Power and Rate Control with Outage

Constraints in CDMA Wireless Networks

C. FISCHIONE, M. BUTUSSI, AND K. H. JOHANSSON

Stockholm 2007

Automatic Control Group

School of Electrical Engineering

Kungliga Tekniska Hgskolan

IR-EE-RT 2006:016

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Power and Rate Control with Outage

Constraints in CDMA Wireless Networks

C. Fischione, M. Butussi, and K. H. Johansson

Abstract

We investigate a power control strategy to achieve maximum throughput for the up-link of a CDMA

wireless system with variable spreading factor. The system model includes slow and fast fading, rake

receiver, and multi-access interference caused by users with heterogeneous data sources. The quality of

the communication is expressed in terms of outage probability, while the throughput is defined as the sum

of the users transmit rates. The outage probability is accounted for by resorting to a tight approximation

based on the extended Wilkinson moment matching. A mixed integer optimization problem P1, where

the objective function is the throughput under outage probability constraints, is investigated. Problem

P1 is solved in two steps: firstly, we propose a modified problem P2 to provide feasible solutions, and

then the optimal solution is obtained with a branch-and-bound search. Numerical results are presented

and discussed to asses the validity of our approach.

Index Terms: CDMA, Outage, Distributed Computation, Combinatorial Optimization, Branch-and-Bound Search.

Work done in the framework of the HYCON Network of Excellence, contract number FP6-IST-511368. The work is alsopartially funded by the Swedish Foundation for Strategic Research and Swedish Research Council.

C. Fischione is with the Department of Electrical Engineering and Computer Science, University of California atBerkeley. M. Butussi was with University of Padova, Italy, when this work was done. K. H. Johansson is with theAutomatic Control Lab, School of Electrical Engineering, Royal Institute of Techonolgy, Stockholm, Sweden. E-mail:fischion@eecs.berkeley.edu, bokassa@dei.unipd.it, kallej@ee.kth.se.

Part of this work is accepted for presentation at IEEE ICC 2007.

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I. INTRODUCTION

In Direct Sequence Code Division Multiple Access (DS-CDMA) wireless systems the transmit

rates can be adapted to channel conditions and traffic load by radio power control, which

keeps multi-access interference (MAI) caused by other co-channel users under acceptable levels.

Maximization of the transmit rate is a major goal in many situations. For instance, the rate

allowed at the physical layer provides a bound to the performance of the Transmission Control

Protocol (TCP), which is widely implemented over wireless systems.

Rate maximization can be cast as an optimization problem, where the objective function is

the sum of the user rates, and the constraints are expressed in terms of the Quality of Service

(QoS) to be guaranteed for each user. First, it is important that QoS constraints are properly

modelled. Second, the QoS models should be allow efficient optimization.

In this paper, we model and solve an optimization problem to maximize the up-link throughput

achievable in CDMA wireless system by power allocation. We express QoS constraints by

outage probability for any user. We provide a detailed overview on existing works in Section II,

and we highlight the original contributions of this paper in Section III. In Section IV, the rate

maximization problem is formulated, and the model is described. The constraints of the problem

are investigated in Section V. In Section VI we propose an approach to solve the optimization

problem. Numerical results are reported and commented in Section VII. Finally, Section VIII

concludes the paper.

II. RELATED WORK

One of the earliest approaches to rate maximization can be found in [1] for a dual-class CDMA

system. Optimal joint rate and power adaptation, subject to peak transmit power constraints,

and a maximum interference constraints, is studied in [2]. Adaptive code rates with multiple

orthogonal codes is considered in [3]. In [4] the power gains achieved by the scheme proposed

in [3] is shown, along with truncated rate adaptation. Extensive studies have been carried out in

the case of down-link throughput maximization and power allocation [5], [6], and [7]. In these

contributions, the goal is the throughput maximization to improve the performance of TCP when

it is employed over wireless channels. Specifically, throughput is studied under joint rate and

power adaptation with bit error rate requirements in a multi-cell VSF (Variable Spreading Factor)

WCDMA system. A dynamic rate variation is achieved by using single-code transmission with

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a VSF that varies inversely with the transmit rate. A general approach can be found in [8],

where the authors consider a single cell scenario: the constraints on the rates are expressed as

thresholds on the instantaneous minimum Signal to Interference plus Noise Ratio (SINR), under

the assumption of perfect channel estimation.

Another line of research has extended these results on instantaneous and average SINR to

the outage probability of the SINR. The outage probability is an interesting QoS measure for

delay-limited scenarios [9]. The problem of radio resource management under outage constraints

dates back at least to the work presented in [10] (see also [11]). The effects of power control

imperfections on the throughput are investigated in [12]–[14]. Specifically, in [14], the authors

propose a maximization of the system capacity (number of users) under outage requirements.

