, k 6= l

(βlFlj)2pj(pl + βlFljpj)2

, if j 6= k, either k or j = l

−∑m

(βlFlm)2p2m/pl

(pl + βlFlmpm)2, if j = k = l

0, otherwise.

Now, the Hessian O2fl(p) is indeed negative definite: forall real vectors z, we have

z>O2fl(p)z = − 1

pl

∑k

(βlFlk)2(plzk − pkzl)2

(pl + βlFlkpk)2< 0.

Another proof is to observe that t log(1 + x/t) is strictlyconcave in (x, t) for strictly positive t, as it is the perspectivefunction of the strictly concave function log(1 + x) [15].Hence, fl(p) is a sum of strictly concave perspective function,and therefore f(p) is strictly concave in p.

We note that any concave self-mapping of K is in K and itis monotone and continuous [16]. Indeed, fl(p) is monotoneincreasing in p as has been shown earlier.

We first state the following key theorem in [10].Theorem 3 (Krause’s theorem [10]): Let ‖ · ‖ be a mono-

tone norm on RL. For a concave mapping f : RL+ → RL+

with f(z) > 0 for z ≥ 0, the following statements hold. Theconditional eigenvalue problem f(z) = λz, λ ∈ R, z ≥ 0,‖z‖ = 1 has a unique solution (λ∗, z∗), where λ∗ > 0, z∗ > 0.Furthermore, limk→∞ f(z(k)) converges geometrically fast toz∗, where f(z) = f(z)/‖f(z)‖.The total and individual power constraints in (4) are themonotone norm ‖p‖1 = P and ‖p‖∞ = p respectively. ByTheorem 3, the convergence of the iteration

p(k + 1) = f(p(k))/‖f(p(k))‖

to the unique fixed point p = f(p)/‖f(p)‖ is geometricallyfast, regardless of the initial point.

Remark 2: To compute Ol(p(k)) in (12), the lthuser measures separately the received interfering power{Fljpj(k)}, j 6= l. The normalization at Step 2 can bemade distributed using gossip algorithms to compute eithermaxl pl(k + 1) or 1

>p(k + 1) [17].

In the following, we first derive useful bounds to O? givenin terms of the problem parameters of (7). Next, we solve (7)analytically in the interference-limited special case withoutany power constraint, and then extend the analysis to thegeneral case with power constraints.

A. Worst Outage Probability Bounds

We now develop lower and upper bounds for the worstoutage probability O? using the certainty-equivalent margin

2555

(CEM) problem, which is defined as the following maximiza-tion of the minimum weighted SINR problem:4

maximize minl

SINRl(p)βl

subject to 1>

p ≤ P , p ≥ 0.(17)

Now, the optimal value and solution of (17) can be obtainedanalytically [11]. We have the following bounds using theCEM analytical optimal value (also in terms of the constantproblem parameters of (9)).

Corollary 1: If P = {p | pl ≤ p ∀ l}, the worst outageprobability O? satisfies

ρ(diag(β)(F+(1/p)ve>i ))

1+ρ(diag(β)(F+(1/p)ve>i ))≤ O? = 1− e−α?

≤ 1− e−ρ(diag(β)(F+(1/p)ve>i )),

(18)

where i = arg maxlρ(diag(β)(F + (1/p)ve

>

l )) (19)

and α? is the optimal value to (9).Proof: Using the inequalities 1 +

∑Ll=1 zl ≤

∏Ll=1(1 +

zl) ≤ e∑Ll=1 zl for nonnegative z (cf. [7]), a lower and upper

bound on α? can be given by 1/(1 + CEM) ≤ α? ≤ 1 −e−1/CEM, where CEM is the optimal value of (17) and isgiven by 1/ρ(diag(β)(F + (1/p)ve

>

i )), where i is given by(19) [11]. The bounds on O? = 1−e−α? can thus be obtained,hence proving Corollary 1.

