Distributed Uplink Power Control with Soft Removal for Wireless Networks

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011 833

Distributed Uplink Power Control withSoft Removal for Wireless Networks

Mehdi Rasti and Ahmad R. Sharafat, Senior Member, IEEE

Abstract—In the well-known distributed target-SIR-trackingpower control algorithm (TPC), when the system is infeasible (aconstrained power vector does not exist to attain target-SIRs),all non-supported users (those who cannot obtain their target-SIRs) transmit at their maximum power. Such users inefficientlyconsume their energy, and cause interference to others, whichincreases the number of non-supported users. To deal withthis, some non-supported users should decrease their transmitpower (the gradual removal problem). We present a distributedpower control scheme with gradual soft removal, by which eitherTPC or OPC (opportunistic power control) is used, dependingon whether the ratio of interference-to-path-gain is below orabove a threshold that is chosen by each user in a distributedmanner. We show that our algorithm converges to a unique fixed-point in both feasible and infeasible systems, and that when thesystem is infeasible, it results in less outage with significantly lessconsumed power, as compared to TPC. We also provide a gametheoretic analysis of our algorithm by introducing a new pricingwhen users are selfish. As our algorithm is fully distributed andrequires only local information, it can be applied to both cellularand ad hoc networks. Simulation results confirm our analysis.

Index Terms—Distributed power control, gradual removalproblem, selective target-SIR-tracking or opportunistic powercontrol algorithm, wireless networks.

I. INTRODUCTION

POWER control in a wireless network is an importantand challenging issue as the network’s capacity depends

highly on how interference and fading are managed. In ad-dition to capacity enhancement, power control is also usedto extend user’s battery life, and to maintain an acceptablequality of service (QoS) in terms of bit-error-rate (BER) and/ortransmission delay for all users by minimizing interferenceto others. A power control algorithm should work well forboth data and voice services. Data services require a highersignal-to-interference ratio (SIR) as compared to the voiceservice, because the former is less tolerant to bit-errors. Also,in contrast to a non-real-time data service, the voice serviceis highly sensitive to transmission delays.

Since the early work on distributed power control in [1],designing distributed power control algorithms has received

Paper approved by K. K. Leung, the Editor for Wireless Network Accessand Performance of the IEEE Communications Society. Manuscript receivedNovember 19, 2009; revised May 10, 2010 and September 4, 2010.

This work was supported in part by Tarbiat Modares University, Tehran,Iran, and in part by Shiraz University of Technology, Shiraz, Iran.

M. Rasti is with the Department of Electrical Engineering, Shiraz Universityof Technology, Shiraz, Iran. Prior to this, he was with the Department ofElectrical and Computer Engineering, Tarbiat Modares Univesity, Tehran,Iran.

A. R. Sharafat (corresponding author) is with the Dept. of Electrical andComputer Engineering, Tarbiat Modares Univesity, P.O. Box 14155-4838,Tehran, Iran (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCOMM.2011.122110.090711

much attention. In distributed algorithms, the information thateach user globally needs to know is minimal. In doing so,two approaches have been considered, namely the target-SIR-tracking scheme that was originally proposed in [2] and hasbeen further studied in [3]–[8], and the opportunistic approachthat was proposed in [9] and [10]. They are different in theway interference (traffic load) and deep fading (low path-gain)are managed.

In the target-SIR-tracking power control algorithm (TPC),each user is tracking its own predefined target SIR. Thiscauses a user to increase its transmit power when the effectiveinterference (i.e., the ratio of interference to path-gain) isincreased because of heavy traffic and/or deep fading and/orfar distance. In contrast to TPC, in an opportunistic powercontrol scheme (OPC), each user increases its transmit poweras the effective interference decreases [9], [10]. The OPCsignificantly increases the aggregate throughput (in terms ofthe aggregate capacity) as compared to TPC by allocatingmore power to users with good channels, and very low powerto users with poor channels (softly removing users with poorchannels). We bring this latter property of OPC (soft removal)into TPC to improve its performance in infeasible systems.

It is well known that in a feasible system, TPC (bothconstrained and unconstrained cases) provides all active userswith their target-SIRs, i.e., zero outage, with minimum powerconsumption [2]. When the system is infeasible, it was shownin [3], [4] and [11] that constrained TPC converges to aunique fixed point, as opposed to unconstrained TPC that doesnot converge. As TPC was originally designed to provide allusers with their target-SIRs in feasible systems, in constrainedTPC, when the system is infeasible, all non-supported users(those who cannot obtain their target-SIRs) transmit at theirmaximum power (without reaching their target-SIRs), andcause interference to others as well. Obviously, this is notdesirable, and can be easily shown that it increases the numberof non-supported users, which can be avoided if only somenon-supported users reduce their transmit power. In otherwords, by gradually removing some non-supported users, theremaining users can be supported and thus, the outage ratio(the ratio of the number of non-supported users to the totalnumber of users) as well as the transmit power consumed byeach user are reduced. This is the gradual removal problemthat we focus on in this paper.

In this paper, we address the gradual removal problemwhere the outage ratio is minimized subject to power con-straint, by removing a minimal number of users in a distributedmanner in an infeasible system. It has been shown in [4] thatminimizing the outage ratio is NP-hard.

To address the gradual removal problem, in [5] and [12]

0090-6778/11$25.00 c⃝ 2011 IEEE

834 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011

TPC was modified to temporarily remove users with a poorchannel, but its convergence is not guaranteed [5], and poweroscillations may ensue [12]. In [13], it was proposed that userswith a poor channel remove themselves permanently with agiven probability. If the effective interference becomes accept-able again for such users, they do not resume transmissionon the same channel, as opposed to a temporarily removeduser as in [5] and [12]. However, temporary removal mayresult in power oscillations [5], [12]. In real-time applications,permanent removal is a severe drawback, particularly becauseunnecessary removals may ensue during initial power updatingiterations (before reaching the steady state), even in feasiblecases. Due to these problems, we focus on power controlalgorithms with the capability of temporary removal.

In order to address the gradual removal problem in adistributed manner that requires minimal information by users,we present a selective target-SIR-tracking or opportunisticpower control algorithm (TOPC). In TOPC, when the effectiveinterference is below a threshold (i.e., a good channel), theuser updates its transmit power using TPC, and when itis above the threshold (poor channel), OPC is used. UnderTOPC, in an infeasible system, a portion of non-supportedusers are identified as candidates for soft removal. For eachof such users, if its interference level is above the threshold,the user would operate according to OPC, i.e., instead oftransmitting at its maximum power as in TPC, or instead ofswitching off as in [5] and [12] that causes power oscillations,it gradually reduces its transmit power, i.e., the user is softlyremoved. In other words, in TOPC, OPC is applied only whenthe channel is poor, resulting in soft removal in order toreduce interference, and consequently increase the number ofsupported users. Note that the objective of employing OPCin TOPC is completely different from the objective of theoriginal OPC that was basically applied in both poor and goodchannels.

We will prove that the proposed algorithm is convergent(in contrast to the algorithms in [5] and [12]), and satisfiesthe objective of gradual removal, i.e., it reduces outage ratioand power consumption (a given user’s energy consumption aswell as the aggregate transmit power consumed by all users) ascompared to TPC. We also provide a game theoretic analysisof our TOPC via a non-cooperative power control game witha new pricing function, when users are assumed to be selfish,and show that it results in a best response function for eachuser, which is the same as our proposed TOPC’s power updatefunction. Simulation results confirm our analysis. Since ourproposed TOPC is fully distributed and always convergent, itcan be applied to cellular as well as to ad hoc networks.

