Generalized Proportional Fair

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SUBMITTED TO IEEE INFOCOM 2006 1

Generalized Proportional Fair Scheduling inThird Generation Wireless Data Networks

Tian Bu Li (Erran) Li Ramachandran Ramjee

Bell Labs, Lucenttbu,erranlli,[email protected]

Abstract In 3G data networks, network operators wouldlike to balance system throughput while serving users in afair manner. This is achieved using the notion of propor-tional fairness. However, so far, proportional fairness hasbeen applied at each base station independently . Such anapproach can result in non-Pareto optimal bandwidth al-location when considering the network as a whole. There-fore, it is important to consider proportional fairness in anetwork-wide context with user associations to base stationsgoverned by optimizing a generalized proportional fairnessobjective. In this paper, we take the rst step in formulat-ing and studying this problem rigorously. We show thatthe general problem is NP-hard and it is also hard to ob-tain a close-to-optimal solution. We then consider a specialcase where multi-user diversity only depends on the num-ber of users scheduled together. We propose efcient of-ine optimal algorithms and heuristic-based greedy onlinealgorithms to solve this problem. Using detailed simulationbased on the base station layout of a large service provide inthe U.S., we show that our simple online algorithm, which as-signs a newly arrived user to a base station that improves thegeneralized proportional fairness objective the most withoutchanging existing users association, is very close to the of-ine optimal solution. The greedy algorithm can achieve sig-nicantly better throughput and fairness in heterogeneoususer distributions, when compared to an approach that as-signs a user to the base station with the best signal strength.

I. INTRODUCTION

Third Generation (3G) wide-area wireless networksbased on the CDMA2000 [1] and UMTS [2] standards arenow being increasingly deployed throughout the world.

As of December 2004, there were over 146 millionCDMA2000 subscribers and over 16 millions UMTS sub-scribers worldwide [3]. Emerging 3G data standards, EV-DO and HSDPA, promise to deliver broadband mobile in-ternet services with peak rates of 2.4 Mbps and 14.4 Mbps,respectively.

In these third generation wireless data networks, Pro-portional Fair (PF) scheduling [4] is employed at the basestation to schedule downlink ows among different users.

Proportional fair is a channel-state based scheduling algo-rithm that relies on the concept of exploiting user diver-sity. Consider a model where there are N active users shar-ing a wireless channel with the channel condition seen byeach user varying independently. Better channel condi-tions translate into higher data rate and vice versa. Eachuser continuously sends its measured channel conditionback to the centralized PF scheduler which resides at thebase station. If the channel measurement feedback delayis relatively small compared to the channel rate variation,the scheduler has a good enough estimate of all the userschannel condition when it schedules a packet to be trans-mitted to the user. Since channel condition varies indepen-dently among different users, PF exploits user diversity byselecting the user with the best condition to transmit dur-ing different time slots. This approach can increase systemthroughput substantially compared to a round-robin sched-uler. However, such a rate maximizing scheme can be veryunfair and users with relatively bad channel conditions can

be starved. Hence, the mechanism used in PF is to weightthe current rate achievable by a user by the average ratereceived by a user.

Specically, at each time slot (every 1.67ms in EV-DO),the decision of the PF scheduler is to schedule the userwith the largest

, where is the rate achievableby user and is the average rate of user . The averagerate is computed over a time window as a moving average:

" !$ #& % ' !) (1 02 # 3 #4 5 0 if scheduled " !$ #& % ' !) (1 02 # 3 # if not scheduled

While PF scheduling achieves high throughput whilemaintaining proportional fairness among all users at a basestation, the association of mobile devices to base stationsin todays networks is not based on any fairness consider-ations. Instead, the mobile device is simply associated tothat base station from which it receives the strongest sig-nal. Clearly, such an association algorithm could createload imbalances where some hotspot base stations areheavily loaded while neighboring base stations are lightly


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loaded. Further, such imbalances will decrease both over-all throughput and also decrease fairness among users atneighboring base stations. In fact, such an approach canresult in non-Pareto optimal bandwidth allocation whenconsidering the network as a whole.

In this paper, we start with the premise that a fairscheduling algorithm should take anetwork-wide view andsupport fairness among all users connected to a network of base stations. To this end, we formulate the general-ized proportional fairness (GPF) problem that takes a moremacro-view of fairness and includes asssignment of usersto base stations as part of an overall strategy to providefairness across all users attached to the network of basestations. We show that the general problem is NP-hardand it is also hard to obtain a close-to-optimal solution. Wethen consider a special case, which roughly holds in prac-

tice, where multi-user diversity only depends on the num-ber of users scheduled together and all users have equalpriority. We propose efcient ofine optimal algorithmsand heuristic-based greedy online algorithms to solve thisproblem. Using detailed simulations based on the base sta-tion layout of a large service provide in the U.S., we showthat our simple online algorithm, which assigns a newlyarrived user to a base station that improves the generalizedproportional fairness objective the most without changingexisting users association, is very close to the ofine op-timal solution. The greedy algorithm can achieve signi-cantly better throughput and fairness, in many cases whereusers are not evenly distributed, when compared to an ap-proach that assigns a user to the base station with the bestsignal strength. Our results also indicate, as expected, thatmax-min fairness sacrices too much overall throughput(less than 60% of what is achieved by our algorithm) forfairness from each users perspective.