In [15], the authors considers the throughput maximization including log-normal fading channels.

The outage probability was expressed by a Gaussian approximation. This approach has later been

further developed in [16], where the channel is modelled with Rayleigh fading distributions. Both

in [15] and [16], it is shown that the problems of power minimization under outage constraints,

and the problem of outage minimization under power constraints, can be cast as a geometric

program. Furthermore, they establish an interesting relation of their approach with the Perron-

Frobeniuos theory for the solution of problems with SINR constraints.

One of the most relevant and interesting approaches to the throughput maximization with

outage constraints is investigated in [17]. In that paper, and reference therein of the same authors,

the problem of joint power minimization and multi-user detection is explored. The authors present

a general method to solve efficiently and iteratively the power allocation problem, under some

cases of mixed slow and fast fading. Furthermore, by relaxing the outage constraints with an

upper bound provided by the Jensen’s inequality on the statistical average, the authors are able

to map the outage constraints over the average SINR.

III. MAIN CONTRIBUTION

The main contribution of this paper consists in providing power allocation policies for rate

maximization in CDMA wireless system with outage constraints. First, we adopt a general model

of the SINR, which is representative of mixed slow and fast fading. The model is representative

also of rake receivers with maximal ratio combining commonly employed in CDMA systems.

Our approach can be used even in presence of the power control mechanisms of WCDMA

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system: a fast inner power control loop and a slow outer loop [18]. The system model includes

VSF under a detailed model of MAI with heterogeneous data sources (video, voice, data, etc.).

We investigate a mixed integer optimization problem P1 which is different from those in the

existing relevant contributions we have mentioned in Section II. In particular, the constraints

on the outage are expressed by resorting to the accurate log-normal approximation in [14] and

[19], which allow us to solve Problem P1 even for low outage probabilities (less than 1%). Our

contribution is original, for example, compared to the interesting works [15] –[17], which are

instead focused on mapping the outage constraint onto the average SINR. They do not include a

detailed model of the MAI and the source behavior. Furthermore, they do not include the general

model of the SINR as we do. We propose a method to solve Problem P1, which is based on

branch-and-bound search. This seems to be unexplored when considering outage constraints in

the general system model we adopt.

IV. SYSTEM DESCRIPTION AND PROBLEM FORMULATION

We consider a system scenario where K mobile users are transmitting toward a Base Station

(BS). Each user j = 1, . . . , K is associated to a traffic source type (voice, video, data, etc.),

employs the same chip time Tc, and transmits with power level Pj . The model of the physical

layer for the up-link of a single-cell asynchronous binary phase shift keying DS/CDMA system

is summarized by the following expression of the SINR [14], [20]:

SINRi(h, ν) =Pihi

N0

2Ti+

∑Kj=1

j 6=i

Pj

Gihjνj

, (1)

where the numerator is the power associated to the useful signal, and the denominator is the power

of the thermal noise plus the MAI. The random variable hj is the channel gain experienced by

the signals of user j. We adopt the Lee and Yeh multiplicative model [21, pag. 91]: hj = ljzjΩj ,

where lj is path loss, zj is the power of the fast fading component, and Ωj is the power of the

shadowing. We consider the Nakagami distribution for the fast fading [21], with correlation 1 and

parameter m, so zj has a Gamma distribution. Note that when m = 1, the fast fading distribution

reduces to a Rayleigh distribution. For any m, zi and zj , i 6= j, are statistically independent.

The shadow fading is expressed by the log-normal model Ωj = exp (ξi), where ξi is a Gaussian

random variable having average µξiand standard deviation σξi

. The random variable ξi may be

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correlated [22], with covariance Ci,j = E [ξi − E ξi][ξj − E ξj]. The binary random variable

νi indicates the activity status (on/off) of the source. Its probability distribution function is such

that Pr[νi = 1] = αi and Pr[νi = 0] = 1 − αi, where αi is the activity factor of source i. We

use vector notation h = [h1, . . . , hK ]T and ν = [ν1, . . . , νK ]T . Independence is assumed between

any pair of processes of the vectors h and ν. Observe that the SINR model (1) is general, and

takes into account also the rake receiver with Maximal Ratio Combining [23].

Let us model the rate of each user as Ri = Ri0ni, where Ri0 = 1/Ti0 is the basic rate

of user i, with basic bit time Ti0, and ni is an integer denoting the assigned rate. The rate

is assumed to be a power of two due to the spreading code structure [21]. Consequently, the

spreading factors are expressed as Gi = Gi0/ni, where Gi0 = Ti0/Tc corresponds to the basic

rate Ri0. We assume that the users’ power is expressed as Pi = pini, where pi is the power

at the basic rate Ri0 (i.e., when ni = 1). This power model is motivated by the fact that users

requiring larger rates need larger transmit powers, as shown in [14]. This model was adopted

also in [7] (and references therein), where it was observed that it has the advantage of keeping

the bit energy to noise ratio constant with respect to variations of ni. We use the notation

p = [p1, . . . , pK ]T and p−i = [p1, p2, . . . , p(i−1)p(i+1) . . . pK ]T , as well as n = [n1, n2, . . . , nK ]T

and n−i = [n1, n2, . . . , ni−1, ni+1, . . . nK ]T .