Remark 3: Note that the lower bound in Corollary 1 is notnecessarily the tightest, but the bounds in Corollary 1 illustratethat the CEM problem, i.e., the spectral information of aconcave self-mapping Tp = diag(β)(F + v)p (cf. [11]) canprovide useful quick bounds to the worst outage probability.Corollary 1 reduces to a result in [7] in the interference-limitedcase. Results similar to Corollary 1 can also be obtained forthe case when P = {p | 1>p ≤ P}, but is omitted here.

Example 1: Figure 1 plots the worst outage probabilityand the bounds for a system with 20 femtocell users usingparameters in [1], where each user has a common SINRthreshold β. Observe that the bounds are concave in β, andthe worst outage probability is very close to its upper bound.In fact, our numerical observations indicate that the optimalpower p? are also close in value to the optimal solution of(17), i.e., x(diag(β)(F + (1/p)ve

>

i )), where i is given by(19) (cf. [11]). Further, for small β, the bounds are veryclose, which suggests that in low-power femtocell networks,the CEM solution can give sufficiently good approximation tothe worst outage probability.

B. Interference-limited Case

We now turn to solve (7) analytically for the interference-limited case, i.e., v = 0. In this case, (15) is in addition aprimitive positive homogeneous function of degree 1. Next,

4The CEM problem replaces the statistical variation in the desired signaland the interference of (3) by their mean values, and is a nonconvex problemthat can also be solved by the nonlinear Perron-Frobenius theory [11].

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

SINR Threshold β

Outa

ge P

robabili

ty

Worst Outage Probability

Lower Bound

Upper Bound

Fig. 1. Outage probability versus the SINR threshold β for a system with20 femtocell users, where each user has a common SINR threshold β.

we define the nonnegative matrix B with the entries (that arefunctions of p):

Blj =

{0, if l = j

plβlpj

log(

1 + βlFljpjpl

), if l 6= j.

(20)

Note that B(p) is irreducible whenever F is. Using (20),we can write the optimal value α? and optimal solution p? of(9) in the interference-limited special case as

α? = ρ(diag(β)B(p?)) (21)

and p? = x(diag(β)B(p?)) (up to a scaling constant) (22)

respectively. Thus, p? is a fixed point of

p =1

ρ(diag(β)B(p))diag(β)B(p)p. (23)

Further, it is interesting to note the following result of p?.Corollary 2: In the interference-limited case, the optimal

power of (9), p?, satisfies

p? = arg maxp>0

ρ(diag(β)B(p)). (24)

Proof: Using (20), we can rewrite the outage constraintsin (9) in matrix form as

diag(β)B(p)p ≤ αp. (25)

Next, we need the following result from [13].Theorem 4 (Theorem 1.6, [13] (Subinvariance Theorem)):

Let A be an irreducible nonnegative matrix, s a positivenumber, and z ≥ 0, a vector satisfying Az ≤ sz. Then, (i)z > 0; (ii) s ≥ ρ(A). Moreover, s = ρ(A) if and only ifAz = sz.

Applying Theorem 4 to (25) (let A = diag(β)B(p), z bea feasible p and s = α), we have ρ(diag(β)B(p)) ≤ α forany feasible p and α. But α? = ρ(diag(β)B(p?)). Hence,ρ(diag(β)B(p)) ≤ ρ(diag(β)B(p?)).

The following method was first proposed in [7] to computethe optimal solution for the interference-limited case withoutany power constraint:

p(k + 1) =1

ρ(diag(β)B(p(k)))diag(β)B(p(k))p(k). (26)

2556

The authors in [7] observed that this iteration convergesnumerically to an acceptable accuracy with a fixed initialvector p(0) = x(diag(β)F) (optimal solution of (17) withoutpower constraints and noise power). The issues of convergenceand existence of a fixed point were however left open in [7].

Our Algorithm 1 in fact reduces to an update similar to (26)when v = 0, and computes a solution that is equal to thatof (26) up to a scaling constant. The scaling factor in (26)tends to ρ(diag(β)B(p?)) with increasing k. From Theorem1, this means that (26) converges from any initial point to thefixed point in (23) geometrically fast, thus resolving an openproblem of convergence in [7].

