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A Distributed Dynamic Target-SIR Tracking PowerControl Algorithm for Wireless Cellular Networks

Mehdi Rasti,Student Member, IEEE, Ahmad R. Sharafat,Senior Member, IEEE, andJens Zander,Member, IEEE

Abstract— The well-known fixed target-SIR tracking powercontrol algorithm (TPC) provides all users with their given feasi-ble fixed target-SIRs, but cannot improve the system throughputeven if additional resources are available. The opportunisticpower control algorithm (OPC) significantly improves the systemthroughput but cannot guarantee the minimum acceptable signal-to-interference-ratio (SIR) for all users (unfairness). To optimizethe system throughput subject to a given lower bound forthe users’ SIRs, we present a distributed dynamic target-SIRtracking power control algorithm (DTPC) for wireless cellularnetworks by using TPC and OPC in a selective manner. Inthe proposed DTPC, when the effective interference (the ratioof the received interference to the path-gain) is less than agiven threshold for a given user, that user opportunistically setsits target-SIR (which is a decreasing function of the effectiveinterference) to a value higher than its minimum acceptabletarget-SIR; otherwise it keeps its target-SIR fixed at its minimumacceptable level. We show that the proposed algorithm convergesto a unique fixed point starting from any initial transmit powerlevel in both synchronous and asynchronous power updatingcases. We also show that our proposed algorithm not onlyguarantees the (feasible) minimum acceptable target-SIRs forall users (in contrast to OPC), but also significantly improvesthe system throughput as compared to TPC. Furthermore, wedemonstrate that DTPC along with TPC and OPC can beutilized to apply different priorities of transmission and servicerequirements among users. Finally, when users are selfish, weprovide a game theoretic analysis of our DTPC algorithm via anon-cooperative power control game with a new pricing function.

Index Terms— Distributed power control, dynamic target-SIRallocation, wireless cellular networks.

I. I NTRODUCTION

POWER control plays an important role in satisfying theincreasing demand for high data rates in wireless data

networks. Enhancing the throughput of a user or of the systemis highly dependent on how interference is managed. Theobjective for power control from a user’s point of view isto support a user with its minimum acceptable throughput;

Copyright (c) 2009 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

Manuscript received January 29, 2009, revised Jun 9, 2009. This work issupported in part by Tarbiat Modares University (TMU), Tehran, Iran, in partby Wireless@KTH, Stockholm, Sweden, where the first author was a visitingPhD student, and in part by Iran Telecommunications Research Center, Tehran,Iran under PhD research grant TMU 86-07-54.

M. Rasti and A. R. Sharafat are with the Department of Electrical and Com-puter Engineering, Tarbiat Modares Univesity, P. O. Box 14155-4838, Tehran,Iran. Corresponding author is A. R. Sharafat (email:sharafat@modares.ac.ir)

J. Zander is with the Wireless@KTH, The Royal Institute of Technology,Electrum 418, 160 40 Stockholm, Sweden.

whereas from a system’s point of view is to maximize theaggregate throughput. These two objectives are completelyopposed to one another, since in the former, it is required tocompensate for the near-far effect by allocating higher powerlevels to users with poor channels as compared to users withgood channels, and in the latter, high power levels are allocatedto a few users with best channels and very low (even zero)power levels to others.

A distributed power control is practically preferred to acentralized one, because in the former, the transmit power levelof a user is decided by the user utilizing the local informationand minimal feedback from the base station. In contrast, acentralized approach needs to have path-gains and throughputrequirements for all users at the base station. To satisfy the twostated objectives for power control, two distributed approacheshave been considered, namely the fixed-target-SIR-trackingproposed in [1], and the opportunistic approaches proposedin [2]–[4].

In the fixed-target-SIR-tracking power control algorithm(TPC), each user is tracking its own predefined fixed target-SIR. The TPC that was proposed in [1] enables users toachieve their fixed target-SIRs at minimal aggregate transmitpower if the target-SIRs are feasible. However, there is amajor drawback in the original TPC [5]. It causes users toexactly hit their fixed target-SIRs in feasible systems even ifadditional resources are still available that can otherwise beused to achieve higher SIRs (and thus better throughputs).Besides, the fixed-target-SIR assignment is suitable only forvoice service for which reaching a SIR value higher than thegiven target value has no practical effect on service quality(due to characteristics of the service and human ears). Incontrast, for data services, a higher SIR results in a betterthroughput, which is desirable. Thus, it is important to designa power control algorithm for wireless data networks by whichthe minimum acceptable target-SIRs (which are assumed to befeasible) are guaranteed for all users, and at the same time, thesystem throughput is increased to the extent that the requiredresources are available by increasing the actual SIRs receivedby some users.

From the system’s point of view, OPC allocates high powerlevels to users with good channels (experiencing high path-gains and low interference levels), and very low power to userswith poor channels. In this algorithm, a small difference inpath-gains between two users may lead to a large differencein their actual throughputs [2], [3]. Since an opportunisticalgorithm always favors those users with better channels, itmagnifies unfairness. For users with low-mobility (when their

channels vary slowly or are static), this might lead to long-termunfairness.

The characteristics of existing distributed power controlschemes are summarized as follows. TPC can provide allusers with their fixed target-SIRs when the system is feasible,but cannot improve the system throughput further even ifadditional resources are available. OPC significantly improvesthe system throughput but cannot guarantee the minimumacceptable SIRs for some users (unfairness).

Motivated by the above drawbacks, in this paper, we for-mally define the problem of system throughput maximizationsubject to a given feasible lower bound for the achievedSIRs of all users in wireless cellular networks; and proposea distributed dynamic power control algorithm to address thisproblem. We will show that when the minimum acceptabletarget-SIRs are feasible, the actual SIRs received by someusers can be dynamically increased (to a value higher thantheir minimum acceptable target-SIRs) in a distributed manner,so far as the required resources are available and the systemremains feasible (meaning that reaching the minimum target-SIRs for the remaining users are guaranteed). This wouldenhance the system throughput (at the cost of higher powerconsumption) as compared to TPC. We will show that our pro-posed algorithm not only guarantees the (feasible) minimumacceptable target-SIRs for all users as is the case by TPC (andin contrast to OPC), but also significantly improves the systemthroughput as compared to TPC. Furthermore, we discuss theapplication of DTPC to cognitive radio networks and multi-service networks; and provide a game theoretic analysis of ourproposed algorithm.