The rest of the paper is structured as follows. In Sec-tion II, we present our motivation for generalized propor-tional fairness and describe our system model. In Sec-

tion III, we present a rigorous formulation for the gen-eralized proportional fairness problem and show that thegeneral problem is NP-hard and it is also hard to obtaina close-to-optimal solution. In Section IV, we present thedetails of our ofine optimal and online greedy algorithms.In Section VI, we present a detailed evaluation of our algo-rithms using simulation. In Section VII, we present relatedwork. Finally, in Section VIII, we present our conclusionswith a discussion of future work.

II . M OTIVATION AND S YSTEM M ODEL

In this section, we rst motivate the need for generalizedproportional fairness and then present our system model.

A. Motivation

Consider a single base station in a 3G data network serv-

ing its associated users. Assume all users have the samepriority. A network operator would like meet users de-mands to the greatest extent possible, given resource con-straints. However, this does not imply maximizing thetotal throughput of all users, as such a policy may leadto starvation of users who have poor channel conditions.Therefore, we need to consider fairness constraints in al-locating shared resources to multiple users. Two fairnessmeasures are commonly used: max-min fairness and pro-portional fairness. Informally, an allocation of bandwidthby a BS is max-min fair if there is no way to give morebandwidth to any user without decreasing the allocationof a user with less or equal bandwidth. Max-min fair-ness achieves ideal fairness from the users perspective andthe system is work-preserving given the constraints on re-source availability to users. However, max-min fairnesscan signicantly sacrice aggregate throughput. For ex-ample, suppose there are two users 6 and 7 associated witha BS . Let the average data rate of 6 and 7 , denoted by

8@ 9$ A and 8$ BC A , be !E D and ! unit respectively. Max-min fair-ness will allocate both users a bandwidth of !E DG FH !I ! (letsignore multi-user diversity for the moment). User 7 with amuch lower rate will be allocated 10 times ( !E DG FH !I ! fraction)more slots than 6 . This clearly sacrices aggregate systemthroughput signicantly. In order to achieve a better trade-off between fairness and throughput, Kelly [5] proposedproportional fairness. In our example, proportional fair-ness is equivalent to time fairness. User 6 and 7 will get abandwidth of P and !$ FR Q respectively. Proportional fairnessachieves much better aggregate throughput than max-minand gives users equal time fairness. For wireless networkswith scarce bandwidth, proportional fairness is much moreappealing to network operators and thus have been imple-mented in 3G data networks such as 3G1x-EVDO.

In CDMA-based 3G networks, a mobile device can hearsignals from multiple BSs. The user is typically associ-ated to the base station with the strongest signal. Pro-portional fairness scheduling is then used independentlyat each base station to schedule downlink transmissions todevices associated with each base station. Using this ap-proach, uneven user distribution will result in uneven loaddistributions at the BSs. In order to handle this load imbal-ance, techniques that control base station coverage such as

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u

102 1

b

v

a

Fig. 1A WIRELESS NETWORK WITH S BS S , T AND U , AND TWO

USERS , V AND W .

cell breathing or user association have been proposed [6],[7]. Although Sang et al. [6] consider load balancing andfairness in an integrated framework. The load balancingmetric does not consider the fairness objective. In fact,the load balancing techniques may reduce network-widefairness objective since highly loaded cells will reduce theamount of bandwidth allocated to users at the cell bound-ary. Their rationale is that this may trigger users to hand-off to other less loaded cells. However, without the ex-plicit consideration of fairness, it is unclear whether theywill maintain a unied fairness measure. Thus, we arethe rst to consider load balancing with the network-wide proportional fairness as the goal . We argue that propor-tional fairness should be used when allocating all shared system resources (all BSs) to users rather than a single sys-tem resource (a single BS). This enables optimal tradeoff between system utilization as a whole and serving users

fairly. We achieve this by controlling user associations tobase stations.

We now illustrate the benet of considering user associ-ation and proportional fairness jointly using an example 1.In this example, 8X 9$ AY %` !E D , 8 9$ a %b D , 8@ BC Ac %d Q , 8 BC a % ! .If user association is based on strongest signal, then both

6 and 7 will associate with BS . If the system is usingproportional fairness scheduling at BS , then 6 will get

P and 7 will get ! . However, if we consider user associ-ation and proportional fairness jointly, users 6 and 7 willbe associated with BS and e respectively. This gives 6 a

bandwidth of !E D

and 7 a bandwidth of !

. This certainly isa better allocation than the previous one which considersuser association and proportional fairness separately. Theallocation of PH f ! is not even Pareto optimal when com-pared with the allocation of !E D f ! . A Pareto optimal allo-cation is one such that, there does not exist another feasi-ble allocation where at least one user gets more bandwidth,and all others get at least the same bandwidth. Now, if weconsider max-min fairness allocation in conjunction with

user association, this will result in both 6 and 7 being as-sociated with BS , and they will both get a bandwidth of

!E DG FR g .