We express the problem of rate maximization as an optimization problem where the cost

function is the sum of the user rates, and the constraints are bounds on the maximum value

allowed for the outage probability, transmission powers and rates:

P : maxp,n

1Tn

s.t Pr [SINRi(h, ν) < γi] ≤ Oi , i = 1, . . . , K

nTp ≤ PT ,

pi ≥ pi0 , i = 1, . . . , K

1 ≤ ni ≤ Gi0, ni ∈ N , i = 1, . . . , K

The K size vector 1 is the all-one vector. The decision variables are the powers p and rates n.

The parameter Oi represents the maximum allowed outage probability of the SINR for user i

with respect to the threshold γi. The constraint on the total power PT is motivated by the fact that

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the receiver input of the BS can receive only a maximum amount of power [14]. For physical

reasons, powers cannot be smaller than a given value pi0. The rate constraint is motivated by

that the spreading factor is Gi = Gi0/ni ≥ 1. Moreover, N ⊂ 2N denotes the set of rates.

Remark 1: We assume that an admission control policy is used (see e.g. [20], [24]), so that

all K users can be accommodated in the system, and an interior point solution of Problem P is

given by the basic powers and rates. Therefore, P is feasible.

Solving Problem P is not easy: firstly, the outage constraints are complicated functions of

the rates and power; secondly, the power constraint PT introduces extra difficulties; thirdly, the

rates are powers of two. We approach these issues in the next sections.

V. OUTAGE PROBABILITY CONSTRAINTS

Knowledge of the outage probability is required to express the first constraint in Problem P .

Such a probability is expressed by the probability distribution function (pdf) of the SINR.

However, the expression of the SINR pdf is in general unknown. The uncertainty concerns

the statistics of the MAI, which are a mixture of the on-off activity of the sources and the

fadings. Consequently, they have to be approximated. We adopt a log-normal approximation of

the SINR pdf. Then the outage probability can be expressed as

pi

Ii(n−i,p−i)≥ γi ∀i = 1, . . . , K , (2)

where Ii(n−i,p−i) is defined as the interference function. In the following subsections, we derive

a formula for Ii(n−i,p−i) and motivate (2).

A. Log Normal Approximation

Let us rewrite the SINR in (1) as follows:

SINRi(h, ν) =zi

Li(h, ν), (3)

where

Li(h, ν) =N0

2piGi0Tc

l−1i exp (−ξi) +

K∑

j=1

j 6=i

pjnj

Gi0pi

l−1i ljzjνj exp (ξj − ξi) . (4)

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We resort to an accurate approximation of the SINR in two steps. First, we note that (4) is a

combination of log-normal random variables, weighted by one-sided random variables. Thus,

we can use the extended Wilkinson Moment matching method [14], [19], [25] to model (4)

with a log-normal random variable, so that Li(h, ν) ≈ exp (−Xi) , where Xi is a Gaussian

random variable with average µXi(n−i,p) and standard deviation σXi

(n−i,p) we derive in the

following. Notice that this method provides a tight approximation. The resulting SINR is given

by the product of a gamma random variable, zi, times a log-normal one, exp (−Xi). Then, this

product can be well approximated with an overall log-normal random variable, as proposed

in [21, pag. 92]. In summary, we have that

SINRi(h, ν) ≈ exp (Yi) , (5)

where Yi is a Gaussian random variable with average and standard deviation, respectively, as

µYi(n−i,p) = ψ(m) − ln(m) − µXi

(n−i,p) , σ2Yi

(n−i,p) = ζ(2,m) + σ2Xi

(n−i,p)

where ψ(m) is Euler’s psi function, and ζ(2,m) is Riemann’s zeta function, as defined in [21,

pag. 107].