The rest of this paper is organized as follows. In SectionII, we introduce the system model and throughput measure. InSection III, we review the existing distributed power controlalgorithms, present a formal statement of the problem, andstate the objectives. Section IV contains the proposed method,and an analysis of its convergence and its improved systemthroughput. Application of our proposed algorithm in cognitiveradio or multi-service networks is discussed in Section V.Section VI provides a game theoretic analysis of our proposedalgorithm. Simulation results and conclusions are presented inSections VII and VIII, respectively.

II. SYSTEM MODEL AND SIR FEASIBILITY

We consider a multi-cell wireless CDMA network withKbase stations (cells) andM active users denoted byK ={1, 2, ..., K} andM = {1, 2, ..., M}, respectively. Letpi bethe transmit power of useri. Noise is assumed to be additivewhite Gaussian whose power at the receiver of the base stationk is σ2

k. Let si denote the base station to which the useri isassigned. Denote the set of users assigned to the base stationk by Ck = {i ∈M|si = k}.

The receiver is assumed to be a conventional matched filter.Thus, for a given transmit power vectorp=[p1, p2, ..., pM ]T,the SIR of a useri, denoted byγi is

γi(p) = gihsii

Ii(p)pi, (1)

wheregi is the processing gain for useri (defined as the ratioof chip rate (or the spreading bandwidth) to the transmit datarate), hsii is the path-gain from useri to its assigned basestation andIi(p) =

∑j 6=i

hsijpj + σ2si

is the interference expe-

rienced by useri at its assigned base-station. The interferenceIi(p) can be rewritten asIi(p) = I int

i + Iexti + σ2

siwhere

I inti =

∑j∈Csi

,j 6=i

pjhsij and Iexti =

∑j /∈Csi

pjhsij are the intra-

cell interference and the inter-cell interference, respectively.The effective interference for useri is defined as the ratio ofits experienced interference to its path-gain [3], denoted byRi,

Ri(p) =Ii(p)gihsii

. (2)

The value ofRi represents the channel status for useri, i.e.for a given processing gain, a higher interference and a lowerpath-gain result in a higherRi, implying a poor channel ascompared to a lower interference and a higher path-gain whichresult in a lowerRi, implying a good channel.

Using matrix notations, the relation between the transmitpower vector and the SIR vector can be rewritten as

p = G.p + η (3)

where the(i, j) component ofG is Gi,j = (hsijγi)/(gihsii)if i 6= j, andGi,j = 0 if i = j, and the(i) component ofη isηi = (σ2

siγi)/(gihsii). A SIR vectorγ is feasible if a power

vectorp ≥ 0 exists that satisfies (3). It was shown in [6] thatthe necessary and sufficient condition for feasibility of a givenSIR vectorγ is ρ(G) < 1, whereρ(G) is the spectral radius(maximum eigenvalue) of the matrixG.

From (3) and with some mathematical manipulations, aone-to-one relation between a transmit power vector and theachieved SIR vector is obtained [4], [7], [8]; that is

pi =γi

hsii(γi + gi)× σ2

si+ Iext

i

1− ∑j∈Csi

γj

γj + gj

, for all i ∈M. (4)

Thus a SIR vector is feasible if∑

j∈Ck

γj

γj + gj< 1, for all k ∈ K. (5)

The interesting and useful point from the feasibility constraint(5) is that feasibility can be individually checked in eachcell by checking the sum of γj

γj+gjfor all users in that cell.

Note that for checking the feasibility, using (5) is simpler, ascompared to checkingρ(G) < 1 that requires calculation ofthe spectral radius of the matrixG.

The left-side of the inequality (5) is the Pareto efficiencymeasure for users’ SIR allocations in the sense that for a givenbase-stationk, as

∑j∈Ck

γj

γj+gjapproaches 1, the SIR of no user

assigned to that base-station (i.e.Ck) can be further increasedwithout decreasing the SIRs of some other users inCk. Inother words, a value of

∑j∈Ck

γj

γj+gjcloser to 1 for eachk ∈ K

means a more Pareto efficient use of resources and a highersystem throughput; and a value farther from 1 means moreresources are available to increase the users’ SIRs.

Similar to [4], [7], [9] and [10], we use an informationtheoretic approach to define the throughput for each useri by

Ti(p) = W log2(1 + γi(p)) , (6)

where W is the channel bandwidth. The system throughput(the sum of achievable rates by users) is

T (p) =∑

i

Ti(p). (7)

III. PROBLEM FORMULATION AND BACKGROUND

A. Existing Distributed Power Control Algorithms

In a distributed power control algorithm, each useri updatesits transmit power by a power-updating functionfi(p), i.e.pi(t + 1) = fi(p(t)) wherep(t) is the transmit power vectorat time t. The fixed-point of the power-updating function,denoted byp∗, is obtained by solvingp∗ = f(p∗). If a dis-tributed power control algorithm converges to an equilibriumstate, it will be a fixed-point of the corresponding power-updating function.

The power-updating function of TPC is

f(T)i (p(t)) = γiRi(p(t)) (8)

where γi is the given target-SIR for useri. It was shownin [1] and [11] that if the target-SIR vectorγ is feasible,then TPC converges either synchronously or asynchronouslyto a fixed point at which users attain their target-SIRs withminimum aggregate transmit power, i.e. its fixed point solvesthe following optimization problem

minp≥0

∑

i

pi

subject toγi(p) ≥ γi, ∀i ∈M. (9)

In OPC, the transmit power levels are updated in a manneropposite to TPC, i.e. it is increased when the channel is goodand is decreased when the channel is poor. The power-updatingfunction of OPC is

f(O)i (p(t)) =

ηi

Ri(p(t))(10)

where ηi is a constant for useri. In this algorithm, eachuser i tries to keep the product of its transmit power and itseffective interference to a constantηi, called the target signal-interference product (SIP). This algorithm is convergent, andalthough it does not guarantee optimal system throughputdefined by

maxp≥0

∑

i

Ti(p), (11)

it significantly enhances the system throughput by transmit-ting at high power levels by users with good channels, andtransmitting at low power levels by users with poor channels;but leads to unfairness as well. In TPC and OPC, each usersets its own target-SIR and its own target-SIP, respectively,meaning that each user requires only to know its SIR measuredat the base station (i.e. minimal feedback information). Notethat each useri can obtain the value ofRi(p(t)) in (8) and(10) by knowing its achieved SIR at its assigned base-stationby usingRi(p(t)) = pi(t)/γi(p(t)).