C. Duality by Lagrange and Perron-Frobenius

We now show that the optimal value α? and optimalsolution p? of (9) can be derived analytically from the spectralinformation of a specially constructed rank-one perturbation ofdiag(β)B, where B is given in (20). The following result isobtained based on the nonlinear Perron-Frobenius theory in[9] and the Friedland-Karlin inequality in [12].

Lemma 2: The optimal solution (p?, α?) of (9) satisfies

logα? = log ρ(diag(β)(B(p?) + vc>

∗ ))

= max‖c‖D=1

log ρ(diag(β)(B(p?) + vc>

))

= maxλ≥0,1>λ=1

minp∈P

∑l

λl logβl(B(p)p + v)l

pl(27)

= minp∈P

maxλ≥0,1>λ=1

∑l

λl logβl(B(p)p + v)l

pl, (28)

where the optimal p in (27) and (28) are both given byx(diag(β)(B(p?)+vc

>

∗ )) (which is equal to p? up to a scalingconstant), and the optimal λ in (27) and (28) are both givenby the Hadamard product of x(diag(β)(B(p?) + bc

>

∗ )) andy(diag(β)(B(p?) + bc

>

∗ )).Furthermore, p = x(diag(β)(B(p?) + vc

>

∗ )) is the dual ofc∗ with respect to ‖ · ‖D.5

Proof: Lemma 2 is proved by considering the Lagrangeduality of (10) and then applying nonnegative matrix theoryin [9], [12].

Note that the optimal dual variable in (10) is equal to theoptimal λ in (27) and (28). Using Lemma 2, we can nowgive analytically the optimal value and solution of (9) whenP = {p | pl ≤ p ∀ l}.

Corollary 3: The optimal value and solution of (9) is givenby

α? = ρ(

diag(β)(B(p?) + (1/p)ve>

i ))

(29)

andp? = x

(diag(β)(B(p?) + (1/p)ve

>

i )), (30)

where i = arg maxlρ(

diag(β)(B(p?) + (1/p)ve>

l )). (31)

Furthermore, p?i = p for the i (not necessarily unique) in (31).

5A pair (x,y) of vectors of RL is said to be a dual pair with respect to‖ · ‖ if ‖y‖D‖x‖ = y

>x = 1.

Remark 4: In general, the optimal index i in (31) differsfrom (19). Our simulations show that both are empirically thesame when the CEM solution is close to p?, especially so inthe low-power regime (cf. Fig. 1). Unlike (31), (19) can becomputed a priori from the problem parameters.

Table I summarizes the connection between the nonlin-ear Perron-Frobenius spectrum (of the respectively differentconcave self-mappings) and the optimal value and solutionof the optimization problems under the CEM model andthe Rayleigh-fading model subject to the different powerconstraints (individual and total power constraints).

IV. TOTAL POWER MINIMIZATION AND ADAPTIVE

OUTAGE POWER CONTROL

In this section, we first study the total power minimizationproblem subject to both outage specification and individualpower constraints, and address its feasibility conditions usingour results in previous sections. We then propose an adaptivealgorithm to minimize the total power consumption and guar-antee a min-max fairness in terms of worst outage probability.

The problem of minimizing the total power subject to givenoutage specification under Rayleigh fading and individualpower constraints can be formulated as

minimize 1>

p

subject to 1− e−vlβlpl

∏j

(1 + βlFljpj

pl

)−1

≤ Ol ∀ l,

p ∈ {p | pl ≤ p ∀ l},variables: p,

(32)where 0 < Ol < 1 is a given outage probability bound for thelth user. Depending on the given parameters Ol for all l, (32)may or may not be feasible. This is unlike the worst outageprobability problem in (5), which is always feasible.