B. System Model

The latest 3G standard, called the 3G1X Evolution or

High Data Rate (HDR), is designed for bursty packet dataapplications. It provides a peak downlink data rate of 2.4Mbps and an average downlink data rate of 600Kbpswithin one 1.25MHz CDMA carrier. HDR is commer-cially available. HDR downlink has much higher peak datarate (2.4Mbps), compared with the uplink peak data rate of 153.6Kbps [8]. Users share the HDR downlink using timemultiplexing with time slots of 1.67ms each. At any timeinstant, data frames are transmitted to one specic user,and the data rate is determined by the users channel con-dition. Users monitor the pilot bursts in the downlink chan-nel to estimate the channel conditions in terms of Signal toNoise Ratio(SNR). This SNR is then mapped into a sup-ported data rate, and fed back every time slot to the basestation through the data-rate-request channel in the reverselink. The duration of transmission to each user is deter-mined by the downlink scheduling algorithm. HDR uses ascheduling algorithm called Proportional-Fair Scheduling[8]. The scheduler serves the user with the highest ratioof instantaneous downlink channel rate over the averagereceived data rate.

In this paper, we focus on the downlink scheduling of anetwork of 3G wireless data base stations. When a devicepowers up or becomes active, we assume the device re-ports the set of BSs it can hear and the data rate to each of the BSs. The network, possibly the Radio Network Con-troller (RNC), then determines a users (devices) associ-ation based on this reported information as well as pre-existing information of currently associated users at eachBS. This can be achieved in a reasonable time since fastfades last 5-50 slots depending on the speed of a mobile,and a device can feed up its rate information each time slot(1.67ms). So it is reasonable to obtain a good estimate of the average rate of a user with a BS in a short period of time

( 200ms). Alternatively, the network could also rst asso-ciate a device to the base station with the strongest signal,then instruct the mobile device to report the average ratesto all the base stations the device can hear, and then switchthe mobile device to the optimal base station.

We determine user association upon arrival or handoff due to mobility. Other work such as [6], [7] have assumedmuch more frequent user association changes. This wouldrequire much more coordination among BSs as to which

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packet a BS should transmit at a certain time slot. Thiswould also increase the signaling load to the RNC if RNCdetermines the association change. Both the work of Sanget al. [6] and Das et al. [7] have assumed that a central con-troller can determine the scheduling policy of each basestation. This would require ne-grained feedbacks fromeach BS to the central controller. In this paper, we as-sume that the BS runs an independent proportional fair-ness scheduler and the network (RNC) only determines theuser association to a base station. Sang et al. [6] has alsoassumed that each mobile determines which BS it wantsto be associated with according to its own utility function(which is determined by the throughput it can get fromeach BS). We remark that, independent users action maylead to handoff oscillations. To avoid such problems, wehave assumed the network (RNC) determines when a usershould change its association.

III. P ROBLEM F ORMULATIONIn this section, we present a rigorous formulation of the

network-wide proportional fairness bandwidth allocationproblem. We consider a 3G wireless data network consist-ing of a set of BSs , and a set of users h . We assumeusers are static for now and allow for mobile users in latersections. A user 6 s average data rate if associated with aBS is denoted as 8$ 9X A . Let i 9p %r q$ s 8@ 9$ Ap tu D fw v Y x y .Let 9 be the actual bandwidth allocation to user 6 by thenetwork. It has been shown in [5] that proportional fair-ness allocation of network resources is equivalent to the

optimization of the following objective function:

9I I X

9H #

Typically 3G data users have elastic trafc. The applica-tion trafc in most cases uses TCP as the transport proto-col. TCP will try to achieve the maximum rate allowed bythe system. Therefore, we assume a user will consume allthe bandwidth allocated and the queues are backlogged.In 3G data networks, typically there is a limit on howmany users can be admitted to the system. For example,

in 3G1x-EVDO, there are 60 Walsh codes for orthogonaltransmission. This puts an upper bound of 60 active users(per base station per sector-carrier) at any given time. Forease of explanation, we will assume that any user 6 willbe admitted as long as s i 9 s t D . We will discuss how toextend our framework to the admission control case later.Note that the limit of 60 users is rarely reached in prac-tice since 60 users sharing an average 600Kbps downlink channel provides very little throughput for each user.

Lets denote 9$ A the association variable, i.e. 9$ A % ! if user 6 is associated with BS , 0 otherwise. We assume auser can only be associated with one BS at any given time,i.e., only one BS can transmit data through its downlink tothe user as in the EV-DO standard [4]. As noted earlier, wepnly consider downlink bandwidth allocation in this paper.Since all users must be admitted, we have the following:

AE I $

9$ A % !

Let the bandwidth allocation for users associated witha given BS be proportional fair. Let the number of usersassociated with BS be A %j 9I I 9X A . Lets denotethe set of users associated with a given BS be k A , i.e.

k

Al %m q

6

s 9X Al %j !

f) v 6

x

h y . In the general case, themulti-user diversity gain can depend on the set of users, not just the number of users. Let n be the mapping that, given

6 and , returns 8$ 9$ A . Users may have different schedul-ing priorities for service differentiation. We denote thescheduling priority of user 6 as o 9 . Let be the map-ping that given a user 6 returns 9 . If 9X A % ! for a givenuser 6 and BS , its actual bandwidth 9 allocation by BS

will be a general function of all the users associated with . Denote this function by A k A f 6 f nY fC o # . Of cause,

AG

k

A

f 6 f n fC o

# % D if 6 x k A . We now derive the fol-lowing problem formulation for Generalized ProportionalFairness, GPF1:

AE 9I I zX

A

k

A

f 6 f nY fC o

# # (1)

Subject to

AE I

9$ A %{ !