B. Problem Statement and Objectives

We now introduce the problem of system throughput op-timization subject to a given feasible lower bound for users’SIRs as

maxp≥0

∑

i

Ti(p)

subject to γi(p) ≥ γi, ∀i ∈M. (12)

The constraint on the received SIR, i.e.γi(p) ≥ γi, isequivalent to a constraint on throughput, i.e.Ti(p) ≥ Ti,where Ti = W log2(1 + γi). It can be shown that the aboveoptimization problems is non-convex and thus it cannot besolved by using the conventional methods. Our focus in thispaper is to design a distributed convergent power controlalgorithm that can significantly improve the system throughputwhile guaranteeing the minimum acceptable target-SIRs for allusers.

IV. T HE PROPOSEDMETHOD: DYNAMIC ALLOCATION OF

TARGET-SIRS

When the minimum acceptable target-SIRs are feasible,the actual target-SIRs tracked by some users with betterchannels should be dynamically set to a value higher than theirminimum acceptable target-SIRs to the extent that the requiredresources are available and the system remains feasible (i.e.the constraint (5) is satisfied). This will enhance the systemthroughput (at the cost of consuming more power).

It is well-known that for enhancing the system throughput,users with good channels should transmit at higher powerlevels compared to other users [2], [3]. On the other hand,reducing the outage necessitates that users with poor channelsalso transmit at just enough power to at least reach theirminimum acceptable SIRs. Based on these observations, wepropose that users with good channels set their power levels inan opportunistic manner and users with poor channels transmitin a minimum-acceptable-target-SIR tracking manner. Thiscan be done in a distributed manner if each useri updatesits transmit power according to the following power controlalgorithm

f(D)i (p(t)) =

{ ηi

Ri(p(t)), if Ri(p(t)) < Rth

i

γiRi(p(t)), if Ri(p(t)) ≥ Rthi

(13)

where Rthi is the effective interference threshold,ηi is a

constant, andγi is the minimum-acceptable-target-SIR for useri. Note that this algorithm is a generalized selective scheme ofeither OPC or TPC. At the extreme cases whereRth

i → 0 orRth

i → ∞, the proposed power-updating function turns intoeither TPC or OPC, respectively.

Our proposed dynamic target-SIR tracking power controlalgorithm (DTPC) is explained below. We rewrite (13) as theDTPC power-updating function

f(D)i (p(t)) = Γi(p(t))Ri(p(t)) (14)

in which Γi(p(t)) is a target-SIR dynamically set as

Γi(p(t)) =

{ ηi

R2i (p(t))

, if Ri(p(t)) < Rthi

γi, if Ri(p(t)) ≥ Rthi .

(15)

Fig. 1. Dynamic target-SIR vs. effective interference in DTPC.

For DTPC to be continuous, its three parameters are adjustedas follows. For a givenγi andηi, the value ofRth

i is set as

Rthi =

√ηi

γi. (16)

The minimum acceptable target-SIR in (14) is set exactlyto the same value of the fixed-target-SIR that was set in (8).We usefixed target-SIRand minimum acceptable target-SIRinterchangeably throughout this paper. As seen in Fig. 1, byusing DTPC, a given user sets its target-SIR at the minimumacceptable value when the channel is not good (the value ofthe effective interference is higher than the threshold). Whenits channel is good, it opportunistically sets its target-SIR at avalue higher than the minimum acceptable value. When the setof minimum target-SIRs are feasible in a non-Pareto efficientmanner (i.e. additional resources are available that can be usedto further enhance users’ SIRs, meaning that the sum ofγj

γj+gj

is far from 1), some users with better channels reach higherSIRs as compared to the minimum acceptable value, and theremaining users exactly hit their minimum target-SIRs. Thismeans that the system throughput is increased while the users’minimum target-SIRs are guaranteed, as we will show in thesequel. Such enhancement on the system throughput causesall users to consume more power as compared to the fixed(minimum-acceptable) target-SIR tracking scheme.

A. Convergence of DTPC

A framework for examining the convergence of the so calledstandard power-updating functions such as TPC was providedin [11]. This framework was generalized to a new frameworkin [2], applicable to a wider range of distributed power controlalgorithms (covering both TPC and OPC) with a two-sidedscalable power updating function. Our proposed DTPC fallsinto this generalized framework. Using the key properties ofa two-sided scalable function, we will show that the DTPC’spower-updating function has a unique fixed point to whichit converges. To show this, we define the two-sided scalablefunctions as in [2], and prove that the DTPC’s power-updatingfunction (14) is two-sided scalable.

Definition 1: A power-updating function f(p) =[f1(p), f2(p), ..., fM (p)]T is two-sided scalable if forall a > 1, 1

ap ≤ p′ ≤ ap implies

1afi(p) ≤ fi(p′) ≤ afip) for all i ∈M. (17)

Theorem 1:The DTPC’s power-updating functionf (D)(p)in (14) is two-sided scalable.

Proof: It has been shown in [2] that for the given twosided scalable functionsf(p) and f ′(p), their component-wise maximum or minimum, i.e.max{f(p), f ′(p)} ormin{f(p), f ′(p)}, respectively, are two-sided scalable. Sincethe power-updating functions corresponding to the fixed-target-SIR tracking and the opportunistic power control al-gorithms are two sided-scalable [2], and since the proposedpower-updating function can be rewritten asf

(D)i (p(t)) =

max{ ηi

Ri(p(t)), γiRi(p(t))}, then this theorem is proved.