In the following, we discuss how to derive µXi(n−i,p) and σXi

(n−i,p), and their dependence

on the power and rate coefficients. The extended Wilkinson moment matching approximation

in [19] computes the average and variance of Xi as

µXi(n−i,p) = 2 lnM

(1)i (n−i,p) − 1

2lnM

(2)i (n−i,p)

σ2Xi

(n−i,p) = lnM(2)i (n−i,p) − 2 lnM

(1)i (n−i,p) ,

where

M(1)i (n−i,p) , E h,νLi(h, ν) , (6)

M(2)i (n−i,p) , E h,νL2

i (h, ν) , (7)

where we have denoted with E h,ν· the expectation w.r.t. the distribution of h and ν. The

expressions of (6) and (7) can be derived applying the statistical expectation operator to (4),

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recalling its linear properties, and that the random vectors h and ν are independent:

M(1)i (n−i,p) =

N0

2piGi0Tc

l−1i exp (−µξi

+1

2σ2

ξi)+

K∑

j=1

j 6=i

pjnj

Gi0pi

l−1i ljµzj

zjαj exp (−µξi+ µξj

+1

2σ2

ξi+

1

2σ2

ξj− Cij)

M(2)i (n−i,p) =

K∑

j=1

j 6=i

K∑

k=1

k 6=i,j

pjpknjnk

G2i0p

2i

l−2i ljlkµzj

µzkαjαk·

exp (−2µξi+ µξj

+ µξk+ 2σ2

ξi+

1

2σ2

ξj+

1

2σ2

ξk− 2Cij − 2Cik + Cjk)+

K∑

j=1

j 6=i

p2jn

2j

G2i0p

2i

l−2i l2jρ

2zjαj exp (−2µξi

+ 2µξj+ 2σ2

ξi+ 2σ2

ξj− 4Cij)+

K∑

j=1

j 6=i

N0pjnj

G2i0p

2iTc

l−2i ljµzj

αj exp (−2µξi+ µξj

+ 2σ2ξi

+1

2σ2

ξj− 2Cij)+

N20

4p2iG

2i0T

2c

l−2i exp (−2µξi

+ 2σ2ξi)

In previous expressions, we have denoted with µzjand ρzj

the expectation and the correlation

of zj , respectively, which easily follow from the Nakagami distribution of zj .

With the approximation (5), the SINR in logarithmic units is a Gaussian random variable.

Hence the derivation of the outage probability can be easily accomplished:

Pr[SINRi(h, ν) < γi] = 1 −Q

(

ln γi − µYi(n−i,p)

σYi(n−i,p)

)

, (8)

where Q(x) = 1/√

2π∫ ∞

xe−t2/2dt is the complementary standard Gaussian distribution. From (8),

it is straightforward to show that the constraints on the outage probability in Problem P can be

rewritten asln γi − ψ(m) + ln(m) + µXi

(n−i,p)√

ζ(2,m) + σZi(n−i,p)

≥ qi , (9)

where qi = Q−1(1 −Oi).

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B. Interference Function

Now, we would like to rewrite (9) in a form that will simply the solving of Problem P . Let

us introduce the following definitions:

M(1)i (n−i,p−i) , piM

(1)i (n−i,p) ,

M(2)i (n−i,p−i) , p2

iM(2)i (n−i,p) ,

µXi(n−i,p−i) , 2 ln M

(1)i (n−i,p−i) −

1

2ln M

(2)i (n−i,p−i) ,

σ2Xi

(n−i,p−i) , ln M(2)i (n−i,p−i) − 2 ln M

(1)i (n−i,p−i) .

By looking at the expressions of (6) and (7), it is easy to see that M (1)i (n−i,p−i) and M (2)

i (n−i,p−i)

are functions neither of pi, nor of ni. Using previous definitions, (9) becomes

ln γi − ψ(m) + ln(m) − ln pi + µXi(n−i,p−i)

√

ζ(2,m) + σZi(n−i,p−i)

≥ qi ,

from which it follows that

pi exp (qi√

ζ(2,m) + σZi(n−i,p−i) − µXi

(n−i,p−i) − ln γi + ψ(m) − ln(m)) ≥ γi ,

We can define the interference function as

Ii(n−i,p−i) = exp (−qi

√

ζ(2,m) + σZi(n−i,p−i) + µXi

(n−i,p−i) + ln γi − ψ(m) + ln(m)) ,

(10)

so that it is easy to see that the constraint on the outage probability can be expressed as in (2).

Such an expression allows us to resort to the theory of standard interference function [26] [27],

as we state in the following proposition:

Proposition 1: For any given rate vector n the function

I(n,p) = [I1(n−1,p−1), . . . , Ii(n−i,p−i), . . . , IK(n−K ,p−K)]T

is a standard interference function with respect to p.

Proof: It can be verified by numerical computations that, in the range of parameters of

practical interest, the following properties of the interference function are verified:

1) I(n,p) > 0;

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2) if p ≤ p then I(n,p) ≤ I(n, p);

3) if c > 1, then cI(n,p) > I(n, cp).

Remark 2: It is possible to show that for any given rate vector p the function

I(n, p) = [I1(n−1, p−1), . . . , Ii(n−i, p−i), . . . , IK(n−K , p−K)]T

is monotone non-decreasing with respect to n.