f4 v 6

x

h (2)

k

A %| q

6

s 9$ A % !

f4 v 6

x

h y (3) 9X A %| q$ D

f

!

y (4)

However, as shown in [9], if the relative rate uctu-

ations are statistically identical, the multi-user diversitygain only depends on the number of users associated witha given BS. This assumption on rate uctuation is roughlyvalid when the users have, for example, Rayleigh fad-ing channels and the feasible rate is approximately lin-ear in the SNR (reasonably accurate when the SNR isnot too high). With this assumption, according to [9],

9X A%~ }

A# 8

9$ AF

A if 6 is associated with BS . Thus, weobtain the following restricted version of the GPF1 prob-

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lem. We refer to this problem as GPF2.

9 I AE I $

9$ A

X

8$ 9$ A 9

} AX #

o

A

# (5)

Subject to

AE R X

9$ A %{ !

f v 6

x

h (6)

o

A %

9I A@ R

9X A 9

f v

p x (7)

A %

9I A@ R X

9$ A

f4 v

x (8)

9$ A % q$ D

f

!

y (9)

When all the users have the same priority in GPF2, wehave the following special case, GPF3:

9I I A@ R X

9$ A

$

8@ 9$ A

} AX #

A

# (10)

Subject to

AE I

9$ A % !

f4 v 6

x

h (11)

A %

9I A@ R X

9X A

f4 v

x (12)

9$ A %| q$ D

f

!

y (13)

A. GPF1 and GPF2 are NP-hard and inapproximable

We show that, for GPF1 and GPF2, there does not existan algorithm that can nd the optimal in polynomial timeunless

%

, i.e. the problem is NP-hard. Our re-duction is via 3-dimensional matching which is known tobe NP-complete. The 3-dimensional matching problem isstated as follows.

Denition 1: Given an instance of the following prob-lem: Disjoint sets % q eX @ fE E E @ fC e y , %b q$ $ fE E E $ f y ,

% q$

fE E E X f y , and a family % qE fE E E @ f y of triples with s s %" s 3 s %" s 3 s for %{ ! fE E E X f .The question is: does contain a matching, i.e. a subfam-ily for which s sI %" and

R z

z %

~ ?Theorem 1: The generalized proportional fairness prob-

lem GPF1 is NP-hard.Proof: Our reduction is along the lines of [10].

Clearly, the answer to the 3D matching decision prob-lem is no if m . It is easy to check whether isa matching or not if %d . We simply check whether

%

5 { . So we assume t in ourreduction.

We call the triples that contain eC triples of type . Letbe the number of triples of type for %{ ! fE E E X f . BS

corresponds to the triple 4 for % ! fE E E X f . We have Q

element users, corresponding to the Q elements of .There are ( ! dummy users of type for %r ! fE E E $ f .Note that, the total number of jobs is (" . For BScorresponding to a triple of type , say % e f f # ,the corresponding element users $ and have an aver-age rate of . Their multi-user diversity gain when sched-uled together is } where !5 } Q #Y Q , i.e. each ob-tain a rate of } FR Q . Each of the dummy jobs of type

has an average rate of . However, the multi-user di-versity gain } Q # when an element user and a dummyuser of type is assigned to one of the BSs, is muchsmaller than } . All other average rates between BSs andusers are 0. Except element users corresponding to a ,all other users, dummy or element, when associated witha BS, the multi-user diversity gain is given by function

} # . Assume -

-

if . Similarly the con-dition holds for } . We claim that a matching exists iff the GPF1s optimal objective function where

% Q

X

} FR Q # ( #

X

.Suppose there is a matching. For each w % e f f G #

in the matching, associate element user $ and withBS . For each , this leaves ( ! idle BSs corre-sponds to triples of type not in the matching; asso-ciate the (~ ! dummy users of type with these (~ !

BSs. This assignment has an objective function value of % Q

X

} FR Q # ( #

X

. Conversely, supposethat there is such an assignment with objective function

. Dummy users can only be assigned to the BSof its type. So the

( !

dummy users will be assigned tothe BSs of type . If possible, BSs should not be idle.Suppose a user E has been assigned to a BS with t !

users and there is an idle BS user @ can be assigned, thenwe can obtain a better solution after moving X to the idleBS. This is because -

-

if (for } as well).Therefore, the set of BSs of each type gets at least one el-ement user (otherwise, there will be an idle BS). We claimthat, if possible assignment exists, no BS should get morethan 3 users. This is because reducing the number of userson this BS and assign them to other BS will one user al-

ways improve the solution. Therefore, each BS gets as-signed at most 2 users in the optimal solution if possible.For those BS with 2 users, it is better to assign the two el-ement users corresponding to BS of type , this is becauseof their multi-user diversity gain is better. In addition, notwo BSs of the same type, each gets assigned two users.So the best possible solution is an assignment with an objective function value . In any of these assignment, thereexists exactly one BS for each type , where two element

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users corresponding to type is assigned, and ( ! BSs of type , each get assigned a dummy user. There is a total of

Q element users which implies a 3-dimensional matchingin the original problem.