Theorem 2:If there exists a fixed point for the proposedpower-updating function, i.e. if there exists ap∗(D) so thatp∗(D) = f(p∗(D)), then

I. This fixed point is unique, andII. For any initial power vector, the power control algorithm

p(t + 1) = f(p(t)) converges to this unique fixed-point.

Proof: This theorem is directly concluded from Theorem1 as a key property of a two-sided scalable function [2].

Theorem 3:There exists a fixed-point for the DTPC’spower-updating functionf (D)(p) if the minimum acceptabletarget-SIR vectorγ is feasible.

Proof: It has been proved in [2] that iff(p) is two-sidedscalable, continuous, andf(p) ≤ u for all p (i.e. f(p) is upperbounded) foru > 0, then a fixed point forf exists. Since theproposed power-updating function is two-sided scalable andcontinuous, by employing (16), the existence of a fixed-pointfor the proposed power-updating function is guaranteed if itis upper bounded. Now we show that whenγ is feasible, theDTPC’s power-updating function is upper bounded. Whenγ isfeasible, a fixed-point denoted byp∗(T) exists for the power-updating function of TPC (8). IfRi(p∗(T)) ≥ Rth

i for all i ∈M, thenp∗(T) is also the fixed-point of the proposed power-updating function for DTPC denoted byp∗(D), i.e. p∗(D) =p∗(T). This means that at the fixed-point of DTPC, all usersare in the TPC mode. Otherwise, at the fixed-point of DTPC,there exist some users (at least one user) operating in the OPCmode, i.e. for whichRi(p∗(D)) < Rth

i and the remaining usersare operating in the TPC mode. For TPC, the existence of afixed-point depends only on path-gains and target-SIRs, anddoes not depend on the level of background noise [12]. Onthe other hand, the transmit power of users operating in theOPC mode appear as the background noise to users operatingin the TPC mode. The transmit power levels set by users inthe OPC mode are upper bounded because the OPC’s power-updating function is always upper bounded, and the transmitpower levels set by users operating in the TPC mode are alsoupper bounded because the set of their fixed-target-SIRs isfeasible.

B. Improvement in System Throughput by Using DTPC

In the following theorem, we will show that by usingDTPC for a given feasible set of minimum target-SIRs, thesystem throughput is improved as compared to TPC, whileguaranteeing that all users are supported at least with theirminimum target-SIRs.

Theorem 4:Given a feasible target-SIR vectorγ, let p∗(T)

andp∗(D) be the fixed points of TPC and DTPC, respectively.We haveγi(p∗(D)) ≥ γi for all i ∈ M and thusT (p∗(D)) ≥T (p∗(T)). More specifically,

I. If and only if Ri(p∗(T)) ≥ Rthi for all i, thenγi(p∗(D)) =

γi for all i ∈M and thusT (p∗(D)) = T (p∗(T)).II. If and only if there exists at least one user with

Ri(p∗(T)) < Rthi , then for a non-empty set of users,

we haveγi(p∗(D)) > γi, and for the remaining users wehaveγi(p∗(D)) = γi, and thusT (p∗(D)) > T (p∗(T)).Proof: If Ri(p∗(T)) ≥ Rth

i for all i, then from (8)and (14), it is easy to see that the fixed-point of TPC isthe same as the fixed-point of DTPC, i.e.p∗(D) = p∗(T).Thus, in this case, we haveγi(p∗(D)) = γi(p∗(T)) for alli ∈ M, which proves Part I. For Part II, the (non-empty)set of users withRi(p∗(T)) < Rth

i is denoted byN 6= ∅.In this case, there exists a non-empty subset of users inN , denoted byL, for which we haveΓi(p∗(D)) > γi, i.e.Ri(p∗(D)) < Rth

i for all i ∈ L. If this is not true, thenwe haveRi(p∗(D)) ≥ Rth

i for all i ∈ M, meaning thatp∗(D) = p∗(T) and consequentlyRi(p∗(T)) ≥ Rth

i for alli ∈ M, which contradicts the existence of at least one userwith Ri(p∗(T)) < Rth

i . Thus from this and from (8) and (14),we conclude thatγi(p∗(D)) > γi(p∗(T)) for each useri ∈ L,and γi(p∗(D)) = γi(p∗(T)) for each useri ∈ M \ L, whichproves Part II.

V. A PPLICATION OFDTPC IN COGNITIVE RADIO

NETWORKS ORMULTI -SERVICE NETWORKS

It is important to employ proper distributed power con-trol schemes for different service requirements (e.g. real-time versus non-real time services) or for different networkparadigms (e.g. a network with only primary users versus theone with both primary and secondary users). In this section,we demonstrate that DTPC along with TPC and OPC can beproperly employed to apply different priorities of transmissionand service requirements.

A. Application of DTPC in Cognitive Radio Networks

Cognitive radio is a promising technique for improvingthe efficiency of using the scarce frequency spectrum. Thistechnique allows unlicensed (secondary) users to opportunisti-cally access the frequency spectrum concurrently with licensed(primary) users, provided that the interference caused by sec-ondary users would not deny any primary user its throughputrequirements. In other words, primary users have a higherpriority of transmission than secondary users.

In previous sections, we assumed equal priority of trans-mission for all users. Now we consider a cognitive radionetwork in which primary and secondary users coexist. Let

us denote the sets of primary and secondary users byP andS, respectively. We adjust the effective interference thresholdin DTPC for primary and secondary users so that primaryusers are protected against all secondary users, and at thesame time, secondary users can use the available resourcesopportunistically. In the following theorem, we show thatwhen the minimum-target-SIR requirements for primary usersare feasible, if DTPC is utilized by primary users and OPCis utilized by secondary users, then secondary users do notprevent primary users from hitting their target-SIRs.