The rest of this paper is organized as follows. In Section II,we introduce the system model and present the background. InSection III, we state the gradual removal problem, present ourproposed TOPC, and prove its convergence. In Section IV, wedescribe how to choose the TOPC parameters, and observe thatTOPC reduces both the outage ratio and power consumptionsby users as compared to TPC. A game theoretic analysis ofTOPC is presented in Section V, where the power controlproblem is formulated as a non-cooperative game. Simulationresults and conclusions are presented in Sections VI and VII,respectively.

II. SYSTEM MODEL AND BACKGROUND

A. CDMA Wireless Network Models and Notations

Consider a multi-cell wireless CDMA network with 𝐾base stations (cells) and 𝑀 active users denoted by 𝒦 ={1, 2, ⋅ ⋅ ⋅ ,𝐾} and ℳ = {1, 2, ⋅ ⋅ ⋅ ,𝑀}, respectively. Let 𝑝𝑖be the transmit power of user 𝑖 and 0 ≤ 𝑝𝑖 ≤ 𝑝𝑖, where𝑝𝑖 is the upper limit of the transmit power for user 𝑖. Thepath gain from user 𝑖 to the base station 𝑘 is denoted byℎ𝑘𝑖. Noise is assumed to be additive white Gaussian (AWGN)whose power is 𝜎2

𝑘 at the receiver of the base station 𝑘. Let𝑠𝑖 denote the base station to which user 𝑖 is assigned. Thereceiver is assumed to be a conventional matched filter. Thus,for a given transmit power vector p= [𝑝1, 𝑝2, ⋅ ⋅ ⋅ , 𝑝𝑀 ]T, theSIR of user 𝑖, denoted by 𝛾𝑖 is

𝛾𝑖(p) =𝑔𝑖ℎ𝑠𝑖𝑖𝐼𝑖(p)

𝑝𝑖, (1)

where 𝑔𝑖 is the processing gain for user 𝑖 (defined as the ratioof chip rate (or spreading bandwidth) to transmit data rate),and

𝐼𝑖(p) =∑𝑗 ∕=𝑖

ℎ𝑠𝑖𝑗𝑝𝑗 + 𝜎2𝑠𝑖 (2)

is the interference to user 𝑖’s base station. We also define theeffective interference as the ratio of interference to the productof processing gain and path-gain for a user 𝑖 (as in [9], [12]),denoted by 𝑅𝑖,

𝑅𝑖(p) =𝐼𝑖(p)

𝑔𝑖ℎ𝑠𝑖𝑖. (3)

The value of 𝑅𝑖 represents the channel status for user 𝑖,i.e., given 𝑔𝑖 (assumed to be fixed for all users), a higherinterference and a lower path gain results in a higher 𝑅𝑖. Theterms poor channel or good channel for user 𝑖 imply that thevalue of 𝑅𝑖 is high or low, respectively.

Definition 1: A SIR vector denoted by 𝜸 =[𝛾1, 𝛾2, ⋅ ⋅ ⋅ , 𝛾𝑀 ]T is feasible if a feasible power vector0 ≤ p ≤ p exists that corresponds to the SIR vector, wherethe vector inequality 0 ≤ p ≤ p implies 0 ≤ 𝑝𝑖 ≤ 𝑝𝑖 forall 𝑖. For a given target-SIR vector denoted by 𝜸, we alsosay that the system is feasible if 𝜸 is feasible, otherwise thesystem is called infeasible.

B. Existing Distributed Power Control Algorithms

In TPC, each user 𝑖 tries to maintain its SIR at a target leveldenoted by 𝛾𝑖. The unconstrained power-update function in [2]is

𝑝𝑖(𝑡+ 1) = 𝛾𝑖𝑅𝑖(p(𝑡)) (4)

where the target-SIR 𝛾𝑖 is fixed and 𝑅𝑖(p(𝑡)) is the effectiveinterference to user 𝑖’s base station at time 𝑡. It was shown in[2] and [8] that if and only if the target-SIR vector 𝜸 is feasi-ble, then unconstrained-TPC converges either synchronouslyor asynchronously to a fixed point at which users attain theirtarget-SIRs with minimal aggregate transmit power, i.e., itsfixed point solves the following optimization problem

minp

∑𝑖∈ℳ

𝑝𝑖

subject to 𝛾𝑖(p) ≥ 𝛾𝑖, ∀𝑖 ∈ ℳ. (5)

RASTI and SHARAFAT: DISTRIBUTED UPLINK POWER CONTROL WITH SOFT REMOVAL FOR WIRELESS NETWORKS 835

When the system is infeasible, the above problem has nosolution, because there is no transmit power vector that cansatisfy SIR requirements for all users. When the unconstrainedTPC (that was originally designed to solve (5) in a distributedmanner assuming feasibility of the system) is used in infea-sible systems, since the target-SIRs are rigidly tracked, allusers increase their transmit power at each step and thus, theunconstrained TPC diverges. Algorithms developed in [3] and[4] assume constrained transmit power to deal with divergencein infeasible systems, that is

𝑝𝑖(𝑡+ 1) = min {𝑝𝑖, 𝛾𝑖𝑅𝑖(p(𝑡))} . (6)

Although convergence to a fixed point is guaranteed for con-strained TPC in both feasible and infeasible systems, it suffersfrom a severe drawback in infeasible systems (high outage andhigh power consumption) as described in the following sectionwhere the gradual removal problem is formally stated. In therest of this paper, TPC implies (6) unless stated otherwise.

Convergence of the unconstrained TPC is facilitated by adynamic target-SIR, denoted by 𝑇𝑖 (𝑅𝑖), which is a decreasingfunction of 𝑅𝑖 and is chosen so that 𝑇𝑖 (𝑅𝑖)𝑅𝑖 is an increasingfunction of 𝑅𝑖 [12]. The power update function for a dynamictarget-SIR-tracking power control (DTPC) is

𝑝𝑖(𝑡+ 1)=

{𝑇𝑖 (𝑅𝑖(p(𝑡)))𝑅𝑖(p(𝑡)), if 𝑇𝑖 (𝑅𝑖(p(𝑡))) ≥ 𝛾

𝑖0, if 𝑇𝑖 (𝑅𝑖(p(𝑡))) < 𝛾

𝑖(7)

where 𝛾𝑖is the threshold SIR for user 𝑖. The updated transmit

power for user 𝑖 is 𝑇𝑖 (𝑅𝑖)𝑅𝑖 so far as 𝑇𝑖 (𝑅𝑖) ≥ 𝛾𝑖, and is 0

(switch-off) otherwise. This algorithm dynamically adjusts thetarget-SIR according to the channel condition. If the channel ispoor, the target-SIR is decreased and may even vanish (i.e., theuser is switched off). This facilitates the convergence [12], inthe sense that as users are not rigidly tracking their fixed targetSIRs, some users may switch off at high traffic loads to makethe system feasible. However, in infeasible systems, someusers may switch off when 𝑇𝑖 (𝑅𝑖) < 𝛾

𝑖, and start transmitting

again when 𝑇𝑖 (𝑅𝑖) ≥ 𝛾𝑖, which means power oscillations

[12] and possible inability of the algorithm to converge. Thereason is that the power-update function is neither standardnor continuous over the entire 𝑇𝑖 (𝑅𝑖). Another drawback ofthe algorithm in [12] is that no upper limit for transmit poweris assumed.