Similarly we can prove the following theorem.Theorem 2: The generalized proportional fairness prob-

lem GPF2 is NP-hard.The key idea is to assign low priority to dummy users andhigh priority to element users; then set the average rate of dummy users with its BSs very high, and set the rate of element users low. This will force element users to be as-signed together. Otherwise, they would steal the rates of dummy users which results in a worse objective functionvalue.

Corollary 1: There does not exist a polynomial time al-gorithm that can approximate GPF1 and GPF2 within afactor of f v t D , i.e. they are inapproximable.

Proof: Theorem 1 holds even if we assign rate 8 in-stead of for dummy users of type to its type BSs.We can pick 8 such that %| D . Suppose we have an ap-proximation algorithm of factor . According to the def-inition of approximation, the algorithm outputs a solutionwith objective function value such that H whichmeans j D . Since % D , this means % D , thuswe have found the optimal which contradicts Theorem 1.Similarly, this holds for problem GPF2.

B. Properties of GPF3

We do not yet know whether GPF3 is NP-hard or not.The reduction for GPF1 and GPF2 can not be applied toGPF3 because we can not enforce two element users cor-responding to a triple in 3-dimensional matching are as-signed to the same BS (GPF3 can associate dummy usersand element users together and still achieve optimal). De-spite this fact, we do know interesting properties of GPF3which enable us to design efcient algorithms. We nowshow these properties.

Proposition 1: If we know A in GPF3, then the prob-lem can be solved optimally in polynomial time.

If A

is xed, then

9$ AY %

X

8$ 9$ A

-

#

is xed. Theproblem is then equivalent to nding a maximum weightedbipartite matching with 9$ A as the weight. The maximumweighted matching problem can be solved optimally inpolynomial time.

Proposition 2: If two users 6 f 7 are associated with BS

fC e respectively in an assignment and

C

, then wecan always swap 6 and 7 s association and improve theobjective function.

Algorithm OfineOPT- BSInput: Network % f h # , multi-user diversitygain } # , 8$ 9$ A f v x f 6 x h

for each E fE E E $ f # such that

%

if t s k s where k 4 %| q 6 s 8$ 9@ w t D f v 6 x h y ,then next iteration

if 8$ 9@ % !

and t D

9@ 4 %

X

8@ 9@

-

#

else 9@ 4 %" D

MatchingAlgo( Y f q 9@ y )end

Fig. 2A FORMAL DESCRIPTION OF THE OFFLINE OPTIMAL

ALGORITHM

This is easily obtained by comparing the two objectivefunction values before and after 6 and 7 are swapped withtheir associated BSs.

IV. A LGORITHMS

A. Ofine Algorithms

We rst design an ofine algorithm for computing theoptimal user assignments to the base stations. If the num-ber of BSs is a constant , based on Proposition 1, wecan obtain a polynomial time algorithm to nd the optimalassociation. The idea is to guess all possible

A

congu-rations, then solve the the maximum weighted matchingproblem for each conguration. The algorithm is poly-nomial because the number of congurations is (let

% s

h

s ) and maximum weighted matching has a run-ning time of 4 @ # . The total running time is thus

)

3 '

# . The OfineOPT algorithm is presented inFigure 2.

However, when the number of BSs is large, OfineOPTwill be very inefcient. We note that, in our context, usersare spatially distributed; users inside a certain region will

not be able communicate with BSs further away from theregion. Therefore, there is a natural partition. We ex-ploit this spatial property and partition the network intosmaller connected components where the number of pos-sible edges (associations) between components is small.We rst solve our generalized proportional fairness prob-lem within each component. We then assign users, whoseedges cross components, greedily. The formal descriptionof the algorithm is in Figure 3. The algorithm is refered

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Algorithm KComponentInput: Network % f h # , multi-user diversitygain } # , mapping n # s.t. 6 f # % 8X 9$ A f v p x f 6 x hRun MinK-Cut Algorithm to obtain

fE E E X f3

connected componentsfor each f v %{ ! fE E E X f3

OfineOPT-

BS(

,}

, )for each user 6 whose edges cross componentsGreedily assign 6 to BS that improvesthe objective function the most

end

Fig. 3A FORMAL DESCRIPTION OF THE KC OMPONENT

ALGORITHM

to as KComponent. We use the approximation algorithmin [11] to compute a cut. We would like to minimize thenumber of edges across components. Formally, the mini-mal -cut problem is dened as follows: a set of edgeswhose removal leaves connected components is called a

-cut. The -cut problem asks for a minimal weight -cut.We can tune to trade off optimality with computationtime.

In a large network with many users, Algorithm ofi-neOPT, even when running KComponent, can result in ahigh computational overhead. In addition, KComponent

(also OfineOPT) only works for GPF3. We thus considerthe design of a heuristic algorithm based on efcient localsearch. We dene two operations: Swap and Change. Atany given time of the algorithm, if swapping two users canimprove the objective function, we then do the swap. If changing a users association can help, then we carry outthe Change operation. To make sure the algorithm runs inpolynomial time, we place a lower bound constant suchthat each improvement operation, or improvement oper-ations, improves the objective function by at least . Thealgorithm is guaranteed to terminate in time

X

# it-

erations. Its running time is

X

#

#

. We refer tothis algorithm as local search (LS).