Next, using (15), (32) can be rewritten as

minimize 1>

p

subject tofl(p)pl≤ αl ∀ l,

p ∈ {p | pl ≤ p ∀ l},

(33)

where αl = − log(1− Ol) for all l.If (32) is feasible, it can be shown that all the L outage

constraints in (32) are tight at optimality [8]. We deduce thefeasibility condition of (32) from (33) in the following result.

Lemma 3: There is a unique and finite optimal p in (32) ifand only if

maxp∈{p|pl≤p ∀ l}

ρ(diag(β/α)B(p)) < 1. (34)

Furthermore, ρ(diag(β/α)B(p)) < 1 if α? < minl αl, i.e.,1− e−α? < Ol for all l.

Proof: To show the feasibility condition, we examine thecondition under which there is a fixed point to

fl(p) = βl((B(p)p)l + vl) = αlpl for all l,

which can be rewritten in matrix form as

(I− diag(β/α)B(p))p = diag(β/α)v. (35)

2557

TABLE INONLINEAR PERRON-FROBENIUS CHARACTERIZATION OF THE CEM PROBLEM AND THE WORST OUTAGE PROBABILITY PROBLEM: THE SECOND AND

THIRD ROW TABLUATE THE CEM CASE FOR INDIVIDUAL AND TOTAL POWER CONSTRAINTS RESPECTIVELY. THE FOURTH AND FIFTH ROW TABLUATE

THE WORST OUTAGE PROBABILITY CASE FOR INDIVIDUAL AND TOTAL POWER CONSTRAINTS RESPECTIVELY.

Concave Self-mapping (Tp)l Perron eigenvalue α? Perron eigenvector p? Remark

(diag(β)(Fp + (1/p)v))l, ρ(

diag(β)(F + (1/p)ve>i ))

x(

diag(β)(F + (1/p)ve>i ))

i = arg maxl ρ(

diag(β)(F + (1/p)ve>l ))

[11]

(diag(β)(Fp + (1/P )v))l ρ(

diag(β)(F + (1/P )v1>

))

x(

diag(β)(F + (1/P )v1>

))

[18]

vlβl +∑j pl log

(1 +

βlFljpjpl

)ρ(

diag(β)(B(p?) + (1/p)ve>i ))

x(

diag(β)(B(p?) + (1/p)ve>i ))

Herein,

i = arg maxl ρ(

diag(β)(B(p?) + (1/p)ve>l ))

Corollary 3

vlβl +∑j pl log

(1 +

βlFljpjpl

)ρ(

diag(β)(B(p?) + (1/P )v1>

))

x(

diag(β)(B(p?) + (1/P )v1>

))

Herein

We first state the following result from [13].Theorem 5: A necessary and sufficient condition for a so-

lution z ≥ 0, z 6= 0 to the equations (I −A)z = c to existfor any c ≥ 0, c 6= 0 is that ρ(A) < 1. In this case there isonly one solution z, which is strictly positive and given byz = (I−A)−1c.

Since diag(β/α)B is an irreducible nonnegative matrix,it follows from Theorem 5 (letting A = diag(β/α)B, c =diag(β/α)v) that p in (35) is unique and strictly positive ifand only if ρ(diag(β/α)B(p)) < 1 for all p ∈ {p | pl ≤p ∀ l}. This is equivalent to stating

maxp∈{p|pl≤p ∀ l}

ρ(diag(β/α)B(p)) < 1,

thus proving the first part.To show the second part, we note that ρ(diag(β/α)B(p)) ≤

(1/minl αl)ρ(diag(β)B(p)) ≤ (1/minl αl)α?. Thus, a suf-ficient condition that (1/minl αl)α? < 1 implies thatρ(diag(β/α)B(p)) < 1.

A. Feasibility Bounds

From (34) in Lemma 3, we see that verifying the feasi-bility of (32) requires solving a Perron-Frobenius eigenvaluemaximization problem (a nonconvex problem). We next pro-vide useful (tight) bounds on this nonconvex optimal valueρ(diag(β/α)B(p)) in (34) that exploit the optimal value andsolution of the worst outage probability problem in SectionIII.