Proposition 1 and Remark 2 will enable to find candidate solutions for Problem P through

a modified optimization problem. Afterwards, a branch-and-bound search will allow us to find

the optimal solution among the candidate set. We approach these issues in the next section.

VI. OPTIMAL SOLUTION

In previous section, we have characterized the outage probability (10). Now, we include such

an expression in Problem P , which thus can be rewritten as

P1 : maxp,n

1Tn

s.t.pi

Ii(n−i,p−i)≥ γi , i = 1, . . . , K

nTp ≤ PT ,

pi ≥ pi0 , i = 1, . . . , K

1 ≤ ni ≤ Gi0, ni ∈ N i = 1, . . . , K

Problem P1 is difficult to solve because the constraint on the total power prevents to apply

iterative algorithms as in [28] and [26]. However, since the rates vector n belongs to a discrete

set N , combinatorial optimization techniques are a good option for solving Problem P1. Whit

this goal in mind, let us note that (n,p) is the solution of Problem P1 only if the first set of

constraints of P1 holds with equality:

Theorem 1: If the pair (n,p) is a solution of Problem P1, then:

pi

Ii(n−i,p−i)= γi ∀ i = 1, . . . , K . (11)

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Proof: The proof is given by contradiction. Assume that the pair (n, p) is the optimal

solution of Problem P1 and that it does not satisfy equation (11), it follows that there exist at

least one index j such that pj

Ij(n−j ,p−j)

> γj . Then, there exists pj < pj such that the powers

vector p = [p1, . . . , pj−1, pj, pj+1, . . . , pK ]T fulfils the following equations

pj

Ij(n−j, p−j)>

pj

Ij(n−j, p−j)= γj

pi

Ii(n−i, p−i)>

pj

Ij(n−j, p−j)≥ γi ∀ i 6= j ,

nT p < PT .

Since Ij(n−j,p−j) is not function of the j-th rate, and Ii(n−i,p−i) is a continuous function

with respect to n−i, it follows that there exists nj > nj such that defining the rates vector

n = [n1, . . . , nj−1, nj, nj+1, . . . , nK ]T verifies the equations

pj

Ij(n−j, p−j)= γj

pi

Ii(n−i, p−i)> γi ∀ i 6= j ,

nT p < PT .

Therefore, starting from (n, p) we have found a pair (n, p), with rates vector n > n. This means

the objective function of the new pair is larger than before: 1T n > 1

T n, which is impossible

because we have assumed that (n, p) solves Problem P1.

Previous theorem allows us to rewrite the first constraints of Problem P1 at the equality. From

this, we propose an approach to solve Problem P1 in two steps. Firstly, we find feasible solutions

by a modified problem P2. Secondly, we solve P1 looking at the set of feasible solutions by

branch-and-bound search. We approach these tasks in the following subsections.

A. Modified problem

Let us consider the following definition:

Definition 1 (Feasible Rates Vector): A rates vector n is feasible if there exists a power vector

p such that the pair (n,p) verifies all the constraints of Problem P1.

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Algorithm 1 Solution of Problem P2.1: t := 0;2: n(t− 1) := 1 ;3: p(t− 1) := 0;4: n(t) := 1 ;5: p(t) := p0 ;6: while ‖p(t) − p(t− 1)‖2 ≥ ε do7: for i := 1 : K do8: pi(t) := Ii(n−i(t− 1),p−i(t− 1))γi

9: end for;10: t := t+ 1;11: end while;

Feasible rates vector for Problem P1 can be found by the solutions of a modified program,

which we define as

P2 : minp

nTp

s.t.pi

Ii(n−i,p−i)= γi , i = 1, . . . , K

pi ≥ pi0 , i = 1, . . . , K

1 ≤ ni ≤ Gi0, ni ∈ N , i = 1, . . . , K

Note that in this problem the decision variable is only p. Consider a pair of vectors p and n

that solve P2. These vectors provide a feasible rate vector for P1 if the cost function is less than

PT . Therefore we can explore the feasibility of a rate vector n just by solving Problem P2.

Problem P2 can be efficiently solved by recalling the properties of the interference function.

Indeed, these properties allow us to use contraction mappings, which give a low computational

cost distributed algorithm that solves Problem P2 by sequences of asynchronous powers (see

e.g. [28, Pag. 431] and [26]). The solution of Problem P2 can be found by applying Algorithm 1.

In the algorithm, p0 is a starting transmit power vector, and ε accounts for the precision of

the solution. Notice that, since the algorithm is an implementation of a contractive mapping,

convergence is ensured.

One could use Algorithm 1 to solve Problem P2 for each possible rate vector in N . Thus, the

rate vector n∗ that solves the Problem P1 could be found using a exhaustive search among all

feasible rate vectors. However, such a technique has large computational costs, as the number

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of feasible rate vectors may be very high. In the next subsection, we will present some useful

properties of the rate vectors, which together with a branch-and-bound search, allow to find the

optimal solution with reduced computational cost.