Note that Algorithm LS can get stuck in local optima.The following is an example. We have three BS fC eX fand three users 6 f 7 f . We have the following rate values,

8@ 9$ A % !@ QI , 8 9$ a % G I Q , 8@ 9$ % I I , 8$ BC A % ! P , 8 B a %r Q !$ ,8@ BC % g

P

, 8@ A % P QI g , 8 a % P g , 8$ BC % I g . It is easy toverify that the optimal assignment 4-a yields an objectivefunction value of !@ P . Algorithm LS outputs the assign-

t

c

u

b

v

a

t

c

u

b

v

a

(a) Optimal assignment (b) LS assignment

Fig. 4A LGORITHM LS MAY GET STUCK IN LOCAL OPTIMA

ment in 4-b with a value of !@ D . We can verify that anysingle Swap or Change operation will not yield a better as-signment. We can replicate this example so that LS searchwill fail even if we consider Swap operations with s h sE ( !

users.

B. Online Algorithms

While ofine algorithms are useful in computing opti-mal assignments to compare against, they cannot be usedin a dynamic network-setting with mobile users. We there-fore consider two greedy online algorithms with very littlecomputational overhead. Our rst algorithm assumes thatonce a user is assigned with a BS, it can not change toassociate with a different BS unless due to handoff or con-nection failure. We greedily associate with the BS suchthat the objective function improves the most. We refer tothis algorithm as Greedy-0.

Given that users association to base stations can changeoften in a dynamic setting, we can also potentially changeusers association if it improves the system objective func-tion. In the second online algorithm, we assume we canchange the association of at most existing users. Because

is a small number, we can try out all possible cases andpick the one that improves the objective function the most.We refer to this algorithms as Greedy-k

V. E XTENSIONS

We discuss possible extensions to our problem. In prac-tice, a BS can only admit a nite number of users due tothe number of available Walsh codes for uplink communi-cation. Let this bound be A f v x . We then have thefollowing constraint:

A

A

Since some users may be blocked, we need to remove theconstraint A@ R X 9X Ap %b ! . However, this is not enough.In order to optimize the objective function using this for-mulation, a system may reject a user 6 even if there exists

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[ k m

]

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Meshed Cell Sites, Service Provider X, Northern NJ

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Fig. 5G EOGRAPHICAL LOCATIONS OF BASE STATIONS

a BS such that 6 x i A and A A . To make the sys-tem work conserving, we dene the indicator variable A ;

A % D

if A % A

, 1 otherwise. We derive the followingconstraint in order for the system to be work conserving:

AE R X

9$ A %{ !

f if A A f 6 x i A

It is easy to see that the GPF1 and GPF2 prob-lem remains to be NP-hard and inapproximable. TheOfineOPT- BS can be extended to our current setting bysimply observing the capacity constraint in the algorithmshown in Figure 2.

VI . E VALUATIONWe have demonstrated the general proportional fairness

problem is NP hard and presented some online and ofineheuristic algorithms. In this section, we evaluate our al-gorithms using simulations. We rst evaluate the qualityof the LS algorithm by comparing to the optimal algo-rithm for a small problem size. We then investigate theadvantages of associations based on our GPF3 problemformulation (hereafter, simply referred to as GPF) overBest-Signal and Max-min associations. Finally, we eval-uate the performance of two online algorithms, Greedy-

0 and Greedy-k. We remark that, except OfineOPTand KComponent, all our other algorithms (LS, Greedy-0 and Greedy-k) work with all three problem formulations(GPF1,GPF2 and GPF3).

A. Simulation setup

The map of base station layout that we use for the per-formance evaluation of our algorithms is presented in Fig-ure 5. It is part of a 3G network operated by one of the

0 0.2 0.4 0.6 0.8 110

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15Slow Fading Ec/Nt

Distance from BS (Cell Radius)

d B

Fig. 6SLOW FADING

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10Fast Fading Ec/Nt

Time Slot (1.67ms)

d B

Fig. 7FAST FADING

major service providers in United States. There are a to-tal of base stations within P D R D . The longitudeand latitude distances are all relative to a reference pointpicked for good visualization.

In our simulation, we assume all base stations have auniform radius of km and there are a total of !E DI DI D 3GHDR mobiles in the network. User requests for radio chan-nel arrives according to a Poisson process with an averagerate D DI D ! /second. We assume that the average radio chan-

nel holding time is exponentially distributed with mean of R DI D seconds. We divide the map in Figure 5 into P D c R D

zones where each zone is one square kilometer. We as-sume mobiles migrate from one zone to another with anexponentially distributed staying time of mean gR D seconds.We further assume mobiles have the same rates within thesame zone and a re-association may be performed when anactive mobile moves from one zone to another and its rateschange.

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0 0.2 0.4 0.6 0.8 10

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1

1.5

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2.5x 10

6 HDR Downlink Channel Rate

Distance from BS (Cell Radius)

D a

t a R a

t e ( b / s )

Average Channel RateInstantaneous Channel Rate

Fig. 8HDR CHANNEL RATE

The HDR downlink channel is modeled according to thepublished experimental data in [8], [12]. The HDR down-link channel quality is determined by both slow fading andfast fading. Slow fading is modeled as a function of theclients distance from the HDR base station, as shown inFigure 6. Fast fading is modeled by Jakes Rayleigh fading[13] as shown in Figure 7. The combined F for bothslow and fast fading is then mapped to a table of supporteddata rate with 1% error [12]. Figure 8 presents a snapshotof HDR downlink instantaneous channel rates, and the av-erage rate over a long time period for clients with differentdistances from the base station. In our simulations, we usethe average rate for determining throughput and base sta-

tion associations and do not simulate the fast fading or thezero mean shadow fading on a per-mobile basis.