Theorem 6: Let α? and p? be given in (21) and (22),respectively. Then, we have∏

l

(αl)−xl(B(p?))yl(B(p?))α? ≤

maxp∈{p|pl≤p ∀ l}

ρ(diag(β/α)B(p)) ≤ maxl

(α?/αl).(36)

Further, equality is achieved in the lower and upper boundswhen αl’s are equal for all l.

Proof: By applying the Friedland-Karlin inequalities in[12], the function ρ(diag(β/α)B(p)) can be bounded by∏

l

(αl)−xl(B)yl(B)ρ(diag(β)B) ≤ ρ(diag(β/α)B)

≤ maxl(1/αl)ρ(diag(β)B)(37)

for any feasible p ∈ {p | pl ≤ p ∀ l}. Now, from the lowerbound in (37), we have∏

l

(αl)−xl(B(p?))yl(B(p?))ρ(diag(β)B(p?))

≤ maxp∈{p|pl≤p ∀ l}

{∏l

(αl)−xl(B(p))yl(B(p))ρ(diag(β)B(p))

}≤ max

p∈{p|pl≤p ∀ l}ρ(diag(β/α)B(p)),

(38)where p? ∈ {p | pl ≤ p ∀ l} is given by (22). Using α? givenin (21), we thus establish (36). The condition under whichequalities in (37) are achieved follows from the application ofthe Friedland-Karlin inequalities in [12].

These bounds can be easily computed in the two-user case.6

Example 2: In the two-user case, we have

α?√α1α2

≤ maxp∈{p|pl≤p ∀ l}

ρ(diag(β/α)B) ≤ max{α?

α1,α?

α2

}.

Combining Theorem 6 with Corollary 1, simplified boundsin terms of the CEM solution (and more directly in terms ofthe problem parameters) can be obtained:∏

l

(αl)−xl(B(p?))yl(B(p?)) log(1 + ρ(diag(β)(F + (1/p)ve>

i )))

≤ maxp∈{p|pl≤p ∀ l}

ρ(diag(β/α)B)

≤ maxl(1/αl)ρ(diag(β)(F + (1/p)ve>

i )),

where i is given by (19) and we have made use of the fact thatmaxl ρ(diag(β)(B(p?) + (1/p)ve

>

l )) ≤ maxl ρ(diag(β)(F+(1/p)ve

>

l )) in the last inequality.

6The Schur product of the Perron and left eigenvectors of a zero-diagonal2× 2 positive matrix equals [1/2, 1/2], simplifying the computation in (36).

2558

We now state the following algorithm proposed in [8].

Algorithm 2 (Total Power Minimization):

pl(k + 1) = min {− log (1−Ol(p(k))) pl(k)/αl, p} ∀ l.(39)

We now establish the necessary and sufficient condition underwhich Algorithm 2 converges. This condition is also necessaryand sufficient for (32) to be feasible.

Corollary 4: Starting from any initial point p(0), p(k)in Algorithm 2 converges geometrically fast to the optimalsolution of (32) if and only if ρ(diag(β/α)B(p)) < 1 for allp ∈ {p | pl ≤ p ∀ l}.

Proof: The necessary and sufficient condition underwhich (32) is feasible is given in Lemma 3. If (32) is feasible,the convergence proof for Algorithm 2 can be found in [8].Hence, Corollary 4 is proved.

B. Adaptive Outage-based Power Control

In this section, we propose an adaptive outage-basedpower control algorithm for total energy minimization.

Algorithm 3 (Adaptve Outage-based Power Control (AOPC)):

1) Update the auxiliary variable z(k + 1):

zl(k + 1) = − log (1−Ol(z(k))) zl(k) ∀ l. (40)

2) Normalize z(k + 1):

z(k + 1)← z(k + 1) · p/maxjzj(k + 1). (41)

3) Update the transmit power p(k + 1):

pl(k+1) = min{− log (1−Ol(p(k))) pl(k)

max{αl,− log (1−Ol(z(k)))}, p

}∀ l.