B. Branch-and-bound search

In previous section, we have seen that n is feasible if and only if Problem P2 has a solution p,

and a cost function that verifies nT p ≤ PT . Using local knowledge about specific values of n, the

set N can be reduced, namely one can find a set of a smaller size over which perform a search

to the optimal solution. We propose the reduction of N using two criteria. The first uses the

cutting-planes idea [28], whereas the second one is based on considerations about Problem P1.

Specifically, the criteria can be summarized in the following propositions.

Proposition 2: If n is feasible, a pair (n,p) is an optimal solution of P1 only if n ≥ n.

Proof: The simple proof is by contradiction. Assume that n < n. Since n is feasible it

follows that there exists a vector p such that (n, p) verifies all the constraints of the Problem P1.

But since n < n, then the cost function associated to the pair (n, p) is larger than the cost function

of (n,p), and then (n,p) cannot be a optimal solution of Problem P1.

Using Proposition 2 it follows that, if we find a feasible rate vector n the search of optimal

solution can be restricted to the set NΣ ⊂ N , where

NΣ(n) = N \ n such that n ≤ n . (12)

Proposition 3: If n is infeasible, then any n such that n ≥ n is infeasible too.

Proof: If n is not feasible, then there is no p such that the pair (n,p) verifies all constraints

of Problem P1. Let us study the cases for each constraint.

Consider the case when n does not verify the first constraint of Problem P1, then there is not

any p such that the pair (n,p) verifies such constraint. Indeed, since for any n ≥ n we have

that Ii(n−i,p−i) ≤ Ii(n−i,p−i) ∀p it follows that also for any n there is not any p so that n

verifies the first constraint.

Consider the case when n does not verify only the second constraint of Problem P1 for any

p. Assume by contradiction that there is n ≥ n which verifies the second constraints of P1, i.e.

nTp ≤ PT ≤ nT p. Then, the power vector p that solves P2 must be less than p, since n ≥ n

and nTp ≤ nT p. Now, since the first constraints of P1 are verified at the equality, we have that

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Ii(n−i,p−i) ≤ Ii(n−i, p−i), and recalling that the interference function is not decreasing with p

(given n), it follows that Ii(n−i, p−i) ≤ Ii(n−i,p−i) ≤ Ii(n−i, p−i), but this is a contradiction,

since we have that Ii(n−i, p−i) ≤ Ii(n−i, p−i), while we assumed that n ≥ n and the interference

function is not decreasing with n, (given p).

Consider the case when n does not verify the fourth constraint of Problem P1, then it obviously

follows that the same constraint is not verified for any n such that n ≥ n. This concludes the

proof.

Previous proposition allows us to say that if n is infeasible, then the set N can be reduced

by defining a set of infeasible solutions NI ⊂ N according to the following rule:

NI(n) = N \ n such that n ≥ n . (13)

In practice, Propositions 2 and 3 are useful to reduce the set of feasible rates by using a

neural network architecture, where computation and information distribution is optimized. In the

following, we define the neural network and how to use it to compute the optimal solution of

Problem P1.

We model a neural network by using an oriented graph. First, let us introduce an equivalent

representation of the vector n. Since ni = 2ri , ri ∈ N, we define the vector r with elements ri

and collect all vectors r in the set Rg, namely

Rg = r = [r1, . . . , rN ]T | ri ∈ N and 1 ≤ ri ≤ gi .

where gi is such that 2gi = G0i. Using such a vector, we define the following oriented graph:

Definition 2 (Rate Graph): The Rate Graph is an oriented graph G = (Rg, E) where Rg is

the set of nodes and E ⊂ Rg ×Rg is the set of oriented edges with (r, r) ∈ E if ‖r − r‖1 = 1

and ‖r‖1 < ‖r‖1.

The Rate Graph is a structure over which branch-and-bound distributed techniques can be

implemented. Indeed, an efficient algorithm can be implemented so that a graph’s node is able

to evaluate its own feasibility solving Problem P2 for its associated rate vector. In particular,

node r computes its own feasibility function:

f(r) =

1 if solving P2 r is feasible

0 otherwise.