Theorem 5:Suppose that each primary user applies DTPCwith its given effective interference threshold, and each sec-ondary user employs OPC. If the set of minimum-target-SIRs for primary users are feasible, then for any number ofsecondary users:

I. A unique fixed-point exists, to which the set of transmitpower levels for all users converge.

II. All primary users are supported at least with their mini-mum target-SIRs, i.e.γi(p∗(D)) ≥ γi, and for secondaryusers we haveγi(p∗(D)) = Γi(p∗(D)).Proof: Since the power-updating function corresponding

to either DTPC or OPC is two-sided scalable as shown inTheorem 3 and in [2], respectively, we conclude that if thereexists a fixed-point, then it is unique, and to which, theset of transmit power levels for all primary and secondaryusers converge. Thus, to prove Part I, we need only to showthe existence of a fixed-point. This can be proved similarto Theorem 2 by noting that the transmit power levels ofsecondary users operating in the OPC mode appear as thebackground noise to primary users, and for the DTPC, theexistence of the fixed-point does not depend on the backgroundnoise. Part II can be easily proved as follows. Since primaryusers employ DTPC, thus at the fixed-point (whose existenceand uniqueness is guaranteed as shown in Part I), each primaryuser obtains a SIR higher than its minimum acceptable target-SIR, i.e. γi(p∗(D)) ≥ γi. Also, since each secondary useremploys OPC, a given secondary useri obtains a SIR equal to

ηi

R2i (p∗(D))

, which may be higher or lower than its minimum

SIR.In the above scheme, so long as the set of minimum

acceptable target-SIRs for primary users is feasible, eachprimary user is satisfied and a given secondary user, dependingon its path-gain and its experienced interference (which inturn depends on the number of active primary users and theirtransmit power), obtains an acceptable or a non-acceptableSIR. We will see in simulation results that as the numberof primary users is decreased (increased), the transmit powerlevels and the achieved SIRs for secondary users are increased(decreased).

B. Application of DTPC in Multi-Service Networks

Service requirements for voice and data applications aredifferent with respect to their tolerance against delay and biterror rates. We consider the following three classes of service.

• Class I. (voice service): tolerant to bit error rates andsensitive to delays.

• Class II. (real-time data services): sensitive to both biterror rates and delays.

• Class III. (non-real-time data services): sensitive to biterror rates and tolerant to delays.

For a data service that is sensitive to bit error rates, its target-SIR is higher than that of a voice service. For users of voiceservice, TPC is suitable. In contrast, for data services, a higherSIR results in a better throughput. We consider two types ofdata services, namely real-time (e.g. online video) or non-real-time (e.g. Email). A real-time data service and voice servicehave transmission priority as compared to a non-real-time dataservice, implying that if the system is not feasible, the users inclass III should be softly removed (i.e. decrease their target-SIRs) in favor of users in classes I and II. Thus, to satisfytransmission priority and SIR requirements, users in classes Ito III should apply TPC, DTPC, and OPC, respectively.

VI. GAME-THEORETICANALYSIS OF DTPC

In this section, we deal with selfish users who do not coop-eratively follow a predefined strategy (in contrast to what wehave assumed so far) unless their own utilities are maximized.In what follows, assuming selfish and non-cooperative users,we investigate how and with what pricing scheme, at theNash equilibrium (NE) of the power control game, the systemthroughput is improved similar to DTPC.

Game theory is a powerful mathematical tool to model andanalyze such selfish and interactive decision making. In a gametheoretic view of the power control problem [4], [13], [14],users are players. Each useri ∈ M chooses its strategy bysetting its own transmit power level from its strategy spacePi = [0,∞), resulting in a utility value that represents thethroughput of that user, as well as its associated costs, denotedby Ui(pi,p−i), wherep−i denotes the transmit power vectorfor all users except useri. The commonly used concept insolving game theoretic problems is the NE at which no usercan improve its utility by unilaterally changing its transmitpower.

It is well established that in contrast to the case in whichno pricing is applied, the pricing scheme could affect theindividual user’s decision in such a way that the efficiencyof NE from a given goal’s point of view (e.g. from fairnessor system’s point of view) is improved. In [4], for a concavethroughput function that is increasing with respect to SIR, wehave shown that the SIR-based pricing can be utilized to satisfysome specific goals (such as fairness, aggregate throughputoptimization, or trading off between these two goals) at NE.In [2], [3] it was shown that the best response function for theutility function Ui(pi,p−i) =

√γi − αipi corresponds to the

opportunistic power control scheme. Note that the throughputfunction defined in [2], [3] has no physical meaning.

A pricing-based utility function for a given useri in manynon-cooperative power control games (NPCGs) is

Ui(pi,p−i) = Ti(pi,p−i)− ci(pi,p−i), (18)

whereTi is the function representing useri’s throughput, andci is its pricing function. We use the logarithmic function ofSIR as the throughput defined in (6), but our results can be

applied to any other throughput function that is an increasingand concave function of SIR. In what follows, we set upa NPCG with a pricing scheme that is linearly proportionalto SIR, and analytically show that with a proper choice ofpricing (proportionality constant), its outcome is the same asthat of our proposed DTPC, implying that at NE, the systemthroughput is improved similar to DTPC.

We propose a SIR-based pricing scheme in which the utilityfunction for useri is

Ui(pi,p−i) = Ti(pi,p−i)− αiγi(pi,p−i), (19)

whereαi ≥ 0 is the price per unit of the actual SIR at thebase station for useri. We will show that this pricing schemeenables us to adequately influence the best response functionof each user for improving the throughput (similar to DTPC)by a proper choice of pricing introduced in the followingtheorem.