In OPC, the transmit power is updated in a manner oppositeto TPC. The opportunistic power update function proposed in[10] is

𝑝𝑖(𝑡+ 1) =𝜂𝑖

𝑅𝑖(p(𝑡))(8)

where 𝜂𝑖 is a constant for user 𝑖. As (8) implies, the transmitpower is increased when the channel is good and is decreasedwhen the channel is poor. This algorithm is always conver-gent, and significantly increases the aggregate throughput bytransmitting more power by users with good channels, andtransmitting less power by users with bad channels, but leadsto unfairness as well. In Section III, we employ OPC to reducethe transmit power of users only when their channel is poor(i.e., soft removal), as their effective interference levels areincreased (in contrast to TPC, in which higher transmit powerlevels are set for poor channels).

III. PROBLEM STATEMENT AND PROPOSED ALGORITHM

A. Problem Statement and Motivation

TPC is suitable for attaining target-SIRs of users in feasiblesystems. Using TPC in infeasible systems causes all non-supported users to transmit at their maximum power becauseall users (including the non-supported ones) rigidly track theirrespective SIRs. This unnecessarily drains the batteries of non-supported users and increases interference to others. As wewill see in the following example and in simulation results, thisalso increases the number of non-supported users, which canbe avoided if some non-supported users reduce their transmitpower.

As an example, consider an infeasible system withfour users whose path gain and target-SIR vectors are[0.15, 0.40, 0.50, 0.60], and [50, 40, 50, 40], respectively. Sup-pose AWGN power is 0.1 Watts, processing gain is 100,and the maximum transmit power for each user is 1 Watt.The fixed-point power-vector of the constrained TPC is[1, 1, 1, 0.767] Watts and thus, their achieved SIRs at the equi-librium are [10, 33, 45, 40], meaning that only one user (user4) attains its target-SIR using TPC. One can easily see thatthe transmit power vector [0.22, 1, 0.93, 0.67] Watts results inthe target-SIR vector [2.4, 40, 50, 40] which means that users2, 3, and 4 can be supported and their required transmit powerto attain their target-SIRs are also reduced if user 1 refrainsfrom transmitting at maximum power and instead, transmitsat 0.22 Watts or less. This shows that in an infeasible system,the policy of reducing transmit power by some non-supportedusers has at least two advantages: 1) the remaining ones canbe supported, and consequently the number of supported usersis increased, and 2) the aggregate transmit power is reducedas well. For the DTPC and the algorithm proposed in [5],although a surge in the transmit power of non-supported usersis avoided, they nevertheless suffer from power oscillations[12] (see also simulation results).

Motivated by the stated drawbacks of TPC and the advan-tages of softly removing a portion of non-supported users ininfeasible systems, we now formally state the gradual removalproblem. Given a transmit power vector p, we denote theset of supported users by 𝒮(p) = {𝑗 ∈ ℳ∣𝛾𝑗(p) ≥ 𝛾𝑗},and the total number of its members by ∣𝒮(p)∣. The outageratio is (∣ℳ∣ − ∣𝒮(p)∣)/∣ℳ∣. We define the problem ofminimum-outage-based removal, or equivalently the problemof maximum number of supported users by

max0≤p≤p

∣𝒮(p)∣, (9)

which is applicable to both feasible and infeasible systems.When the system is feasible, all users can be supported (i.e.,the minimum outage ratio is zero) and its solution is given byTPC consuming minimal transmit power. When the systemis infeasible, a minimal number of users should be removed,which is a NP-hard problem [4]. In this case, as stated earlier,TPC results in some avoidable non-supported users.

In what follows, we present a distributed power controlalgorithm for addressing the gradual removal problem, i.e.,reducing the outage that not only prevents transmission atmaximum power in a poor channel (instead, reduces thetransmit power), but also is convergent and stable.

836 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011

Fig. 1. The proposed TOPC algorithm. The regions corresponding to𝑅𝑖 < 𝑅th

𝑖 or to 𝑅𝑖 > 𝑅th𝑖 are called TPC mode or OPC mode of TOPC,

respectively.

B. Proposed Algorithm: Selective Target-SIR-Tracking or Op-portunistic Power Control Algorithm (TOPC)

The observations made in Section II-B on TPC and OPCmotivate us to design a distributed power control algorithmthat adjusts the transmit power according to either TPC orOPC, depending on the channel conditions to address thegradual removal objectives. We call this strategy TOPC, asdefined and formulated below.

Definition 2: A power control algorithm is TOPC if thepower-update function 𝑓𝑖(p) is continuous and is either anincreasing function of 𝑅𝑖 for 𝑅𝑖 values below a given thresh-old, or a decreasing function of 𝑅𝑖 for 𝑅𝑖 values above thatthreshold (see Fig. 1) for all 𝑖.

In TOPC, when a given user 𝑖 operates in the target-SIR-tracking mode, an increase in 𝑅𝑖 would increase its transmitpower, until the user’s threshold for 𝑅𝑖 is reached, upon whichfurther increases in 𝑅𝑖 would reduce the user’s transmit power(opportunistic mode).

We propose TOPC’s power-update function as

𝑝𝑖(𝑡+1) = 𝑓(C)𝑖 (p(𝑡)) ≜

{𝛾𝑖𝑅𝑖(p(𝑡)), if𝑅𝑖(p(𝑡)) ≤ 𝑅th

𝑖𝜂𝑖

𝑅𝑖(p(𝑡)), if𝑅𝑖(p(𝑡)) ≥ 𝑅th

𝑖

(10)where 𝑅th

𝑖 is the effective interference threshold, 𝜂𝑖 is aconstant, and 𝛾𝑖 is the target-SIR for user 𝑖 (see Fig. 1). Thesethree parameters of TOPC are adjusted so that 𝑓 (C)

𝑖 (p(𝑡)) iscontinuous, i.e.,

𝜂𝑖 = 𝛾𝑖(𝑅th𝑖 )2. (11)

Obviously, the performance of TOPC highly depends on thethreshold values of the effective interference set by users. Animportant question is how each user 𝑖 should adjust its 𝑅th

𝑖

in a distributed manner to satisfy the stated objectives. InSection IV we will propose a distributed method for choosingthe effective interference thresholds to reduce both the outageratio and the power consumption as compared to TPC. Inwhat follows, for a given 𝑅th

𝑖 for each user 𝑖, we show thatTOPC converges to a unique fixed point in both feasible andinfeasible systems.

C. Convergence of TOPC

The work presented in [8] provides a framework to examinethe convergence of TPC. This framework is generalized toa new framework in [9] applicable to a wider range ofdistributed power control algorithms including TPC and OPC.Our proposed TOPC falls into this generalized framework. Wewill show that TOPC’s power update function has a uniquefixed point to which it converges. To show this, we first definetwo-sided scalable functions as in [9] and prove that TOPC’spower update function (10) is two-sided scalable.

Definition 3: A power update function f(p) =[𝑓1(p), 𝑓2(p), ⋅ ⋅ ⋅ , 𝑓𝑀 (p)]

T is two-sided scalable if forall 𝑎 > 1, 1

𝑎p ≤ p′ ≤ 𝑎p implies

1

𝑎𝑓𝑖(p) ≤ 𝑓𝑖(p

′) ≤ 𝑎𝑓𝑖(p) for all 𝑖 ∈ ℳ. (12)

Theorem 1: TOPC’s power update function f (C)(p) in (10)is two-sided scalable.