15

Algorithm 2 Branch-and-bound Search1: k := 0;2: Pick any A(k) ⊂ Rg;3: S(k) := Rg \ A(k);4: D := ∅;5: for any r ∈ A(k) do6: if f(r) = 1 then7: D := D ∪ r ∪ B(r);8: else9: D := D ∪ r ∪ F(r);

10: end if;11: end for;12: S := Rg \ D;13: A(k + 1) := ∅;14: for any r ∈ A(k) do15: if f(r) = 1 then16: A(k + 1) := A(k + 1) ∪ [S ∩ B(r)];17: else18: A(k + 1) := A(k + 1) ∪ [S ∩ F(r)];19: end if;20: end for;21: if A(k + 1) 6= ∅ then22: k = k + 1;23: go to step 5;24: else25: the search of the optimum is concluded;26: end if;

Each node sends the result of the feasibility function to its adjacent nodes, and, by exploiting

Propositions 2 and 3, this information exchange reduces the set of possible solutions, without

the need to solving Problem P2 for all possible rate vectors in Rg. Specifically, let us introduce

the definition of forward and backward nodes:

Definition 3 (Forward and Backward Nodes): Given a Rate Graph G and a node r ∈ Rg, the

set of Forward Nodes F(r) ⊂ Rg and Backward Nodes B(r) ⊂ Rg are defined, respectively, as:

F(r) = r ∈ Rg|(r, r) ∈ E ,

B(r) = r ∈ Rg|(r, r) ∈ E .

The branch-and-bound search is described in Algorithm 2. According to Algorithm 2, the graph

16

nodes are split in three sets:

A : Active Node Set. It is the set of the nodes that are computing their own feasibility.

S : Stand-By Set. It is the set of the nodes that are waiting to compute their own feasibility.

D : Deleted Node Set. It is the set of the nodes that are deleted from the graph. A node

can be deleted from the graph by itself, if it has already computed its feasibility, or by

other nodes coherently to Propositions 2 and 3.

Lines 1–4 initialize the algorithm. In particular, defines A as any set in Rg (line 2). Then, define

S as the complementary of A, and D as the empty set (lines 3 and 4). Now, any feasible active

node in B(r) has an objective function of Problem P1 smaller than node r. From Proposition 2,

such nodes cannot improve the solution, then they are put in the deleted node set D graph (line

7). Furthermore, each active node which is not feasible has forward nodes clearly not feasible,

since nodes in F(r) have higher rates and Proposition 3 applies. Such nodes are put in the

deleted node set D (line 9). Next, nodes in D are removed from the graph, and a stand by node

set is defined in line 12. Now, a new set of active nodes has to be defined for the time step k+1

(line 13). All nodes who are in the stand-by set and are forward nodes of a feasible node, are put

in the active set at time k+ 1 (line 16), as they may exhibit a larger cost function. Furthermore,

all nodes who are in the stand-by set and are backward nodes of an infeasible node, are put

in the active set at time k + 1 (line 18). Indeed such nodes may be feasible, since they have a

reduced cost function. It is easy to see that this procedure is repeated until at the time k the

active set is emptied, when all (few) nodes in the set A(k − 1) are candidate to provide the

optimal solution of Problem P1. Note that Propositions 2 and 3 play a major role in lines 14–20,

since they avoid the computation of the solution of Problem P2 for any possible rate vector.

By applying Algorithm 2, we are able to find the exact solution of the optimization Problem P1,

since we reduce the set of candidate rate vectors to a smaller set. It is then easy to check within

that set for the optimal solution, since it has few nodes.

VII. NUMERICAL RESULTS

In this section we apply our method to the relevant case of a wireless system of third

generation, where inner loop and outer loop power control mechanisms are used [18]. It is

important to remark here that we do not compare our analytical results with simulations, since

17

the outage approximations we have adopted are very good. In the following, we describe the

system parameters.

Operations of inner loop and outer loop power control are envisaged according to [14], [29]

and [18]. Specifically, the model of the SINR (1) can be used with the Rayleigh fading and the

path loss coefficients set to 1, and imposing that ψ(m)− lnm = 0 and ζ(2,m) = 0. Furthermore,

ξi assumes the meaning of residual fluctuation of the inner loop power control, so that all µξi

are 0. We consider a system with K = 4 users. The system parameters are taken from 3GPP

specifications [18]: we have used a chip time Tc = 2.610−7s (corresponding to a bandwidth of

5MHz), a power spectral density of the thermal noise has been set to N0 = Tc/10, and the

maximum spreading factors is Gi0 = 256. We assumed a value of the SINR threshold γi = 3.1.

Furthermore we assumed that PTTi0/N0 = 22dB, thus getting PT = 0.0391.

Six scenarios are discussed, denoted with A, B, C, D, E, and M. Scenarios from A to E have

uniform standard deviation of the power fluctuation ξi and the activity of the sources αi. The

sixth scenario, denoted M, has mixed values. In Tab. I, the values adopted for σξiand αi are

reported for each case.

In Fig. 1, the solution of Problem P2 for the case M is plotted as obtained with the iterative

Algorithm 1. The outage probability Oi has been set to 0.01. As it can be observed, the

convergence is very fast, and takes less than 5 iterations. For different settings of the parameters,

the behavior of the convergence remains the basically the same.