(42)

Corollary 5: Starting from any initial point z(0) andp(0), p(k) in Algorithm 3 converges geometrically fastto the optimal solution of (33), where the righthandside of the outage constraints in (33) are replaced bymax{αl, ρ(diag(β)(B(p?) + (1/p)ve

>

i ))}, where p? and iare given by (30) and (31) respectively, for all l.

Proof: Theorem 1 proves the convergence of z(k) inStep 1 and 2 of Algorithm 3, and also limk→∞− log(1 −Ol(z(k)))→ ρ(diag(β)(B(p?)+(1/p)ve

>

i )). From Corollary4, p(k) converges to a point p′ that satisfies − log(1 −Ol(p′)) = α′ = max{α, ρ(diag(β)(B(p?) + (1/p)ve

>

i ))},which always satisfies ρ(diag(β/α′)B(p)) < 1. This provesCorollary 5.

Remark 5: If αl < ρ(diag(β)(B(p?)+(1/p)ve>

i ))} for alll, then limk→∞ z(k) = limk→∞ p(k) = p? in Algorithm 3.

Macro Access

Point

Macro Access

Terminal

Home Access

PointHome Access Terminal

X (dB)

X (dB)

X (dB)

Y (dB)

Z (dB)

Fig. 2. A macro-femtocell model. The parameter X dB, Y dB and Z dBdenote pass gain between MAP and HAP, between HAP and HAT and betweenHAP and MAT respectively (cf. Table 2 of [1] for values of X,Y, Z).

0 5 10

10-3

iteration

Pow

er

(log s

cale

)

User 1

User 2

User 3

User 4

User 5

Fig. 3. Experiment 1. Convergence in power for five femtocell users usingAlgorithm 1.

V. NUMERICAL EVALUATION

In this section, we evaluate the performance of Algorithms1, 2 and AOPC. Fig. 2 illustrates the basic macro-femtocellinterference network model used in [1] for our simulations. Afemtocell user is a link between the home access terminal andaccess point (box of Fig. 2), and interference comes from themacrocell base station/access terminals and other femtocellusers. We consider a single macrocell base station with 50macro access terminals and vary the number of femtocellusers. We use the dense-urban propagation parameters in Table2 of [1].

A. Expt. 1 (Convergence of Algorithm 1)

Fig. 3 shows the convergence of Algorithm 1 for fivefemtocell users. Simulations show that convergence happensin less than ten iterations even for thousands of users and alarge power range, e.g., 125mW to 2W (maximum output ofUMTS/3G Power Class 4 to Class 1 mobile phone respec-tively). From Theorem 3, Algorithm 1 can be viewed as anonlinear power method in linear algebra. It is well knownthat the convergence rate of the power method is determinedby the ratio of the second dominant eigenvalue to the Perron-Frobenius eigenvalue [13]. The method converges slowly ifthis ratio is close to one. Now, this ratio for B(p?) (cf. fourthand fifth row of Table I) determines only the local convergencerate of Algorithm 1 near the fixed point p?. Since the CEMsolution is numerically observed to give good approximationto p? (especially in the regime of low-power and small β), thisratio is conceivably close to those computed using the CEM

2559

0 20 40 60 80 1000

50

100

150

200

iteration

Pow

er

(W)

Algo. 2 (100)

Algo. AOPC (100)

Algo. 2 (200)

Algo. AOPC (200)88%, Algo. 2: 0AOPC : 0 71%,

Algo. 2: 0.11AOPC : 0.05

126%, Algo. 2: 1AOPC : 0.06

121%, Algo. 2: 1AOPC : 0.2

64%, Algo. 2: 1AOPC : 0.87

58%, Algo. 2: 0.68AOPC : 0.32

0 20 40 60 80 1000

2

4

6

8

10

12

14

16

18

20

22

iteration

Pow

er

(W)

Algo. 2 (100)

Algo. AOPC (100)

Algo. 2 (200)