Proof: If both 𝑅𝑖(p) and 𝑅𝑖(p′) are either smaller or

greater than 𝑅th𝑖 , then for p and p′, a user 𝑖 operates in

either the target-SIR-tracking mode or the opportunistic mode,respectively. This means that for user 𝑖, (12) holds, becausethe TPC’s and the OPC’s power update functions are two-sided scalable [9]. If 1

𝑎p ≤ p′ ≤ 𝑎p for a given 𝑎 > 1, thenby simple mathematical manipulations, it is easy to show that

1

𝑎𝑅2

𝑖 (p′) ≤ 𝑅𝑖(p

′)𝑅𝑖(p) ≤ 𝑎𝑅2𝑖 (p), (13)

and1

𝑎𝑅2

𝑖 (p) ≤ 𝑅𝑖(p′)𝑅𝑖(p) ≤ 𝑎𝑅2

𝑖 (p′). (14)

If 𝑅𝑖(p) ≤ 𝑅th𝑖 ≤ 𝑅𝑖(p

′), then from (13) we have

1

𝑎

(𝑅th

𝑖

)2 ≤ 𝑅𝑖(p′)𝑅𝑖(p) ≤ 𝑎

(𝑅th

𝑖

)2, (15)

or equivalently

1

𝑎(𝑅th

𝑖

)2 ≤ 1

𝑅𝑖(p′)𝑅𝑖(p)≤ 𝑎(

𝑅th𝑖

)2 . (16)

We substitute (11) into (16), multiply (16) by 𝑅𝑖(p), and get

1

𝑎𝛾𝑖𝑅𝑖(p) ≤ 𝜂𝑖

𝑅𝑖(p′)≤ 𝑎𝛾𝑖𝑅𝑖(p) (17)

which is equivalent to (12). If 𝑅𝑖(p′) ≤ 𝑅th

𝑖 ≤ 𝑅𝑖(p), thenfrom (14) we have

1

𝑎

(𝑅th

𝑖

)2 ≤ 𝑅𝑖(p′)𝑅𝑖(p) ≤ 𝑎

(𝑅th

𝑖

)2. (18)

We substitute (11) into (18), divide (18) by 𝑅𝑖(p), and get

1

𝑎

𝜂𝑖𝑅𝑖(p)

≤ 𝛾𝑖𝑅𝑖(p′) ≤ 𝑎

𝜂𝑖𝑅𝑖(p)

, (19)

which is equivalent to (12).Theorem 2: a) TOPC’s power update function f (C)(p)

has a fixed point, i.e., there exists a transmit power vectorp∗ such that p∗ = f (C)(p∗). Besides, the fixed point isunique.

b) For any given initial power vector, TOPC convergesto the fixed point of f (C)(p) in both synchronous andasynchronous cases.

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Proof: The followings have been proved in [9].

1. If a given f(p) is two-sided scalable and there exists afixed point p∗ so that p∗ = f(p∗), then

– the fixed point is unique, and– for any initial power vector, the power control algo-

rithm p(𝑡+ 1) = f(p(𝑡)) converges to p∗.2. If f(p) is two-sided scalable, continuous, and f(p) ≤ u

for all p (i.e., f(p) is upper bounded) for a u > 0, thena fixed point exists.

From the above, we conclude that if f(p) is two-sided scalableand continuous, and there is a u > 0 such that f(p) ≤ u forall p, then the fixed point p∗ = f(p∗) exists and is unique.Also, for any initial power vector, the power control algorithmp(𝑡 + 1) = f(p(𝑡)) converges to p∗. Thus, since f (C)(p)is continuous (with constraint (11)) and is upper bounded(𝑓 (C)

𝑖 (p) ≤ 𝛾𝑖𝑅th𝑖 for all 𝑖), and since in Theorem 1, it

was shown that f (C)p) is two-sided scalable, this theorem isproved.

IV. TOPC PARAMETERS

So far we have shown that, given 𝑅th𝑖 for each user 𝑖, TOPC

converges to a unique fixed point. We now show how userscan choose their TOPC’s parameters in a distributed mannerto reduce both the outage ratio and power consumption ascompared to TPC. There are three parameters in TOPC givenby (10), namely 𝛾𝑖, 𝑅th

𝑖 and 𝜂𝑖. The values of these parametersdepend on one another via (11). We choose the values of𝛾𝑖 and 𝑅th

𝑖 , and obtain 𝜂𝑖 from (11). The value of 𝛾𝑖 foreach user 𝑖 is simply decided by its target-SIR. Thus we needonly to propose a distributed method for choosing the effectiveinterference threshold 𝑅th

𝑖 . In what follows we explain howto choose the effective interference thresholds in the proposedalgorithm to reduce the outage ratio and power consumptionas compared to TPC.

Assume that each user 𝑖 with the target-SIR of 𝛾𝑖, choosesits 𝑅th

𝑖 by

𝑅th𝑖 =

𝑝𝑖𝛾𝑖. (20)

where𝑝𝑖𝛾𝑖

is the maximum value of 𝑅𝑖 for which the target-

SIR for user 𝑖 is achievable (by transmitting at its maximumpower). Note that if 𝑅th

𝑖 is chosen at a value higher thanthe above, it may result in maximum transmit power withoutreaching target-SIRs for some users when the system isinfeasible (similar to TPC), which is not desirable, and ifit is chosen at a value lower than the above, it may resultin not reaching target-SIRs even when the system is feasible(because the user may go to the OPC mode without being

required). By substituting (11) into (20), we get 𝜂𝑖 =𝑝2𝑖𝛾𝑖

.

In the following two theorems, we will show that by usingTOPC (as compared to TPC), each user consumes less energy,resulting in a reduced total power consumption, and a loweroutage ratio.

Theorem 3: Given 𝛾𝑖 for all 𝑖, let p∗(T) and p∗(C) asthe fixed points of TPC and TOPC for which the effectiveinterference threshold is chosen by (20), respectively.

a) If and only if the system (i.e., the target-SIR vector 𝜸)is feasible, then p∗(C) = p∗(T).

b) If and only if the system is infeasible, then p∗(C) < p∗(T)

and consequently∑

𝑖 𝑝𝑖∗(C) <

∑𝑖 𝑝𝑖

∗(T).Proof: The first part can be easily proved, so we only

prove the second part. We know that if p(C)(𝑡) ≤ p(T)(𝑡)for a given 𝑡, then from (3) we have 𝑅𝑖(p

(C)(𝑡)) ≤𝑅𝑖(p

(T)(𝑡)) for all 𝑖 and thus p(C)(𝑡 + 1) ≤ p(T)(𝑡 +1) obtained by comparing TOPC’s power-update function(10) for which the effective interference threshold is cho-sen according to (20) (which can be re-written as 𝑝(C)

𝑖 (𝑡 +

1) = min

{𝛾𝑖𝑅𝑖(p

(C)(𝑡)),𝜂𝑖

𝑅𝑖(p(C)(𝑡))

}) with TPC’s power-

update function (6). Thus, if p(C)(𝑡0) ≤ p(T)(𝑡0) for a giventime 𝑡0, then p(C)(𝑡) ≤ p(T)(𝑡) for all 𝑡 ≥ 𝑡0. Since for anygiven initial transmit power p(0), TPC and TOPC convergeto their own unique fixed point ([3], [4] and Theorem 2),we assume the initial transmit power p(0) is the same forboth. Therefore, we have p(C)(𝑡) ≤ p(T)(𝑡) for all 𝑡 ≥ 0and consequently p∗(C) ≤ p∗(T). In what follows, we provethat for an infeasible system, the strict inequality holds, i.e.,p∗(C) < p∗(T). As a direct result of the first part of this theo-rem, and from p∗(C) ≤ p∗(T), there is at least one 𝑗 for which𝑝𝑗

∗(C) < 𝑝𝑗∗(T) in an infeasible case, because otherwise, it

contradicts the first part. Now we prove that if there is at leastone 𝑗 for which 𝑝𝑗∗(C) < 𝑝𝑗

∗(T), then 𝑝𝑖∗(C) < 𝑝𝑖∗(T) for all 𝑖.