Theorem 6:In NPCGG = 〈M, (Pi), (Ui)〉 in which Ui is(19), the best response of useri ∈M to a given power vectorp−i is the same as our proposed power-updating function ofDTPC in (14) with the given values ofγi andRth

i if the pricingin (20) for each useri is set to

αi =

kR2i

R2i + γiRth

i2 , if Ri ≤ Rth

i

k

1 + γi, if Ri > Rth

i ,

(20)

wherek = Wln 2 . Thus, a unique NE for this game exists, and is

the fixed point of DTPC, i.e. the fixed-point of DTPC denotedby p∗ is the solution to

maxpi∈Pi

Ti(pi,p∗−i)− αiγi(pi,p∗−i), for all i ∈M. (21)

Proof: To obtain the best response function denoted bybi(p−i), we use the first and the second derivatives of thepricing-based utility with respect topi

∂Ui

∂pi=

1Ri

(k

1 + γi− αi

), (22)

∂2Ui

∂p2i

= − 1Ri

2

k

(1 + γi)2. (23)

For a givenRi, we note from (22) that∂Ui/∂pi = 0 has a

unique root, that ispi =Rth

i2

Riγi if Ri ≤ Rth

i , and ispi =

γiRi if Ri > Rthi . From (23), we note that∂2Ui/∂p2

i <0, which means the unique root of∂Ui/∂pi = 0 globallymaximizesUi. Thus the best transmit power in response top−i that maximizesUi is also unique and is given by

bi(p) =

Rthi

2

Riγi, for Ri ≤ Rth

i

γiRi, for Ri > Rthi

(24)

which is the same as in DTPC, and its fixed point is the NEfor G = 〈M, (Pi), (Ui)〉. As there exists a unique fixed pointin DTPC, the NE exists and is unique. Thus, the fixed-pointof DTPC is the unique solution to (21).

Note that the proposed pricing in (20) implies that when theeffective interference for a given user is below the threshold,

its pricing is increased as its effective interference is increasedin order to discourage selfish users from transmitting at highpower levels until a threshold for the effective interference isreached upon which, its pricing is fixed.

Another important property of the proposed pricing schemeis that it depends on the information pertinent to that useronly. This enables each user to compute its pricing in (20) ina distributed and iterative manner by

αi(t + 1) =

kpi(t)2

pi(t)2 + γi(t)γiRthi

2 , ifpi(t)γi(t)

≤ Rthi

k

1 + γi, if

pi(t)γi(t)

> Rthi ,

(25)which is obtained from (20) by usingRi(p(t)) =pi(t)/γi(p(t)). This is in contrast to the pricing schemes de-veloped in [13], [14] that require the base station to announcethe pricing to each user.

VII. S IMULATION RESULTS

Now we provide simulation results to get an insight intohow our proposed DTPC improves the system throughputwhile guaranteeing a minimum value for the individual user’sthroughput. We assume a processing gain of 100. The AWGNpower at the receiver, i.e.σ2, is assumed to be -113 dBm. Asin [14], we adopt a simple modelhi = kd−4

siifor the path-gain

wheredsii is the distance between useri and its base stationsi, k is the attenuation factor that represents power variations,and takek = 0.09. In our simulations, for DTPC we take thetarget-SIPηi = 1 (as in [2], [15]) for all users. The values ofγi are the same in TPC and in DTPC.

We first consider a single-cell to investigate how our pro-posed algorithm works as compared to TPC and OPC. Thenwe consider multi-cell networks, first a two-cell network toinvestigate how DTPC deals with users’ movements, andfinally proceed to 4-cell networks each with a different numberof users for two separately simulated cases (with and withouta cognitive network). In this way, we see the performance ofDTPC in multi-cell wireless networks for different numberof users for a primary network only, and for a cognitive(secondary) radio network.

A. Single-Cell Network

Consider 4 users indexed from 1 to 4 with the sameminimum target-SIR of 13 dB in a single-cell environmentwhere their distance vector isd = [300, 530, 740, 860]T m,in which each element is the distance of the corresponding userfrom the base station. Figs. 2-4 show the transmit-power andthe received SIR in each iteration (power-updating step) forusers when TPC, OPC, and DTPC are applied, respectively.In TPC (Fig. 2), all users reach their minimum target-SIRs. InOPC (Fig. 3), the user with the best channel (user 1) transmitsat high power and obtains a high SIR, but the other users obtainvery low SIRs, much lower than their minimum target-SIR. Aswe see in Fig. 4, in our proposed DTPC, in contrast to OPC,the minimum target-SIR for each user is guaranteed, and thesystem throughput is notably improved as compared to TPC.

Fig. 2. Transmit power and SIRs for each user in TPC.

Fig. 3. Transmit power and SIRs for each user in OPC.

Fig. 4. Transmit power and SIRs for each user in DTPC.

Fig. 5. Transmit power and SIRs for each user with a different service inDTPC.

Fig. 6. Average system throughput per Hz per user (∑

i(log2(1 +γi))/W/M , where W and M are the allocated bandwidth and the totalnumber of transmitting users, respectively) achieved by TPC and DTPC vs.target-SIR.

Specifically, by using DTPC, user 1 which has the best channelutilizes the additional available resource (power) and obtainsa SIR of 20 dB which is 7 dB higher than its minimum target-SIR by dynamically increasing its target-SIR, and at the sametime other users hit their minimum target-SIRs, i.e. 13 dB.

Now consider that the above 4 users are running differentservices with different target-SIRs. Suppose that user 3 utilizesa voice service for 6 seconds withγ1 = 13 dB, users 1 and 2utilize a real-time data service withγ2 = γ4 = 15 dB for 2 and4 seconds, respectively, and user 4 utilizes a non-real-time dataservice withγ3 = 18 dB for 6 seconds. All users are activefrom the start-time (t = 0) and each user stops its transmissionimmediately after its respective activity time elapses, i.e. users1-4 are active in time-intervals of[0, 6], [0, 2], [0, 4], and[0, 6],respectively. Note that when all users are active, the system is

infeasible (becausei=4∑i=1

γi

γi+gi= 1.03 > 1), meaning that the

minimum acceptable target-SIRs of users are not reachable atthe same time. As stated in Section V-B, in accordance withthe users’ service requirements, users 1-4 set their transmitpower levels according to DTPC, DTPC, TPC, and OPC,

respectively. Each user updates its transmit power every 1 ms(1 KHz). Fig. 5 shows the transmit-power and the received SIRversus time for users 1-4. Note that for the time interval[0, 2]seconds during which all users are active, the voice-serviceuser obtains its target-SIR, the two real-time data service users,i.e. users 1 and 2 obtain SIRs equal to and higher than theirminimum target-SIRs, respectively, and the non-real-time dataservice user obtains the low SIR of -22.76 dB. During thisinterval, the three real-time (voice and data) service users areprioritized with respect to the non-real time data service user.When user 1 leaves the system (att = 2), the interference isdecreased, the remaining real-time data user (i.e. user 2) andthe non-real time data service user (i.e. user 4) immediatelyincrease their transmit power and obtain higher-SIRs while thevoice service user continues to receive its target-SIR. Similarly,when user 2 leaves the system (t = 4), user 4 obtains anacceptable SIR. Such prioritization is applied in a distributedand automatic manner, as each user employs an appropriatedistributed power control algorithm in accordance with itsservice and priority requirements.