In order to simulate skewed user distributions that aretypical in a real 3G network setting, we assign differ-ent weights to different zones. When a mobile moves, itchoose its next destination zone from one of the neigh-boring zones with probability proportional to the weightsof that zone. In our simulation, we randomly generate aweight for each zone in D f !E D uniformly.

B. Comparison of local search algorithm with optimal

It has been shown that Algorithm OfineOPT can ndthe optimal association in polynomial time. We evaluatethe local search (LS) heuristic algorithm for a small net-work of g base stations where the optimal solution can beobtained in reasonable amount of time using OfineOPT.We picked base stations PH f f f ! f ! P and Q P in Figure 5together with QR DI D users as input and run the simulation for

!E DI DI DI D seconds. A new association is calculated when aradio channel is requested or terminated. A radio chan-

0.98 0.985 0.99 0.995 1 1.00510

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10 2

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Performance Ratio

C D F

Fig. 9CDF OF PERFORMANCE RATIOS

nel request may be triggered by either a mobile user be-coming active or an active mobile user moving into a newzone. A radio channel termination may be triggered bythe inactive timer timeout or a mobile moving away froma zone. We compute each association with both the op-timal algorithm and our LS heuristic algorithm. Figure 9plots the CDF of the performance ratio, which is denedas the ratio between the value of generalized proportionalfairness objective function obtained from the heuristic al-gorithm and that obtained from the optimal algorithm. Weobserve that only less than !$ of the performance ratiosare less than D I . In almost R DG of the cases, the LSheuristic algorithm achieves the same objective functionvalues as the optimal algorithm. Thus, we can approximatethe OfineOPT algorithm with the much more efcient LSheuristic-based algorithm with high condence when computing the optimal bounds on large networks.

C. Comparison of GPF with other association algorithms

In this Section, we compare our GPF association withother association schemes including Best-Signal and Max-Min. We use the LS algorithm to obtain the optimal GPFassociation. The Best-Signal algorithms always assigns amobile to the BS with the strongest signal regardless of

network load. The Max-Min algorithm is the ofine al-gorithm presented in [14]. We do not consider backhaulbottlenecks. Therefore, the max-min fair association com-puted has an approximation factor of 2 coordinate-wisein terms of the throughput vector (in ascending order) allusers get with respect to the optimal max-min fair associ-ation.

We evaluate the algorithms using the achieved network aggregate throughput and fairness metrics. The achieved

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throughput metric measures the throughput efciency atthe network level whereas the fairness metric focuses onthe performance perceived by each individual user.

The fairness metric is measured using fairness indexproposed in [15]. The throughput is rst normalized bythe reference association which we assume an equal band-width share. In this case, the fairness index can be com-puted as following

E

l %

#

The fairness index value is a positive number with maxvalue ! suggesting an equal throughput among all mobiles.The higher the fairness index is, the closer the allocation isto an equal throughput allocation.

We compare the GPF allocation with the other two bynormalizing the achieved throughput using the one ob-tained from the GPF allocation; the fairness index metricis normalized to users receiving equal share. Figures 10and 11 plot the CDF of normalized throughput and fair-ness indexes respectively (thus, GPF has a value of 1 inFigure 10).

We observe from Figure 10 that both Best-Signal andGPF algorithm achieve much higher throughput than Max-Min algorithm. Almost in all cases, the Max-Min algo-rithm achieves throughput less than gR DG of the through-put of GPF. The GPF approach achieves higher throughputin more than R DG cases than Best-Signal. In about gR DG

cases, the GPF approach achieve more than !E DG morethroughput than Best-Signal. Figure 11 shows that thebandwidth allocation of our GPF algorithm is always morefair than the allocation of Best-Signal algorithm whilethe allocation of the Max-Min algorithm is the most fairscheme. However, we observe in Figure 10 the fairnessof Max-min is achieved by signicantly compromisingthroughput. Thus, the GPF association can achieve sig-nicantly better throughput and fairness when compared to an approach that assigns a user to the base station withthe best signal strength. Also, as expected, max-min fair-ness sacrices too much overall throughput (less than 60%

of what is achieved by our algorithm) for fairness.

D. Comparison of online GPF association algorithms

We have demonstrated that an ofine generalized pro-portional fairness association achieves both better fairnessand higher throughput than the currently used Best-Signalapproach. In this section, we compare Best-Signal to twoonline versions of the GPF association,i.e, greedy-0 andgreedy-k; they differ in whether a constant number of the

0.2 0.4 0.6 0.8 1 1.20

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Normalized throughput

C D F

BestSignalMaxMin

Fig. 10CDF OF THROUGHPUT NORMALIZED BY THROUGHPUT

ACHIEVED FROM GPF ASSOCIATION

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Fairness Index (relative to equal share)

C D F

GPFBestSignalMaxMin

Fig. 11CDF OF THE FAIRNESS INDEXES RELATIVE TO EQUAL

SHARES

existing association can be changed. We would like to seeif the gains of using the ofine GPF association can be pre-served in a practical dynamic environment using the onlinealgorithms. As before, we evaluate these algorithms usingboth the throughput and fairness metrics.