This is true because from 𝑝𝑗∗(C) < 𝑝𝑗

∗(T) and p∗(C) ≤ p∗(T),we conclude that for all 𝑖 ∕= 𝑗, 𝑅𝑖(p

∗(C)) < 𝑅𝑖(p∗(T)), which

implies 𝑝𝑖∗(C) < 𝑝𝑖∗(T). Thus, if the system is infeasible, then

p∗(C) < p∗(T). Also from this and from the first part, weconclude that if p∗(C) < p∗(T), then the system is infeasible.The two latter statements complete the proof of the secondpart of this theorem.

Remark 1: Theorem 3 states that in a feasible system, theperformance of TOPC for which the effective interferencethreshold is chosen according to (20) is the same as thatof TPC. This implies that under TOPC, the outage ratio ina feasible system is zero (i.e., target-SIRs are reached andall users operate in the target-SIR-tracking mode), and theminimum transmit power vector required to satisfy the users’target-SIRs is obtained, i.e., both optimization problems in (5)and (9) are solved by TOPC and TPC in a similar manner.

Remark 2: As we see in the above theorem and in its proof,when the system is infeasible, for any initial power vector, inaddition to the equilibrium (the fixed-point), the instantaneouspower consumed by TOPC for each user and consequently theinstantaneous total power for all users are strictly less thanthose of TPC. In simulation results we show that this energyefficiency is significant.

Remark 3: In TOPC for which the effective interferencethreshold is chosen according to (20), if the system is infea-sible, then at the fixed point, there exists at least one useroperating in the opportunistic mode and vice versa. This isbecause, at the fixed point of TOPC in an infeasible system,if all users operate in TPC, Theorem 3 is contradicted. In thiscase, the users that operate in the opportunistic mode are notsupported (do not reach their target SIRs). Such users transmitat less than their maximum power, as opposed to TPC whereall non-supported users transmit at their maximum power.

838 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011

Theorem 4: Given 𝛾𝑖 for all 𝑖, 𝒮(p∗(T)) ⊆ 𝒮(p∗(C)),which implies ∣𝒮(p∗(T))∣ ≤ ∣𝒮(p∗(C))∣, where p∗(T) andp∗(C) are the fixed points of TPC and TOPC (for which theeffective interference threshold is chosen according to (20)),respectively. In other words, if a user reaches its target-SIRusing TPC, then that user also reaches its target-SIR usingTOPC. The inverse is not necessarily true, i.e., there mayexist a user that does not reach its target-SIR by using TPC,although its target-SIR is reachable by using TOPC.

Proof: From Theorem 3 we have p∗(C) ≤ p∗(T) andthus 𝑅𝑖

(p∗(C)

) ≤ 𝑅𝑖

(p∗(T)

)for all 𝑖. Suppose a given user

𝑖 reaches its target-SIR using TPC, i.e., 𝛾𝑖(p∗(T)

)= 𝛾𝑖,

which implies 𝑅𝑖

(p∗(T)

) ≤ 𝑝𝑖𝛾𝑖

. In this case, user 𝑖 reaches

its target-SIR when TOPC is used, i.e., 𝛾𝑖(p∗(C)

)= 𝛾𝑖.

This is because if 𝛾𝑖(p∗(C)

)< 𝛾𝑖, then we must have

𝑅𝑖

(p∗(C)

)>𝑝𝑖𝛾𝑖

, and thus 𝑅𝑖

(p∗(T)

)< 𝑅𝑖

(p∗(C)

), which

contradicts 𝑅𝑖

(p∗(C)

) ≤ 𝑅𝑖

(p∗(T)

).

The following example shows that the inverse is not true.Consider again the example in Section III.A for an infeasiblesystem with four users. At the fixed point of TPC, onlyone user is supported. But in TOPC, one can easily verifythat at the fixed point, the users’ transmit power vector is[0.22, 1, 0.93, 0.67] Watts and their achieved SIR vector is[2.4, 40, 50, 40], respectively. This means that user 1 (thenon-supported user with the worst channel) senses that thesystem is infeasible and reduces its transmit power, instead oftransmitting at its maximum power, and thus users 2 and 3can attain their target-SIRs in addition to user 4. Besides, thetransmit power consumed by each user is reduced as well (seealso simulation results).

Employing TOPC in an infeasible system (in contrast to em-ploying TPC) causes some users to operate in the opportunisticmode (Remark 3 above), meaning that they start reducingtheir transmit power as the effective interference increases.This makes more resources available to other users to reachtheir target-SIRs. Hence, the number of users that reach theirtarget-SIRs using TOPC is always equal to or higher thanthat of using TPC. Equivalently, the outage ratio in TOPC forwhich the effective interference threshold is chosen accordingto (20) is less than that of TPC (see also simulation resultsin Section VI). Thus, although in general, TOPC does notguarantee the minimum-outage ratio defined in (9), but it hasa unique equilibrium to which the algorithm converges, whichis closer to the optimum solution of (9) (in the sense that theoutage ratio as well as the power consumption are less) ascompared to TPC’s equilibrium.

Now we discuss which users in an infeasible system mayoperate in the opportunistic mode in favor of others byadjusting 𝑅th

𝑖 according to (20). Consider the case that 𝑝𝑖and 𝛾𝑖 are the same for all users and the system is infeasible.In this case, 𝑅th

𝑖 is the same for all users. Thus those usersthat operate in the opportunistic mode have high values of𝑅𝑖 (have low path-gains and/or high interference levels). Thismeans that when the system is infeasible, users with a poorchannel operate in the opportunistic mode (they start reducingtheir transmit power levels as their effective interference isincreased), thus favoring users with a good channel, which is

desirable in this case. Similarly, for the case that target-SIRsare not the same for different users, in an infeasible system,those users with low path-gains and/or with high values fortarget-SIRs may operate in the opportunistic mode.

V. GAME-THEORETIC ANALYSIS OF TOPC

Recently, the non-cooperative game theoretic analysis iswidely applied to power control in wireless networks [12],[14]–[18] when users are assumed to be selfish (in contrastto what we have assumed so far). In what follows, assumingselfish and non-cooperative users, we investigate how and withwhat pricing scheme, at the Nash equilibrium (NE) of thepower control game, the gradual removal problem is addressedsimilar to TOPC.

In a game theoretic view of the power control problem, eachuser 𝑖 ∈ ℳ chooses its transmit power level from its strategyspace 𝑃𝑖 = [0, 𝑝𝑖] in a selfish manner to maximize its ownutility denoted by 𝑢𝑖(𝑝𝑖,p−𝑖) in which p−𝑖 is the transmitpower vector for all users except user 𝑖. The utility functionis defined for each user and represents the QoS offerings tothat user, as well as its associated costs. When a user transmitsin a shared medium, that user should pay a price (cost) forreceiving the service and for causing interference to others. Anon-cooperative power control game (NPCG) is denoted by𝐺 = ⟨ℳ, (𝑃𝑖), (𝑢𝑖)⟩ and is formally stated [19] by

max𝑝𝑖∈𝑃𝑖

𝑢𝑖(𝑝𝑖,p−𝑖) for all 𝑖 ∈ ℳ. (21)

The commonly used concept in solving game theoretic prob-lems is the Nash equilibrium (NE) at which no user canimprove its utility by unilaterally changing its transmit power.

Definition 4: A transmit power vector p∗ is the NEpoint for NPCG 𝐺 = ⟨ℳ, (𝑃𝑖), (𝑢𝑖)⟩ if for every user 𝑖,𝑢𝑖(𝑝

∗𝑖 ,p

∗−𝑖) ≥ 𝑢𝑖(𝑝𝑖,p

∗−𝑖), ∀𝑝𝑖 ∈ 𝑃𝑖.