To show that similar improvements in system throughputare obtained for different snapshots of users’ locations andfor different values of target-SIRs, we consider a single-cellwireless network with a radius of 250 m and with 10 fixedusers. The minimum target-SIRs are considered the same forall users, ranging from 2 dB to 10 dB with a step size of 0.5dB. For each target-SIR, we average the corresponding valuesof the system throughput at equilibrium, where the algorithmconverges for both TPC and DTPC, for 1500 independentsnapshots from a uniform distribution of users’ locationswithin a single-cell.

Fig. 6 shows the average system throughput per Hz peruser, i.e.

∑i(log2(1+γi))/W/M obtained by TPC and DTPC

versus target-SIRs. The average system throughput obtained byapplying OPC is 50. Although this is very high as comparedto TPC and DTPC, it does not guarantee an average SIR forall users, and only users with better channels obtain acceptableSIRs.

In Fig. 6 we observe that DTPC results in an improvedsystem throughput compared to TPC, as was proved in Theo-rem 4. At the same time, all users are supported by utilizingDTPC at least with their minimum target-SIRs, which is notthe case when OPC is employed. As expected, increasing theminimum acceptable target-SIRs for users results in decreasingthe difference between system throughputs of DTPC and TPC.The reason is that when the minimum acceptable target-SIRγi for each useri is increased, then

∑i

γi

γi+giis increased and

the feasible system approaches an infeasible system, meaningthat the users’ target-SIRs cannot be increased further, becauseotherwise we may have

∑i

γi

γi+gi> 1, i.e. the system be-

comes infeasible. In other words, for high values of minimumtarget-SIRs (e.g. values close to 10 dB in our simulation),after providing all users with their minimum target-SIRs, theremaining resources are not sufficient to further improve thesystem throughput. For instance, for the target-SIR of 10 dB,we have

∑i

γi

γi+gi= 0.91, which means very little is available

Fig. 7. Distribution of users and base stations in a two-cell wireless network.Users are marked by “©”, and base stations are marked by “4”. Users 1 to6, 8 and 9 are fixed, and user 7 att = 0 starts moving from the starting-pointx = 700 m in cell No. 2 towards the end-pointx = 300 in cell No. 1 alongthe illustrated line at a uniform speed of 20 m/s (72 km/h).

Fig. 8. Transmit-power and SIRs versus time for users 2, 7, and 8 usingDTPC in the two-cell network of Fig. 7.

to further improve the system throughput as compared to thetarget-SIR of 6 dB, corresponding to

∑i

γi

γi+gi= 0.38.

B. Multi-Cell Networks

1) Two-Cell Networks:To show how DTPC works whenusers move, we assume a two-cell wireless network with 9users as shown in Fig. 7. The minimum target-SIR for allusers is 13 dB. Suppose that users 1 to 6, 8, and 9 are fixed,and user 7 att = 0 starts moving from its starting point incell No. 2 towards cell No. 1 along the line in Fig. 7 at auniform speed of 20 m/s (72 km/h). The movement of user 7from the starting point to the end point lasts 20 seconds. Eachuser updates its transmit power every 1 ms (1 KHz) employingDTPC. When user 7 enters cell No. 1, i.e. att = 10, base-station 1 is assigned to it. Excluding the moving user 7, wenote that user 2 in cell No. 1, and user 8 in cell No. 2 are theclosest users to the base station in their corresponding cells.Fig. 8 shows the transmit-power levels and the received SIRsversus time for users 2, 7, and 8.

As we observe in Fig. 8, for the interval that user 7 is theclosest user to the base station in cell No. 2 (i.e.[0, 2]), itobtains the highest SIR (greater than its minimum acceptabletarget-SIR), and the other users in that cell, including user8, receive their minimum acceptable target-SIRs. At the sametime, in cell No. 1, user 2 obtains the highest SIR and theother users in that cell obtain their minimum acceptable target-SIRs. As user 7 moves farther from base station No. 2 anduser 8 becomes the closest user to that base station, user 8increases its transmit power and obtains a SIR higher than

Fig. 9. An example of users’ and base stations’ locations in a 4-cell primarynetwork with a cognitive radio network. Primary and secondary users aremarked by “◦” and “•”, respectively, and primary and secondary base stationsare marked by “4” and “N”, respectively.

Fig. 10. Average system throughput per Hz per user (∑

i(log2(1 +γi))/W/M ) achieved by TPC and DTPC vs. the number of primary users.

its minimum acceptable target-SIR, and user 7 decreases itstransmit power and hits its minimum acceptable target-SIR.When user 7 enters cell No. 1 (i.e. att = 10 seconds), as thenumber of users in that cell is increased (in contrast to cellNo. 2 in which the number of users is decreased), the systemthroughput for cell Nos. 1 and 2 are decreased (as per thereduced SIR for user 2) and increased (as per the increasedSIR for user 8), respectively. Finally when user 7 becomes theclosest user to base station No. 1, it obtains the highest SIR incell No. 1, whereupon user 2 receives its minimum acceptabletarget-SIR. These show that DTPC works well when usersmove. This is achieved by dynamically assigning high target-SIRs to user(s) with better channels, and assigning at leastthe minimum acceptable SIR to the remaining users, in adistributed manner.

2) A 4-Cell Primary Network with and without a CognitiveRadio Network:Now we consider a 4-cell wireless network toinvestigate the performance of DTPC in multi-cell networksand in cognitive radio networks. We first consider a 4-cellnetwork without any secondary user (all users are primary and

Fig. 11. Average transmit-power and average SIRs for 4 secondary users(the sum of transmit power consumed and sum of SIRs received by secondaryusers divided by the total number of secondary users, respectively) versus thenumber of primary users, when the primary and the secondary users employDTPC and OPC, respectively.

have equal priority) and then consider the same 4-cell primarynetwork but along with a secondary (cognitive radio) network.An example of such a 4-cell wireless network and a cognitiveradio network is shown in Fig. 9.