In Figs. 2 and 3, the optimal solution of Problem P1 obtained with our approach are plotted for

each scenario. Each dot of the curves is referred to a value of the outage probability constraint

(which is reported on the x axis, and it is assumed to be the same for each user). On the y

axis, the optimal objective function of Problem P1 is plotted. Algorithm 2 has been initialized in

the worst case possible, with the rates of the users set all to 1. We observed that the maximum

achievable throughput is 22. Furthermore, as the outage becomes less stringent (i.e., higher values

of the outages are allowed) the objective function obviously increases. It is also interesting to

observe that better performance is obtained when the users have low activity factors (αi = 0.2).

In fact, in such a case, the MAI seen from the users is reduced, thus enabling higher transmit

rates. Also, observe that as the standard deviations of the power control errors increase, the total

rate decreases. Once again, the reason can be found in the fact that larger fluctuations of the

power control error increases the MAI.

18

In Tab. II, the number of nodes explored in the Rate Graph is reported in terms of minimum,

maximum, and average number. These figures have been computed over the interval of outage

probability used (Oi = 0.01, . . . , 0.29). An exhaustive search over all the possible rates would

have required to solve Problem P2 for 94 = 6561 times, since there are 9 possible rates per user,

and 4 users. As it can be noticed, the maximum number of nodes for which P2 is solved is 247.

Since our algorithm was initialized with the lower case of the rates, the convergence speed is

the slowest. Therefore, we believe that employing an initial guess closer to the optimal solution,

it could be possible to increase considerably the convergence speed.

VIII. CONCLUSIONS AND FUTURE WORK

In this paper, we have proposed an optimal approach to allocate the user powers to maximize

the system throughput. A general model of the physical layer of CDMA wireless system has

been considered, and quality of service has been expressed by the outage probability.

The problem can be cast as a mixed integer optimization program, where we have expressed the

constraints by the extended Wilkinson Moment matching approximation of the outage probability.

As a relevant contribution, we have shown that the outage constraints can be tracked down to the

standard interference function theory. This result enabled to derive the optimal solution of the

problem in two steps: first a modified program has been investigated to provide feasible solutions,

and then an algorithm, based on branch-and-bound search, has been proposed. Numerical results

in scenarios of practical interest confirm the validity of our approach.

Ongoing work is focused on the extension of our method to systems with limited computational

resources (as wireless sensor networks), where the implementation of simpler algorithms is a

must. We plan to investigate suboptimal solutions, where the integer constraints on the rates

are relaxed, or less accurate outage probability expressions are used. Initial results are being

obtained employing a Geometric Programming approach.

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Scenario σξiαi

A 0.5, 0.5, 0.5, 0.5 0.4, 0.4, 0.4, 0.4B 0.5, 0.5, 0.5, 0.5 0.2, 0.2, 0.2, 0.2C 0.5, 0.5, 0.5, 0.5 0.7, 0.7, 0.7, 0.7D 0.4, 0.4, 0.4, 0.4 0.4, 0.4, 0.4, 0.4E 0.6, 0.6, 0.6, 0.6 0.4, 0.4, 0.4, 0.4M 0.5, 0.6, 0.4, 0.6 0.2, 0.4, 0.2, 0.7

TABLE ISCENARIOS CONSIDERED IN THE NUMERICAL RESULTS.

Scenario min max averageA 46 247 150B 84 247 153C 47 238 137D 84 247 149E 47 247 141M 31 222 103

TABLE IINUMBER OF NODES EXPLORED TO OBTAIN THE SOLUTION OF PROBLEM P1 FOR EACH SCENARIO. A NODE IS EXPLORED

WHEN PROBLEM P2 IS SOLVED USING THE RATE VECTOR ASSOCIATED WITH THAT NODE.

1 2 3 4 510−3

10−2

Iterations

Pow

ers

User 1User 2User 3User 4

Fig. 1. Convergence of the powers for getting the solution of Problem P2 as obtained with Algorithm 1, for the case M. Thepowers are reported in log units.

22

0.05 0.1 0.15 0.2 0.254

6

8

10

12

14

16

18

20

22

24

Outage Constraint

Rat

e S

um

α = 0.2 ; σ =0.5α = 0.4 ; σ =0.5α = 0.7 ; σ =0.5

Fig. 2. Optimal solution of Problem P1 as obtained with Algorithm 2 for the cases A, B, and C. In the legend, σ and α denotethat σξi

and αi are uniformly chosen.

0.05 0.1 0.15 0.2 0.254

6

8

10

12

14

16

18

20

22

24

Outage Constraint

Rat

e S

um

α = 0.4 ; σ =0.4α = 0.4 ; σ =0.6Mixed Environment

Fig. 3. Optimal solution of Problem P1 as obtained with Algorithm 2 for the cases D, E, and M. In the legend, σ and α

denote that σξiand αi are uniformly chosen.