Another commonly used concept in game theory is thebest response function for each player. Formally, the user 𝑖’sbest response function 𝑏𝑖 : 𝑃−𝑖 → 𝑃𝑖, where 𝑃−𝑖 is theCartesian product of 𝑃𝑗 for 𝑗 ∕= 𝑖 (i.e., 𝑃−𝑖 =

∏𝑗 ∕=𝑖

𝑃𝑗), is

a set-valued function that assigns the set of best power levelsin the utility sense to each power vector p−𝑖 ∈ 𝑃−𝑖, thatis 𝑏𝑖(p−𝑖) = {𝑝𝑖 ∈ 𝑃𝑖 ∣∀𝑝′𝑖 ∈ 𝑃𝑖 : 𝑢𝑖(𝑝𝑖,p−𝑖) ≥ 𝑢𝑖(𝑝

′𝑖,p−𝑖)}.

The NE is the fixed point of the best response function set,that is p = b(p), where b(p) = [𝑏1(p), 𝑏2(p), ⋅ ⋅ ⋅ , 𝑏𝑀 (p)]T.Note that 𝑏𝑖(p) and 𝑏𝑖(p−𝑖) are equivalent.

A pricing-based utility function in many NPCGs is

𝑈𝑖(𝑝𝑖,p−i) = 𝑞𝑖(𝑝𝑖,p−i)− 𝑐𝑖(𝑝𝑖,p−i), (22)

where 𝑞𝑖 is the function representing the QoS that user 𝑖receives, and 𝑐𝑖 is the pricing function for user 𝑖. It is wellestablished that in contrast to the case in which no pricing isapplied, the pricing scheme could affect the individual user’sdecision in such a way that the efficiency of NE from agiven goal’s point of view (e.g., from fairness’s or system’spoint of view) is improved. In [18], for a QoS functionwhich is concave and increasing with respect to SIR, we haveshown that SIR-based pricing can be utilized to satisfy specificgoals (such as fairness, aggregate throughput optimization, ortrading off between these two goals) at the NE. In [9], [10] it

RASTI and SHARAFAT: DISTRIBUTED UPLINK POWER CONTROL WITH SOFT REMOVAL FOR WIRELESS NETWORKS 839

was shown that the best response function for utility function𝑈𝑖(𝑝𝑖,p−i) =

√𝛾𝑖 − 𝛼𝑖𝑝𝑖 corresponds to the opportunistic

power control scheme. Note that the QoS function defined in[9], [10] has no physical meaning.

Similar to [16]–[18] and [20], we use an informationtheoretic approach to define a meaningful QoS function interms of channel capacity as the highest rate at which user𝑖’s information can be sent with an arbitrary low probabilityof error [21]. We do not restrict ourselves to a specificchannel model, modulation, or coding scheme. For a givenSIR experienced by user 𝑖 at its base-station, we denote thecorresponding QoS function by 𝑞(𝛾𝑖). Generally, 𝑞(𝛾𝑖) is anincreasing and concave function of 𝛾𝑖 for every channel modelwith average power constraint [22]; thus the following twoproperties hold.

Property 1: 𝑞′(𝛾𝑖) > 0 for all 𝛾𝑖Property 2: 𝑞′′(𝛾𝑖) < 0 for all 𝛾𝑖,

where 𝑞′(𝛾𝑖) and 𝑞′′(𝛾𝑖) are the first and the second derivativesof 𝑞(𝛾𝑖) with respect to 𝛾𝑖, respectively. We rely only on thesetwo general properties, and our developments hold for anyother QoS function that satisfies Properties 1-2 above.

As two examples for the QoS function, consider a loga-rithmic function of SIR, defined in [16] and [20] denoted by𝑞G(𝛾), and channel capacity for a binary symmetric channel

(BSC) as in [17] denoted by 𝑞BSC

(𝛾), and write

𝑞G(𝛾) =

𝑏

2log2(1 + 𝛾) bits/second, (23)

and

𝑞BSC(𝛾)=𝑏(1 + 𝑝e(𝛾) log2 𝑝e(𝛾)

+(1− 𝑝e(𝛾)) log2(1− 𝑝e(𝛾)))

bits/second,(24)

where 𝑏 is the data rate, and 𝑝e(𝛾) is the cross error probabilitydefined in [21]. Note that these two examples for QoS functionsatisfy Properties 1 and 2 above. The logarithmic function (4)is the capacity of a Gaussian channel, provided that noise plusinterference for each user is Gaussian [16].

In what follows, we set up a NPCG with a pricing schemethat is a function of SIR, and analytically show that witha proper choice of pricing units, its outcome is the sameas that of our proposed TOPC, implying that at the NE,the gradual removal problem is addressed. Let 𝑐𝑖(𝛾𝑖) be thepricing function of user 𝑖 for 𝛾𝑖 at the base station, and thepricing-based utility function for user 𝑖 be

𝑈𝑖(𝑝𝑖,p−𝑖) = 𝑞(𝛾𝑖(𝑝𝑖,p−i))− 𝛼𝑖𝛾𝑖(𝑝𝑖,p−i), (25)

where 𝛼𝑖 ≥ 0 is the price per unit of the actual SIR at thebase station for user 𝑖. We will show that this pricing schemeenables us to adequately influence the best response functionof each user to address the gradual removal (similar to TOPC)by a proper choice of pricing units introduced in the followingtheorem.

Theorem 5: In NPCG 𝐺 = ⟨ℳ, (𝑃𝑖), (𝑈𝑖)⟩ in which 𝑈𝑖 is(25), the best response of user 𝑖 ∈ ℳ to a given power vectorp−𝑖 is the same as our proposed power update function ofTOPC in (10) with the given parameters of 𝛾𝑖 and 𝑅th

𝑖 if the

pricing units in (26) for each user 𝑖 is set to

𝛼𝑖 =

⎧⎨⎩𝑞′(𝛾𝑖), if 𝑅𝑖 ≤ 𝑅th

𝑖

𝑞′((𝑅th

𝑖

𝑅𝑖)2𝛾𝑖), if 𝑅𝑖 > 𝑅th

𝑖 .(26)

Thus, a unique Nash equilibrium for this game exists, and isthe fixed point of TOPC, i.e., the fixed-point of TOPC denotedby p∗ is the solution to

max𝑝𝑖∈𝑃𝑖

𝑞(𝛾𝑖(𝑝𝑖,p∗−i))− 𝛼𝑖𝛾𝑖(𝑝𝑖,p

∗−i), for all 𝑖 ∈ ℳ. (27)

Proof: See Appendix I.Note that as the QoS function is concave, its first derivative,

i.e., 𝑞′(⋅), is a decreasing function. Thus the proposed pricingin (26) implies that when the effective interference for a givenuser is below the threshold, the pricing unit for that useris fixed at a value given by 𝑞′(𝛾𝑖), and when the effectiveinterference exceeds the threshold, its pricing unit is increasedas its effective interference is increased in order to discourageselfish users from transmitting with high power. Anotherimportant property of the proposed pricing scheme is thatthe pricing unit in (26) can be obtained by each user in adistributed manner, because it depends on the informationpertinent to that user only.

VI. SIMULATION RESULTS

Now we show that TOPC outperforms other existing powercontrol algorithms in terms of convergence, outage ratio, andenergy efficiency1. The outage ratio in one iteration (onepower update step) is the ratio of the number of non-supportedusers to the total number of active users in that iteration, asdefined in Section III. A given user 𝑖 with target-SIR 𝛾𝑖 issupported if 𝛾𝑖 ≥ 𝛿𝛾𝑖, where 0 < 𝛿 ≤ 1 is the utilized fademargin. We use 𝛿 = 0.88 that corresponds to -0.5 dB, andassume a data rate of 104 bits/second and a chip rate of 106

bits/second for each user as in [15], which means that theprocessing gain is 100. The AWGN power at the receiver,i.e., 𝜎2, is assumed to be 5× 10−15 Watts as in [15].