Algo. AOPC (200)82%Algo. 2: 0.77AOPC : 0.24 71%

Algo. 2: 0.97AOPC : 0.32

138%Algo. 2: 0.97AOPC : 0.39

142%Algo. 2: 0.99AOPC : 0.43

67%Algo. 2: 1AOPC : 0.7

48%Algo. 2: 1AOPC : 0.6

0 20 40 60 80 1000

0.5

1

1.5

2

iteration

Pow

er

(W)

Algo. 2 (100)

Algo. AOPC (100)

Algo. 2 (200)

Algo. AOPC (200)

56%Algo 2: 1AOPC: 0.84

55%Algo 2: 0.99AOPC: 0.7

65%Algo 2: 0.97AOPC: 0.44

113%Algo 2: 0.95AOPC: 0.63

115%Algo 2: 0.87AOPC: 0.28

84%Algo 2: 0.8AOPC: 0.2

0 20 40 60 80 1000

0.05

0.1

0.15

0.2

iteration

Pow

er

(W)

Algo. 2 (100)

Algo. AOPC (100)

Algo. 2 (200)

Algo. AOPC (200)

76%Algo 2: 1AOPC: 0.81

55%Algo 2: 0.99AOPC: 0.86

80%Algo 2: 0.82AOPC: 0.17

113%Algo 2: 0.99AOPC: 0.52

120%Algo 2: 0.91AOPC: 0.44

82%Algo 2: 0.81AOPC: 0.17

Fig. 4. Experiment 2. We plot the evolution of the total power consumption for twonetworks with 100 and 200 femtocell users as we vary p. A time slot on the x-axisof each graph refers to a power update iteration. Ten and fifty percent of the users areremoved at time slot 30 and 70 respectively. The top left graph, top right graph, bottomleft graph and bottom right graph show the case for p = 1W , p = 0.1W , p = 10mWand p = 1mW respectively. Power savings between Algorithms 2 and AOPC, and thefraction of the number of users meeting the given threshold α are in text form.

solution (cf. second and third row of Table I), which on theother hand determines a global convergence rate for the CEMcase. We empirically determine this ratio to be in the rangeof 0.2− 0.4 using the parameters in [1]. This thus determinesnumerically the overall convergence rate of Algorithm 1.

B. Expt. 2 (Comparison between Algorithms 2 and AOPC)

We next provide numerical examples to compare the totalpower consumption using Algorithms 2 and AOPC. Fig. 4shows the total power evolution in a network with initially100 and 200 femtocell users. Then, ten and fifty percent of theusers are removed at time slot 30 and 70 respectively. On eachgraph, the difference in total power and the fraction of usersthat satisfy their outage probability threshold α are recorded.As illustrated, Algorithm 2 can produce power savings of 50%or more in all cases at the expense of a smaller number of usersmeeting α. On the other hand, problem infeasibility causesusers who run Algorithm 2 and unable to achieve α to transmitat p. By enforcing a worst outage probability fairness acrossall users, Algorithm AOPC computes power that are typicallysmaller than p, thus leading to a smaller total power.

VI. CONCLUSION

We solved analytically the worst outage probability problemhaving either a total or individual power constraints in amultiuser Rayleigh-faded network. We then proposed a ge-ometrically fast convergent algorithm to solve it optimally ina distributed manner. As a by-product, we solved an openproblem of convergence for a previously proposed algorithm

in the interference-limited case. We also established a tightrelationship between the worst outage probability problemand its certainty-equivalent margin counterpart, and utilizedthe connection to find useful bounds and convergence rate.We then addressed a total power minimization problem withoutage specification constraints and its feasibility condition.We proposed a dynamic algorithm that adapted its outageprobability specification to minimize the total power in aheterogeneous network. This permitted a graceful handling ofoutage infeasibility in a distributed manner, and guaranteed amin-max fairness in terms of the worst outage probability.

ACKNOWLEDGEMENT

The work in this paper was partially supported by grantsfrom the Research Grants Council of Hong Kong, Project No.RGC CityU 112909, CityU 7200183 and CityU 7008087.

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