The primary users are distributed in an area 1000 m×1000 m covered by 4 base stations each covering a 500 m× 500 m cell. The minimum acceptable target-SIR for eachprimary user is 10 dB. We consider the range of 1 primaryuser/cell (a total of 4 users) to 9 primary users/cell (a totalof 36 users) with a step size of 4 users. For each snapshot,TPC and DTPC are separately applied and their correspondingsystem throughput is computed at the equilibrium. We averagethe corresponding values of the system throughput for 1500snapshots of uniform distribution of users’ locations. Fig. 10shows the average system throughput per Hz per user (the sumof throughput for all primary users divided by the allocatedbandwidth and the corresponding total number of users) versusthe total number of users, for TPC and DTPC. As can be seen,DTPC outperforms TPC in terms of system throughput.

Now we consider the same primary network together witha single base-station (located at the center of the whole area)with 4 secondary users (who coexist with primary users). Inthis scenario, primary users employ DTPC and secondary usersemploy OPC as discussed in Section V. Fig. 11 shows the av-erage transmit power and SIR for each secondary user. Fig. 12shows the average system throughput for the primary networkand for the cognitive radio network versus the total number ofprimary users. Note that as the number of primary users in-creases, secondary users sense it by experiencing an increasedeffective interference at their receiver and consequently reducetheir transmit power levels in accordance with OPC, resultingin lower SIRs and a lower system throughput. Note that bycomparing the average system throughput achieved for the twocases (with and without the cognitive radio network shown inFigs. 10 and 12, respectively), we see that secondary usersdo not prevent primary users from obtaining their expectedsystem throughput.

Fig. 12. Average system throughput of the primary network per Hz perprimary user (the sum of throughputs for all primary users divided by theallocated bandwidth and the total number of primary users), and the averagesystem throughput of the cognitive radio network per Hz per secondary user(the sum of throughputs for all secondary users divided by the allocatedbandwidth and the total number of secondary users) vs. the total numberof primary users, when the primary and the secondary users employ DTPCand OPC, respectively.

VIII. C ONCLUSIONS

We studied the problem of system throughput optimizationsubject to a given lower bound for the users’ SIRs in wirelesscellular networks. To address this problem in a distributedmanner, we introduced our proposed DTPC. In the proposedDTPC, when the effective interference for a given user is lessthan a given threshold, that user dynamically sets its target-SIRwhich is a decreasing function of the effective interference, toa value higher than the minimum acceptable target-SIR; oth-erwise it keeps its target-SIR fixed at its minimum acceptablevalue.

We showed that the proposed algorithm converges to aunique fixed point starting from any initial transmit power inboth synchronous and asynchronous power updating cases. Wealso showed that our proposed algorithm not only guaranteesthe (feasible) minimum acceptable target-SIRs for all usersas is the case by TPC (and in contrast to OPC), but alsosignificantly improves the system throughput as compared toTPC. Furthermore, we discussed the application of DTPCalong with OPC in cognitive radio networks and in wirelesscellular networks with different services. For the case in whichusers are assumed to be selfish, a game theoretic analysisof our DTPC algorithm was presented via a non-cooperativepower control game with a new pricing function. It was shownthat the best response function for each user is the same asour proposed power-updating function of DTPC.

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Mehdi Rasti (S’08) is a Doctoral Candidate atthe Department of Electrical and Computer Engi-neering,Tarbiat Modares University, Tehran, Iran.He received his B.Sc. and M.Sc. degrees both inElectrical Engineering from Shiraz University, Shi-raz, Iran and Tarbiat Modares University, in 2001and 2003 respectively. From November 2007 toNovember 2008, he was a visiting PhD student atthe Wireless@KTH, Royal Institute of Technology,Stockholm, Sweden. His current research interestsinclude resource allocation in wireless networks, and

application of game theory and pricing in wireless networks.

Ahmad R. Sharafat (S’75-M’81-SM’94) is aprofessor of Electrical and Computer Engineeringat Tarbiat Modares University, Tehran, Iran. Hereceived his B.Sc. degree from Sharif Universityof Technology, Tehran, Iran, and his M.Sc. andhis Ph.D. degrees both from Stanford University,Stanford, California, all in Electrical Engineeringin 1975, 1976, and 1981, respectively. His researchinterests are advanced signal processing techniques,and communications systems and networks. He is aSenior Member of the IEEE and Sigma Xi.

Jens Zander (S’82-M’85) received the M.S degreein Electrical Engineering and the Ph.D Degree fromLinkping University, Sweden, in 1979 and 1985respectively. From 1985 to 1989 he was a partnerof SECTRA, high-tech company in telecommuni-cations systems and applications. In 1989 he wasappointed professor and head of the Radio Commu-nication Systems Laboratory at the Royal Institute ofTechnology, Stockholm, Sweden. Since 1992 he alsoserves as Senior Scientific Advisor to the SwedishNational Defence Research Institute (FOI). Between

2001-2002 he was Scientific Director and since 2003 he is the Director ofthe Center for Wireless Systems (Wireless@KTH) at the Royal Institute ofTechnology, Stockholm. Dr Zander has published numerous papers in the fieldof Radio Communication, in particular on resource management aspects ofPersonal Communication Systems. He has also co-authored four textbooks inRadio Communication Systems, including the English textbooks ”Principlesof Wireless Communications” and ”Radio Resource Management for WirelessNetworks”. He was the recipient of the IEEE Veh Tech Soc ”Jack NeubauerAward” for best systems paper in 1992. Dr Zander is a member of theRoyal Academy of Engineering Sciences. He is the chairman of the IEEEVT/COM Swedish Chapter. Dr Zander is associate editor of the ACM WirelessNetworks Journal and Area Editor of Wireless Personal Communications.His current research interests include future wireless infrastructures and inparticular related resource allocation and economic issues.