We adopt a simple and well known model ℎ𝑠𝑖𝑖 = 𝑘𝑑−4𝑠𝑖𝑖

forthe path-gain as in [23] where 𝑑𝑠𝑖𝑖 is the distance betweenuser 𝑖 and its base station 𝑠𝑖, and 𝑘 is the attenuation factorthat represents power variations due to the shadowing effect,assumed to be 𝑘 = 0.09 as in [15]. The upper bound on thetransmit power for all users is 1 Watt. We first consider asingle cell wireless network, and then proceed to a multi-cellone.

A. Single Cell Networks

We consider a single-cell wireless network, first with fixedlocations of users to study and track the performance of thealgorithms in detail, and then proceed to different snapshotsof users’ locations, to verify that the results do not depend onspecific user-locations.

Consider 6 users indexed from 1 to 6 in a singlecell environment where their distance vector is d =

1Comparing TPC, DTPC, and TOPC with OPC with respect to the statedperformance measures does not make much sense, because their objectivesare different. However to show how OPC works as compared to TOPC, wehave included it in our simulation results.

840 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011

Fig. 2. Transmit power and SIR of each user vs. iteration number, usingTPC. Users 5 and 6 become active after iterations 30 and 60, respectively.

Fig. 3. Transmit power and SIR of each user vs. iteration number, usingDTPC. Users 5 and 6 become active after iterations 30 and 60, respectively.

[200, 320, 460, 530, 550, 800]T m, in which each element is

the distance of the corresponding user from the base station.The target-SIR vector is 𝜸 = [24, 20, 30, 28, 24, 30]T. Onecan easily verify that the system is infeasible. The systembecomes feasible if either user 6 or users 5 and 4 switch off.Without loss of generality, we assume that initially, users 1 to 4are active (from the first iteration), and then user 5 followed byuser 6 become active after 30 and 60 iterations, respectively.

The parameter values for TPC, DTPC, OPC, and TOPCare as follows. For DTPC (7), 𝛾

𝑖= 𝛾𝑖 and 𝑇𝑖(𝑅𝑖) = 𝛾𝑖 −

1𝑎𝑖

ln

((𝑎𝑖𝛾𝑖 − 1)

(𝑎𝑖

2𝛼𝑖𝑅𝑖− 1−

√(𝑎𝑖

2𝛼𝑖𝑅𝑖− 1)2

− 1

))as

in [12], where 𝑎𝑖 = 200, 𝛼𝑖 = 𝛼ℎ𝑠𝑖𝑖 for all 𝑖 and 𝛼 = 3000.In TOPC (10), for a given target-SIR for each user 𝑖, the valueof 𝑅th

𝑖 is set according to (20) and 𝜂𝑖 is obtained from (11).For OPC, the value of 𝜂𝑖 is set to 10 for all 𝑖.

The transmit power and SIR for each user versus theiteration number are shown in Figs. 2 to 5 for TPC, DTPC,OPC, and TOPC, respectively. As expected, since OPC aimsto increase the aggregate throughput, it sacrifices fairness andthose users with a good channel transmit at high power (Fig.4). As seen in Figs. 2, 3 and 5 for a feasible system (whenboth users 5 and 6 are switched off, i.e., before iteration 30),TPC and TOPC work well, and DTPC oscillates (albeit thisoscillation can be avoided by increasing the pricing coefficient

Fig. 4. Transmit power and SIR of each user vs. iteration number, usingOPC. Users 5 and 6 become active after iterations 30 and 60, respectively.

Fig. 5. Transmit power and SIR of each user vs. iteration number, usingTOPC with the effective interference threshold of (20). Users 5 and 6 becomeactive after iteration 30 and 60, respectively.

𝛼 for oscillatory users, meaning a centralized decision makingthat uses a trial and error approach, which contradicts thenotion of a distributed power control algorithm).

When user 5 becomes active but user 6 is still switchedoff (i.e., from iteration 30 to iteration 60), TPC and TOPCwork well and all users reach their target-SIRs, and DTPCbecomes stable but users 3 and 4 do not reach their target-SIRs. When user 6 starts transmitting (after iteration 60), thesystem becomes infeasible (as shown in Fig. 2). As user 6 hasa poor channel, under TPC, it transmits at its maximum powerbut does not reach its target-SIR. This increases interferenceto other users, resulting in users 4 and 5 not reaching theirtarget-SIRs although they transmit at their maximum power,and users 1 to 3 have to consume more power to reach theirtarget-SIRs. In this case, if user 6 reduces its transmit power,the interference to other users is decreased, resulting in users4 and 5 to reach their target-SIRs and users 1 to 3 to attaintheir target-SIRs at lower transmit power. This is achieved ifTOPC is applied as shown in Fig. 5. In an infeasible system,in contrast to TPC and DTPC, TOPC for which the effectiveinterference threshold is chosen according to (20) causes thoseusers with a poor channel to operate in the opportunistic modeof TOPC, where the transmission power is reduced with anincrease in the effective interference levels (transmitting atvery low power and thus experiencing a lower SIR than its

RASTI and SHARAFAT: DISTRIBUTED UPLINK POWER CONTROL WITH SOFT REMOVAL FOR WIRELESS NETWORKS 841

Fig. 6. Average outage ratio vs. target-SIR for TPC, for TOPC with theeffective interference threshold chosen by (20) and for the optimum solutionobtained via exhaustive search. The outage ratio in TOPC is closer to theoptimum solution as compared to TPC.

target value to make target-SIR requirements for the remainingusers achievable), while other users operate in the target-SIR-tracking mode of TOPC and attain their target-SIRs.

In summary, the outage ratio for OPC is the worst one, aswe expected. The outage ratio for DTPC increases from 0 to0.4 and 0.5 when users 5 and 6 are switched on, respectively.We see that the outage ratio for both TOPC and TPC is zerowhen the system is feasible, but after both users 5 and 6 areswitched on (i.e., after iteration 60), it reaches 0.16 (onlyuser 6 does not reach its target-SIR) in TOPC, and gets to0.5 in TPC, indicating that the outage ratio for TOPC isless than those of other algorithms. This outage reductionis achieved while each user consumes less power and thusthe total power consumption is significantly reduced by usingTOPC as compared to using TPC (as shown in Figs. 2 and 5,and in Theorem 3).

To show that the better performance of TOPC as comparedto TPC in reducing the outage is not dependent on specific lo-cations and target-SIRs of users, we compare the performancesof TOPC with both TPC and the optimum solution obtainedby exhaustive search for different snapshots of users’ locationsand for different values of target-SIRs2. To do so, we considera single cell network with a radius of 1 Km and with 12 fixedusers.

To compare the performances of TOPC with that of TPCand the minimum outage, the target-SIRs are considered thesame for all users, ranging from 5 to 25 with step size 5. Thelower and higher values of SIRs correspond to the feasibleand the infeasible systems, respectively. For each target-SIR,we average the corresponding values of outage ratio (forTPC, TOPC and also for the optimum one), and averagethe transmit power consumed by each user (the average totaltransmit power over the entire snapshots divided by the totalnumber of users) for 1000 independent snapshots for a uniformdistribution of users’ locations within a single-cell. The initialtransmit power for each user is uniformly set from the interval[0, 1] for each snapshot. Figs. 6 and 7 show the average outageratio and the average total transmit power versus target-SIR,respectively. Note that TOPC outperforms TPC with respect to

2As DTPC may not converge for some snapshots, it is not included.