We normalize the throughput of each mobiles by its GPFallocation, where we approximate the GPF allocation us-ing values obtained from the LS algorithm. Let us denotethe throughput achieved by GPF for mobile being

-

and the throughput achieved by association scheme .We compare the fairness of Best-Signal and on-line GPFassociation relative to the ofine GPF association. Thefairness index relative to GPF association for is calcu-

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0.6 0.7 0.8 0.9 1 1.1 1.20

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Normalized throughput

C D F

BestsignalGreedy2Greedy

Fig. 12CDF OF THROUGHPUT NORMALIZED BY THROUGHPUT

ACHIEVED FROM GPF ASSOCIATION

lated from

-

# %

d

FX

-

#

FX

-

#

where is total number of active mobiles in the network.We also normalize the throughput from different asso-

ciations by the throughput achieved by algorithm LS andpresent the CDF of normalized throughput in Figure 12.We observe that both online algorithms achieve higherthroughput than the Best-Signal. There is very little differ-ence between the two online algorithm. In more than DG

cases, the online algorithms achieves throughput greater than P of that achieved by the ofine GPF .

Figure 13 presents the CDF of the fairness indexes fordifferent association algorithms. We observe that both on-line algorithms are more fair than Best-Signal with theGreedy-2 slightly more fair than the Greedy-0 . The fair-ness indexes for both online algorithms are almost alwaysgreater than D .

VII . R ELATED W OR K

The Proportional fair scheduler was rst presentedin [4], [16] for the third generation EV-DO wireless datasystem. Since then, there has been a lot of work on the

topic of proportional fair scheduling.The performance of PF has been analyzed by several

The proportional fair algorithm can be shown to maximizethe sum of the logarithm of the throughput of the usersattached to each base station [17]. The user-level perfor-mance of the proportional fair algorithm in a dynamic set-ting was analyzed in [9]. Extensions to proportional fairalgorithm to support quality of service have been proposedin [18], [19], [20], [21]. The authors in [19], [20] show that

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Fairness Index (Relative to PF)

C D F

BestSignalGreedy2Greedy

Fig. 13CDF OF FAIRNESS INDEXES RELATIVE TO GPF

ASSOCIATION

an exponential rule performs best in gracefully trading-off delay versus throughput.

Several researchers have also examined the fairness as-pect of the proportional fair algorithm [22], [23]. It hasbeen shown that, with users experiencing heterogeneouschannel quality, the differences in variances of the channelquality can result in unfairness using the proportional fairalgorithm [9]. In [23], the author describes a monitoring-based feedback algorithm to correct the unfairness of theproportional fair algorithm under heterogeneous channelconditions. The authors in [22] propose a new denitionof fairness that extends the absolute fairness bound mea-sure derived from the Generalized Processor Sharing dis-cipline [24]. They propose an opportunistic proportionalfair algorithm that is a weighted combination of the maxrate scheduling [19] and proportional fair scheduling. Notethat all the above algorithms, like the proportional fairscheduling algorithm, only support fairness among usersattached to a single base station. In this paper, we take anetwork-wide approach for supporting fairness while ex-ploiting user diversity to maximize throughput.

As we have discussed in Section II, user associationchanges have been used as one of the techniques for load

balancing in third generation wireless data networks. Theobjective of their work [6], [7] is load balancing given cer-tain denition of load at the BS. However, they do notconsider fairness in a network-wide view. We remark thata network-wide proportional fairness formulation enablesbetter tradeoff between system utilization and fairness, andis a natural objective of load balancing. As shown in [5],proportional fairness optimizes the aggregate user utili-ties (total user satisfaction) if each users utility function

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is an increasing, strictly concave and continuously differ-entiable function of its throughput. Load balancing andfairness have been considered jointly in the wireless LANcontext in [14]. However, the fairness metric consideredin [14] was max-min fairness. We remark that, in thirdgeneration wireless data networks, a better fairness metricis proportional fairness which is currently used in deployedthird generation data networks.

VIII. C ONCLUSION AND FUTURE WORK

In todays networks, users are associated to base stationswith the strongest signal strength and each base station in-dependently executes the proportional fairness schedulingalgorithm. This approach can result in non-Pareto optimalbandwidth allocation when considering the network as awhole. Therefore, we formulate and study a generalizedproportional fairness problem where user associations tobase stations are based on optimizing a generalized pro-portional fairness objective.

We show that this problem is NP-hard and hard to ap-proximate in general. For the special case, which roughlyholds in practice, we obtain efcient ofine and online al-gorithms. Our results show that the throughput and fair-ness can both be improved in heterogeneous user distribu-tions when compared to an approach that assigns a user tothe base station with the best signal strength.

For future research, we would like to incorporate otherload balancing techniques such as cell breathing into theconsideration of network-wide proportional fairness. Inaddition, we would like to consider uplink scheduling.

IX. A CKNOWLEDGEMENT

We are grateful to the helpful discussions with SemBorst which was instrumental in arriving at the generalizedproportional fairness formulation. We would also like tothank Chandra Chekuri for pointing out the possible con-nection between 3-dimensional matching and our problem.

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