Fig. 7. Average transmit power vs. target-SIR for TPC and for TOPC with theeffective interference threshold chosen by (20). The average transmit powerconsumed by users in TOPC is significantly lower as compared to TPC.

Fig. 8. Users and base stations locations. Users are marked “×” with theirindex next to them and base stations are marked “△”.

the outage ratio and the consumed power and its performanceis closely following that of the optimum one obtained viaexhaustive search. For instance, for the target-SIR of 15, byusing TOPC, the outage ratio is reduced by 13% and theaverage transmit-power is significantly reduced by 54%, ascompared to using TPC. As seen in Figs. 6 and 7 for othertarget-SIRs that make the system infeasible, either similar orbetter improvements are achieved by using TOPC as comparedto using TPC.

B. Multi-Cell Networks

Now we consider a multi-cell wireless system with 20 usersdistributed in an area covered by 4 base stations. The locationsof users for a uniformly generated set marked by “×” and thebase stations marked by “Δ” are shown in Fig. 8, where eachuser is assigned to its nearest base station. Each cell covers500 m × 500 m. Assume that for each user, the target-SIR is24 and the value of 𝑅th

𝑖 is obtained from (20). To show howTOPC works when users move in a multi-cell environment, weassume that initially, user 20 is at its starting point in cell No.4, and after 30 iterations3, it moves towards cell No. 3 along

3User 20 is assumed to be fixed in the initial 30 iterations to showinfeasibility of the system when it stays at the starting point in Fig. 8.

842 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011

Fig. 9. Transmit power and SIR of remotely located users vs. iterationnumber, using TOPC. User 20 after iteration 30 starts moving from the startpoint and at uniform speed reach the end point at iteration 250.

Fig. 10. Outage ratio for each cell in TOPC vs. iteration number. The reasonfor the brief transition in the outage ratio for cell 3 is the hand-over of user20 from cell 4 to cell 3.

the line in Fig. 8 at a uniform speed, so that its movement fromthe start point to the end point takes 220 iterations (equivalentto a speed of 38.21 m/s and transmit power updating every 50ms (20 Hz)). When user 20 enters cell No. 3 coverage area,base station 3 is assigned to it.

This system is infeasible when user 20 is at its startingpoint, but becomes feasible when it reaches towards the endpoint. Note that (ignoring user 20), user 5 in cell No. 1, user10 in cell No. 2, user 14 in cell No. 3, and user 19 in cell No.4 are the farthest away users in each cell from their respectivebase stations. The transmit power and SIRs for these usersand for the moving user 20 are shown in Fig. 9. The outageratio for each cell versus the iteration number are shown inFig. 10. As we see in Fig. 9, prior to start of moving by user20 (i.e., before iteration 30), only users 10, 19 and 20 fail toreach their target-SIRs and thus the outage ratio for cell Nos.1 to 4 are 0, 0.2, 0.33, and 0, respectively.

When user 20 starts moving (at iteration 30) and gets faraway from cell No. 2, as its interference to cell No. 2 isreduced, user 10 in cell No. 2 switches to target-SIR-trackingmode, and reaches its target SIR, and thus the outage ratiofor cell No. 2 becomes zero. As user 20 moves farther and ishanded over from cell No. 4 to cell No. 3, it switches to TPCand reaches its target-SIR, but user 19 still fails to reach its

target-SIR, thus the outage ratio for cell No. 4 is 0.2. However,when user 20 gets closer to base station 3 and far from basestation 4, since its interference to cell No. 4 is reduced, user19 can also be supported, thus the outage ratio for cell No.3 is reduced to zero. These numbers show that TOPC alsoworks well when users move, and supports more users thanTPC by causing some users (those that cause bottlenecks) tooperate in the opportunistic mode.

VII. CONCLUSIONS

We proposed TOPC for distributed power control in wirelessnetworks. The algorithm was proved to converge to a uniquefixed point (as opposed to DTPC), and its power consumptionis significantly less than that of TPC for individual users andfor the aggregate of all users. This significant energy saving isachieved while our scheme reduces the outage ratio as well,compared to existing algorithms. Thus TOPC outperformsTPC with respect to addressing the minimum-outage problem(less outage) by consuming less power. For the case that usersare selfish, we also provided a game theoretic analysis ofour proposed algorithm by presenting a new pricing scheme.Due to its fully distributed power updating scheme and itsguaranteed convergence, TOPC can be used in both cellularand ad hoc networks.

APPENDIX APROOF OF THEOREM 5

To obtain the best response function denoted by 𝑏𝑖(p−𝑖),we use the first and the second derivatives of the pricing-basedutility with respect to 𝑝𝑖

∂𝑈𝑖

∂𝑝𝑖=

1

𝑅𝑖(𝑞′(𝛾𝑖)− 𝛼𝑖) , (28)

∂2𝑈𝑖

∂𝑝2𝑖=

(1

𝑅𝑖

)2

𝑞′′(𝛾𝑖). (29)

For a given 𝑅𝑖, we note from (28) that ∂𝑈𝑖/∂𝑝𝑖 = 0 hasa unique root 𝛾𝑖 = 𝑞′−1(𝛼𝑖), that is 𝛾𝑖 = 𝛾𝑖 if 𝑅𝑖 ≤ 𝑅th

𝑖

and 𝛾𝑖 = (𝑅th

𝑖

𝑅𝑖)2𝛾𝑖 if 𝑅𝑖 > 𝑅th

𝑖 . From Property II in Section

V, we have 𝑞′′(𝛾𝑖) < 0 for all 𝛾𝑖, thus ∂2𝑈𝑖/∂𝑝2𝑖 < 0 which

means the unique root of ∂𝑈𝑖/∂𝑝𝑖 = 0 globally maximizes𝑈𝑖. For a given effective interference 𝑅𝑖, a one-to-one relationexists between SIR and the transmit power, and thus the besttransmit power in response to p−𝑖 that maximizes 𝑈𝑖 is alsounique and is equal to 𝑝𝑖= 𝑞′−1(𝛼𝑖)𝑅𝑖, i.e., the best responsefunction is

𝑏𝑖(p) = 𝑞′−1(𝛼𝑖)𝑅𝑖 =

⎧⎨⎩𝛾𝑖𝑅𝑖, for 𝑅𝑖 ≤ 𝑅th𝑖

(𝑅th𝑖 )2

𝑅𝑖𝛾𝑖, for 𝑅𝑖 > 𝑅th

𝑖

(30)

which is the same as in TOPC and its fixed point is theNash equilibrium for 𝐺 = ⟨ℳ, (𝑃𝑖), (𝑈𝑖)⟩. As there existsa unique fixed point in TOPC, the Nash equilibrium existsand is unique. Thus (27) holds.

RASTI and SHARAFAT: DISTRIBUTED UPLINK POWER CONTROL WITH SOFT REMOVAL FOR WIRELESS NETWORKS 843

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Mehdi Rasti received his B.Sc. degree from ShirazUniversity, Shiraz, Iran, and the M.Sc. and Ph.D. de-grees both from Tarbiat Modares University, Tehran,Iran, all in Electrical Engineering in 2001, 2003and 2009, respectively. From November 2007 toNovember 2008, he was a visiting researcher atthe Wireless@KTH, Royal Institute of Technology,Stockholm, Sweden. He is now with Shiraz Univer-sity of Technology, Shiraz, Iran. His current researchinterests include resource allocation in wireless net-works, 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.